Abstract

We study the dynamics of bound solitons of the complex cubic-quintic Ginzburg– Landau equation under the influence of external modulation. We consider periods of modulation being either smaller or larger than the soliton separation and the amplitude of modulation being either real or imaginary. For each case, we observe bifurcation and hysteresis phenomena in the parameters of the pair when changing the amplitude of modulation. Namely, soliton separation and phase difference between the solitons may take two or more values for the same modulation amplitude. In the case of gain-loss modulation, two solitons may split and be positioned in the two equilibrium states of the periodic potential. The complicated dynamics of this process is illustrated with numerical examples.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
    [CrossRef]
  2. K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
    [CrossRef]
  3. F. Krausz and T. Brabec, “Passive mode locking in standing-wave laser resonators,” Opt. Lett. 18, 888-890 (1993).
    [CrossRef] [PubMed]
  4. E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159-170 (1994).
    [CrossRef]
  5. O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156-158 (1984).
    [CrossRef] [PubMed]
  6. F. X. Kärtner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  7. R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653-662 (2001).
    [CrossRef]
  8. G. Palmer, M. Emons, M. Siegel, A. Steinmann, M. Schultze, M. Lederer, and U. Morgner, “Passively mode-locked and cavity-dumped Yb:KY(WO4)2 oscillator with positive dispersion,” Opt. Express 15, 16017-16021 (2007).
    [CrossRef] [PubMed]
  9. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
    [CrossRef]
  10. N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
    [CrossRef]
  11. S. Wabnitz, “Suppression of soliton interaction by phase modulation,” Electron. Lett. 29, 1711-1712 (1993).
    [CrossRef]
  12. L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
    [CrossRef]
  13. G. R. Huggett, “Mode locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
    [CrossRef]
  14. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
    [CrossRef]
  15. J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, and W. H. Knox, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
    [CrossRef]
  16. J. D. Moores, “Oscillating solitons in a novel integrable model of asynchronous mode locking,” Opt. Lett. 26, 87-89 (2001).
    [CrossRef]
  17. J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
    [CrossRef]
  18. N. D. Nguyen and L. N. Binh, “Generation of bound solitons in actively phase modulation mode-locked fiber ring resonators,” Opt. Commun. 281, 2012-2022 (2008).
    [CrossRef]
  19. W.-W. Hsiang, C.-Y. Lin, and Y. Lai, “Stable new bound soliton pairs in a 10 GHz hybrid frequency modulation mode-locked Er-fiber laser,” Opt. Lett. 31, 1627-1629 (2006).
    [CrossRef] [PubMed]
  20. W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
    [CrossRef]
  21. W.-W. Hsiang, C.-Y. Lin, M.-F. Tien, and Y. Lai, “Direct generation of a 10 GHz 816 fs pulse train from an erbium-fiber soliton laser with asynchronous phase modulation,” Opt. Lett. 30, 2493-2495 (2005).
    [CrossRef] [PubMed]
  22. I. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99-143 (2002).
    [CrossRef]
  23. H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
    [CrossRef]
  24. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
    [CrossRef]
  25. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
    [CrossRef]
  26. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  27. J. D. Moores, “On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65-70 (1993).
    [CrossRef]
  28. S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88, 073903 (2002).
    [CrossRef] [PubMed]
  29. J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757-1759 (2003).
    [CrossRef] [PubMed]
  30. J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
    [CrossRef]
  31. J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
    [CrossRef]
  32. H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801-2804 (1989).
    [CrossRef] [PubMed]
  33. B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954-6957 (1991).
    [CrossRef] [PubMed]
  34. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15, 515-523 (1998).
    [CrossRef]
  35. D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
    [CrossRef]
  36. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
    [CrossRef]
  37. Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12, 3184-3189 (2004).
    [CrossRef] [PubMed]
  38. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27, 966-968 (2002).
    [CrossRef]
  39. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Relative phase locking of pulses in a passively mode-locked fiber laser,” J. Opt. Soc. Am. B 20, 863-870 (2003).
    [CrossRef]

2009 (1)

W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
[CrossRef]

2008 (2)

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

N. D. Nguyen and L. N. Binh, “Generation of bound solitons in actively phase modulation mode-locked fiber ring resonators,” Opt. Commun. 281, 2012-2022 (2008).
[CrossRef]

2007 (3)

D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[CrossRef]

G. Palmer, M. Emons, M. Siegel, A. Steinmann, M. Schultze, M. Lederer, and U. Morgner, “Passively mode-locked and cavity-dumped Yb:KY(WO4)2 oscillator with positive dispersion,” Opt. Express 15, 16017-16021 (2007).
[CrossRef] [PubMed]

