Abstract

We show numerically the existence of vibrating soliton pairs that are consistent with observations performed with a passively mode-locked fiber laser. These vibrating pairs are new types of multisoliton complexes that exist in the vicinity of the phase-locked soliton pairs discovered a few years ago [Opt. Lett. 27, 966 (2002) ]. The pairs are found numerically with a laser propagation model that includes nonlinear dissipation and cavity periodicity, and they can appear following a Hopf-type bifurcation when a cavity parameter is tuned.

© 2006 Optical Society of America

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  1. J. M. Soto-Crespo and Ph. Grelu, in Dissipative Solitons, N.Akhmediev and A.Ankiewicz, eds. (Springer-Verlag, 2005), pp. 207-240.
    [CrossRef]
  2. N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
    [CrossRef]
  3. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, Opt. Lett. 27, 966 (2002).
    [CrossRef]
  4. D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
    [CrossRef]
  5. N. H. Seong and D. Y. Kim, Opt. Lett. 27, 1321 (2002).
    [CrossRef]
  6. A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
    [CrossRef]
  7. Ph. Grelu and N. Akhmediev, Opt. Express 12, 3184 (2004).
    [CrossRef] [PubMed]
  8. M. Olivier, V. Roy, M. Piché, and F. Babin, Opt. Lett. 29, 1461 (2004).
    [CrossRef] [PubMed]
  9. V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
    [CrossRef]
  10. Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 20, 863 (2003).
    [CrossRef]
  11. N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
    [CrossRef]
  12. J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
    [CrossRef]

2004 (3)

2003 (2)

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 20, 863 (2003).
[CrossRef]

2002 (2)

2001 (2)

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
[CrossRef]

1998 (1)

1991 (1)

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

Akhmediev, N.

Ph. Grelu and N. Akhmediev, Opt. Express 12, 3184 (2004).
[CrossRef] [PubMed]

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
[CrossRef]

Ankiewicz, A.

Babin, F.

Belhache, F.

Brunel, M.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Chartier, T.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Drummond, P. D.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Grelu, Ph.

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Ph. Grelu and N. Akhmediev, Opt. Express 12, 3184 (2004).
[CrossRef] [PubMed]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 20, 863 (2003).
[CrossRef]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, Opt. Lett. 27, 966 (2002).
[CrossRef]

J. M. Soto-Crespo and Ph. Grelu, in Dissipative Solitons, N.Akhmediev and A.Ankiewicz, eds. (Springer-Verlag, 2005), pp. 207-240.
[CrossRef]

Gutty, F.

Hideur, A.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Kim, D. Y.

Leblond, H.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Man, W. S.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Matsas, V.

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

Newson, T.

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

Olivier, M.

Ortaç, B.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Payne, D.

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

Piché, M.

Richardson, D.

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

Roy, V.

Sanchez, F.

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Seong, N. H.

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 20, 863 (2003).
[CrossRef]

Ph. Grelu, F. Belhache, F. Gutty, and J. M. Soto-Crespo, Opt. Lett. 27, 966 (2002).
[CrossRef]

N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
[CrossRef]

N. N. Akhmediev, A. Ankiewicz, and J. M. Soto-Crespo, J. Opt. Soc. Am. B 15, 515 (1998).
[CrossRef]

J. M. Soto-Crespo and Ph. Grelu, in Dissipative Solitons, N.Akhmediev and A.Ankiewicz, eds. (Springer-Verlag, 2005), pp. 207-240.
[CrossRef]

Tam, H. Y.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Tang, D. Y.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Town, G.

N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Electron. Lett. (1)

V. Matsas, T. Newson, D. Richardson, and D. Payne, Electron. Lett. 28, 1391 (1991).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Commun. (1)

A. Hideur, B. Ortaç, T. Chartier, M. Brunel, H. Leblond, and F. Sanchez, Opt. Commun. 225, 71 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

D. Y. Tang, W. S. Man, H. Y. Tam, and P. D. Drummond, Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Phys. Rev. E (2)

N. N. Akhmediev, J. M. Soto-Crespo, and G. Town, Phys. Rev. E 63, 056602 (2001).
[CrossRef]

J. M. Soto-Crespo, M. Grapinet, Ph. Grelu, and N. Akhmediev, Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Other (1)

J. M. Soto-Crespo and Ph. Grelu, in Dissipative Solitons, N.Akhmediev and A.Ankiewicz, eds. (Springer-Verlag, 2005), pp. 207-240.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Experimental recordings of the optical spectrum and autocorrelation of (a) a stable soliton pair at pump power of 210 mW and (b) an unstable soliton pair at pump power of 220 mW : the blurring of fringe contrast and the widening of autocorrelation side peaks are attributed to vibrational motion of the soliton pair.

Fig. 2
Fig. 2

Typical limit cycle for a vibrating soliton pair, with Q sat = 0.6 and θ = 64 ° . The diagram represents the phase relationship versus the temporal separation of the two solitons. Initial conditions are represented by large dots. After each cavity round trip, phase and separation are calculated, adding a new point to a given trajectory. The limit cycle is the accumulation line of all displayed trajectories.

Fig. 3
Fig. 3

Starting from stable solitons pairs, limit cycles for vibrating solitons pairs can appear as supercritical Hopf bifurcations when a cavity parameter is varied. In two examples bifurcation appears along with (a) an increase of the saturation energy at Q sat = 0.55 and (b) a decrease of the polarizer angle at θ = 67.6 ° . (c) Averaging of optical spectra for Q sat = 0.60 and θ = 66 ° , featuring significant blurring of spectral fringes.

Equations (2)

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i U z + γ e , s U + ( D e , s 2 i κ e , s ) U t t + Γ e , s ( U 2 U + 2 3 V 2 U + 1 3 V 2 U * ) = i g e , s ( Q U ) U ,
i V z γ e , s V + ( D e , s 2 i κ e , s ) V t t + Γ e , s ( V 2 V + 2 3 U 2 V + 1 3 U 2 V * ) = i g e , s ( Q V ) V ,

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