2006 (1)

2005 (2)

2004 (2)

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12, 3184-3189 (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (4)

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27, 966-968 (2002).
[CrossRef]

J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
[CrossRef]

S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

I. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

2001 (3)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

J. D. Moores, “Oscillating solitons in a novel integrable model of asynchronous mode locking,” Opt. Lett. 26, 87-89 (2001).
[CrossRef]

1998 (1)

1997 (2)

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, and W. H. Knox, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
[CrossRef]

1995 (1)

1994 (1)

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159-170 (1994).
[CrossRef]

1993 (3)

S. Wabnitz, “Suppression of soliton interaction by phase modulation,” Electron. Lett. 29, 1711-1712 (1993).
[CrossRef]

J. D. Moores, “On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65-70 (1993).
[CrossRef]

F. Krausz and T. Brabec, “Passive mode locking in standing-wave laser resonators,” Opt. Lett. 18, 888-890 (1993).
[CrossRef] [PubMed]

1991 (1)

B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954-6957 (1991).
[CrossRef] [PubMed]

1989 (1)

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801-2804 (1989).
[CrossRef] [PubMed]

1984 (1)

1977 (1)

K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
[CrossRef]

1975 (3)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

1968 (1)

G. R. Huggett, “Mode locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

1966 (1)

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[CrossRef]

1964 (1)

L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
[CrossRef]

Akhmediev, N.

W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case of 2+1,” Opt. Express 12, 3184-3189 (2004).
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757-1759 (2003).
[CrossRef] [PubMed]

S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[CrossRef]

Akhmediev, N. N.

Ankiewicz, A.

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15, 515-523 (1998).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
[CrossRef]

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[CrossRef]

Aranson, I.

I. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

Belhache, F.

Bergman, K.

Binh, L. N.

N. D. Nguyen and L. N. Binh, “Generation of bound solitons in actively phase modulation mode-locked fiber ring resonators,” Opt. Commun. 281, 2012-2022 (2008).
[CrossRef]

Brabec, T.

Brand, H. R.

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801-2804 (1989).
[CrossRef] [PubMed]

Chang, W.

W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
[CrossRef]

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Collings, B. C.

Cundiff, S. T.

Deissler, R. J.

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801-2804 (1989).
[CrossRef] [PubMed]

DeMaria, A. J.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[CrossRef]

Devine, N.

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[CrossRef]

Emons, M.

Fork, R. L.

O. E. Martinez, R. L. Fork, and J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156-158 (1984).
[CrossRef] [PubMed]

L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
[CrossRef]

Gordon, J. P.

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Grelu, Ph.

Gutty, F.

Hargrove, L. E.

L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

Heynau, H.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[CrossRef]

Hsiang, W. -W.

Huggett, G. R.

G. R. Huggett, “Mode locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

Ippen, E. P.

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159-170 (1994).
[CrossRef]

Isenor, N. R.

K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
[CrossRef]

Kärtner, F. X.

Keller, U.

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

F. X. Kärtner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16-18 (1995).
[CrossRef] [PubMed]

Knox, W. H.

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

Kramer, L.

I. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

Krausz, F.

Kutz, J. N.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. T. Cundiff, and W. H. Knox, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
[CrossRef]

Lai, Y.

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

Lederer, M.

Lin, C. -Y.

Malomed, B. A.

B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954-6957 (1991).
[CrossRef] [PubMed]

Martinez, O. E.

Moores, J. D.

J. D. Moores, “Oscillating solitons in a novel integrable model of asynchronous mode locking,” Opt. Lett. 26, 87-89 (2001).
[CrossRef]

J. D. Moores, “On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65-70 (1993).
[CrossRef]

Morgner, U.

Nguyen, N. D.

N. D. Nguyen and L. N. Binh, “Generation of bound solitons in actively phase modulation mode-locked fiber ring resonators,” Opt. Commun. 281, 2012-2022 (2008).
[CrossRef]

O'Neil, J. J.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
[CrossRef]

Palmer, G.

Paschotta, R.

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

Pollack, M. A.

L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
[CrossRef]

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Richardson, M. C.

K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
[CrossRef]

Sala, K.

K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
[CrossRef]

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

Sandstede, B.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
[CrossRef]

Schultze, M.

Siegel, M.

Soto-Crespo, J. M.

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Relative phase locking of pulses in a passively mode-locked fiber laser,” J. Opt. Soc. Am. B 20, 863-870 (2003).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, Ph. Grelu, and F. Belhache, “Quantized separations of phase-locked soliton pairs in fiber lasers,” Opt. Lett. 28, 1757-1759 (2003).
[CrossRef] [PubMed]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, “Phase-locked soliton pairs in a stretched-pulse fiber laser,” Opt. Lett. 27, 966-968 (2002).
[CrossRef]

S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Stable soliton pairs in optical transmission lines and fiber lasers,” J. Opt. Soc. Am. B 15, 515-523 (1998).
[CrossRef]

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
[CrossRef]

Steinmann, A.

Stetser, D. A.

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[CrossRef]

Tien, M. -F.

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Tsuda, S.

Turaev, D.

D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
[CrossRef]

Vladimirov, A.

D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
[CrossRef]

Wabnitz, S.

W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
[CrossRef]

S. Wabnitz, “Suppression of soliton interaction by phase modulation,” Electron. Lett. 29, 1711-1712 (1993).
[CrossRef]

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Zelik, S.

D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
[CrossRef]

Appl. Phys. B (2)

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159-170 (1994).
[CrossRef]

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653-662 (2001).
[CrossRef]

Appl. Phys. Lett. (3)

L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He-Ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. 5, 4-5 (1964).
[CrossRef]

G. R. Huggett, “Mode locking of CW lasers by regenerative RF feedback,” Appl. Phys. Lett. 13, 186-187 (1968).
[CrossRef]

A. J. DeMaria, D. A. Stetser, and H. Heynau, “Self mode locking of lasers with saturable absorbers,” Appl. Phys. Lett. 8, 174-176 (1966).
[CrossRef]

Electron. Lett. (1)

S. Wabnitz, “Suppression of soliton interaction by phase modulation,” Electron. Lett. 29, 1711-1712 (1993).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. Sala, M. C. Richardson, and N. R. Isenor, “Passive mode locking of lasers with the optical Kerr effect modulator,” IEEE J. Quantum Electron. 13, 915-924 (1977).
[CrossRef]

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. 11, 736-746 (1975).
[CrossRef]

H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

J. J. O'Neil, J. N. Kutz, and B. Sandstede, “Theory and simulation of the dynamics and stability of actively mode-locked lasers,” IEEE J. Quantum Electron. 38, 1412-1418 (2002).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049-3058 (1975).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (2)

J. D. Moores, “On the Ginzburg-Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65-70 (1993).
[CrossRef]

N. D. Nguyen and L. N. Binh, “Generation of bound solitons in actively phase modulation mode-locked fiber ring resonators,” Opt. Commun. 281, 2012-2022 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

Phys. Rev. A (3)

B. A. Malomed, “Bound solitons in the nonlinear Schrodinger-Ginzburg-Landau equation,” Phys. Rev. A 44, 6954-6957 (1991).
[CrossRef] [PubMed]

W. Chang, N. Akhmediev, and S. Wabnitz, “Effect of external periodic potential on pairs of dissipative solitons,” Phys. Rev. A 80, 013815 (2009).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Phys. Rev. E (5)

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: CGLE approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

D. Turaev, S. Zelik, and A. Vladimirov, “Chaotic bound states of localized structures in the complex Ginzburg-Landau equation,” Phys. Rev. E 75, 045601 (2007).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, “Bifurcations and multiple period soliton pulsations in a passively mode-locked fiber laser,” Phys. Rev. E 70, 066612 (2004).
[CrossRef]

J. M. Soto-Crespo, Ph. Grelu, N. Akhmediev, and N. Devine, “Soliton complexes in dissipative systems: vibrating, shaking, and mixed soliton pairs,” Phys. Rev. E 75, 016613 (2007).
[CrossRef]

Phys. Rev. Lett. (3)

H. R. Brand and R. J. Deissler, “Interaction of localized solution for subcritical bifurcations,” Phys. Rev. Lett. 63, 2801-2804 (1989).
[CrossRef] [PubMed]

N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, “Multisoliton solutions of the complex Ginzburg-Landau equation,” Phys. Rev. Lett. 79, 4047-4051 (1997).
[CrossRef]

S. T. Cundiff, J. M. Soto-Crespo, and N. Akhmediev, “Experimental evidence for soliton explosions,” Phys. Rev. Lett. 88, 073903 (2002).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

I. Aranson and L. Kramer, “The world of the complex Ginzburg-Landau equation,” Rev. Mod. Phys. 74, 99-143 (2002).
[CrossRef]

Other (1)

N. Akhmediev and A. Ankiewicz, Dissipative Solitons (Springer, 2005).
[CrossRef]

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 (13)

Fig. 1
Fig. 1

External phase modulation (dashed curve) with three different frequencies (a) Ω = Ω 0 , (b) Ω = 2 Ω 0 , and (c) Ω = 3 Ω 0 applied to the soliton bound state (solid curve) formed at D = 1 , ϵ = 1.4 , β = 0.5 , δ = 0.2 , μ = 0.5 , and ν = 0.075 . The initial separation ρ i of the pair is 8.923 and the phase difference ϕ is π / 2 .

Fig. 2
Fig. 2

[(a),(c),(e)] Soliton pair separation ρ and [(b),(d),(f)] phase difference between the pair ϕ versus the modulation depth | α | for the three cases shown in Fig. 1.

Fig. 3
Fig. 3

External phase modulation with the frequency Ω = 0.125 Ω 0 (dashed curve), Ω = 0.25 Ω 0 (dotted curve), and Ω = 0.5 Ω 0 (dashed-dotted curve) is applied to the soliton pair solution (solid curve).

Fig. 4
Fig. 4

Soliton pair separation ρ versus the modulation depth α for the three cases shown in Fig. 3. Namely, Ω = 0.125 Ω 0 (dashed curve), Ω = 0.25 Ω 0 (dotted curve), and Ω = 0.5 Ω 0 (dashed-dotted curve).

Fig. 5
Fig. 5

[(a),(c)] Soliton pair separation ρ and [(b),(d)] phase difference between the pair ϕ versus the modulation depth | α | for modulation frequencies Ω = 2 Ω 0 and Ω = 3 Ω 0 , respectively.

Fig. 6
Fig. 6

Soliton separation ρ versus | α | for Ω = 1.25 Ω 0 . Phase difference is zero in the whole interval except for a region of small | α | , which cannot be resolved in the scale of this figure.

Fig. 7
Fig. 7

Location of two solitons (solid curve) in the pair relative to the pattern of periodic gain-loss modulation (dashed curve) for Ω = 1.25 Ω 0 and α = 0.4 i .

Fig. 8
Fig. 8

Soliton separation ρ and phase difference between the solitons versus | α | for Ω = 0.8 Ω 0 . The crosses on the vertical axes where | α | = 0 represent unperturbed soliton separation and phase difference. Very small modulation | α | < 0.002 shifts these parameters to new values ρ 11.153 and ϕ = π that stay almost fixed until the bifurcation threshold is reached. Splitting into two types of solitons occurs at | α | 0.24 . Soliton pairs above this threshold have branches of 0, π / 2 , and π phase difference between the pulses. The arrows show the sequence of changes when increasing or decreasing | α | .

Fig. 9
Fig. 9

Location of two solitons (solid curve) in the pair relative to the pattern of periodic modulation (dashed curve) for Ω = 0.8 Ω 0 and α = 0.3 i . Solitons prefer certain equilibrium positions in the pattern thus comprising two types of bound states. (a) The solution representing the upper branch and (b) the lower branch in Fig. 8a.

Fig. 10
Fig. 10

(a) Soliton pair separation ρ and (b) the phase difference ϕ versus the gain-loss modulation depth | α | for period of modulation Ω = Ω 0 .

Fig. 11
Fig. 11

Equilibrium positions of the soliton pairs and single solitons shown by solid curves when (a) α = 0.1 i and (b) α = 0.2 i . Dashed curve shows the periodic linear gain-loss function with Ω = 0.125 Ω 0 .

Fig. 12
Fig. 12

Equilibrium state of single solitons when α = 0.3 i . An additional soliton appears in the middle due to the instability of the background. Its position is not fixed and may vary along t in evolution. Dashed curve shows the periodic linear gain-loss function with Ω = 0.125 Ω 0 .

Fig. 13
Fig. 13

Soliton pair evolution under the effect of the gain-loss modulation with Ω = 0.125 Ω 0 (dashed line). (a) α = 0.1 i ; the pair remains bounded and occupies the equilibrium position as a bound state. (b) α = 0.2 i ; the pair splits and each soliton occupies the equilibrium position. (c) α = 0.3 i ; the pair splits and each soliton occupies the equilibrium position in the periodic potential well. The background between the two solitons is unstable creating additional chaotically moving solitons.

Equations (1)

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

i ψ z + D 2 ψ t t + | ψ | 2 ψ + ν | ψ | 4 ψ = i δ ψ + i ϵ | ψ | 2 ψ + i β ψ t t + i μ | ψ | 4 ψ + α   cos ( Ω t ) ψ .

Metrics