Abstract

We consider experimentally and theoretically a refined parameter space in a laser system near the transition to multi-pulse mode-locking. Near the transition, the onset of instability is initiated by a Hopf (periodic) bifurcation. As the cavity energy is increased, the band of unstable, oscillatory modes generates a chaotic behavior between single- and multi-pulse operation. Both theory and experiment are in good qualitative agreement and they suggest that the phenomenon is of a universal nature in mode-locked lasers at the onset of multi-pulsing from N to N+1 pulses per round trip. This is the first theoretical and experimental characterization of the transition behavior, made possible by a highly refined tuning of the gain pump level.

© 2009 Optical Society of America

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References

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  1. H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  2. J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
    [CrossRef]
  3. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
    [CrossRef]
  4. B. Collings, K. Berman, and W. H. Knox, "Stable multi-gigahertz pulse train formation in a short cavity passively harmonic mode-locked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
    [CrossRef]
  5. M. E. Fermann and J. D. Minelly, "Cladding-pumped passive harmonically mode-locked fiber laser," Opt. Lett. 21, 970-972 (1996).
    [CrossRef] [PubMed]
  6. A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
    [CrossRef]
  7. M. J. Guy, P. U. Noske, A. Boskovic, and J. R. Taylor, "Femtosecond soliton generation in a praseodymium fluoride fiber laser," Opt. Lett. 19, 828-830 (1994).
    [CrossRef] [PubMed]
  8. M. Horowitz, C. R. Menyuk, T. F. Carruthers, and I. N. DulingIII, "Theoretical and experimental study of harmonically mode-locked fiber lasers for optical communication systems," J. Lightwave Technol. 18, 1565-1574 (2000).
    [CrossRef]
  9. R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
    [CrossRef]
  10. M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N. Akhmediev, and J. M. Soto-Crespo, "Multipulse operation of a Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror," J. Opt. Soc. Am. B 16, 895-904 (1999).
    [CrossRef]
  11. Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
    [CrossRef]
  12. M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
    [CrossRef]
  13. Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
    [CrossRef]
  14. H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
    [CrossRef]
  15. A. N. Pilipetskii, E. A. Golovchenck, and C. R. Menyuk, "Acoustic effect in passively mode-locked fiber ring lasers," Opt. Lett. 20, 907-909 (1995).
    [CrossRef] [PubMed]
  16. S. Namiki, E. P. Ippen, H. Haus, and C. X. Yu, "Energy equations for mode-locked lasers," J. Opt. Soc. Am. B 14, 2099-2111 (1997).
    [CrossRef]
  17. 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]
  18. J. N. Kutz and B. Sandstede, "Theory of passive harmonic mode-locking using wave-guide arrays," Opt. Express 16, 636-650 (2008).
    [CrossRef] [PubMed]
  19. T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of Pulses in the Master-Mode-locking Equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
    [CrossRef]
  20. T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
    [CrossRef]
  21. B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
    [CrossRef]
  22. J. Proctor and J. N. Kutz, "Theory and Simulation of Passive Mode-Locking with Waveguide Arrays," Opt. Lett. 13, 2013-2015 (2005).
    [CrossRef]
  23. J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
    [CrossRef]
  24. K. Kieu and M. Mansuripur, "Femtosecond laser pulse generation with a fiber taper embedded in carbon nanotube/ polymer composite," Opt. Lett. 32, 2242-2244 (2007).
    [CrossRef] [PubMed]

2009 (1)

B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
[CrossRef]

2008 (1)

2007 (1)

2006 (1)

J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
[CrossRef]

2005 (1)

J. Proctor and J. N. Kutz, "Theory and Simulation of Passive Mode-Locking with Waveguide Arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

2004 (2)

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
[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]

2002 (1)

2000 (2)

1999 (2)

M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N. Akhmediev, and J. M. Soto-Crespo, "Multipulse operation of a Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror," J. Opt. Soc. Am. B 16, 895-904 (1999).
[CrossRef]

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

1998 (3)

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

B. Collings, K. Berman, and W. H. Knox, "Stable multi-gigahertz pulse train formation in a short cavity passively harmonic mode-locked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
[CrossRef]

1997 (3)

S. Namiki, E. P. Ippen, H. Haus, and C. X. Yu, "Energy equations for mode-locked lasers," J. Opt. Soc. Am. B 14, 2099-2111 (1997).
[CrossRef]

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

1996 (1)

1995 (1)

1994 (1)

1992 (1)

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

1991 (1)

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

1983 (1)

J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
[CrossRef]

Akhmediev, N.

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]

Akhmediev, N. N.

Bale, B. G.

B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
[CrossRef]

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Berman, K.

Boskovic, A.

Carruthers, T. F.

Chai, L.

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Collings, B.

Collings, B. C.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Davey, R. P.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

Duling, I. N.

Ferguson, A. I.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

Fermann, M. E.

Golovchenck, E. A.

Gordon, J. P.

J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
[CrossRef]

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.

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]

Grudinin, A. B.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Guy, M. J.

Haus, H.

Haus, H. A.

H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

Horowitz, M.

Ippen, E. P.

Jagadish, C.

Kapitula, T.

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of Pulses in the Master-Mode-locking Equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
[CrossRef]

Kieu, K.

Kinoshita, S.

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Kitano, H.

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Knox, W. H.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

B. Collings, K. Berman, and W. H. Knox, "Stable multi-gigahertz pulse train formation in a short cavity passively harmonic mode-locked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
[CrossRef]

Kutz, J. N.

B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
[CrossRef]

J. N. Kutz and B. Sandstede, "Theory of passive harmonic mode-locking using wave-guide arrays," Opt. Express 16, 636-650 (2008).
[CrossRef] [PubMed]

J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
[CrossRef]

J. Proctor and J. N. Kutz, "Theory and Simulation of Passive Mode-Locking with Waveguide Arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of Pulses in the Master-Mode-locking Equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Lai, M.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Langford, N.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

Lederer, M. J.

Lee, K. F.

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Luther-Davis, B.

Mansuripur, M.

Menyuk, C. R.

Minelly, J. D.

Namiki, S.

Nicholson, J.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Noske, P. U.

Payne, D. N.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Pilipetskii, A. N.

Proctor, J.

J. Proctor and J. N. Kutz, "Theory and Simulation of Passive Mode-Locking with Waveguide Arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

Richardson, D. J.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Rudolph, W.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Sandstede, B.

B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
[CrossRef]

J. N. Kutz and B. Sandstede, "Theory of passive harmonic mode-locking using wave-guide arrays," Opt. Express 16, 636-650 (2008).
[CrossRef] [PubMed]

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of Pulses in the Master-Mode-locking Equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
[CrossRef]

Soto-Crespo, J. 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]

M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N. Akhmediev, and J. M. Soto-Crespo, "Multipulse operation of a Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror," J. Opt. Soc. Am. B 16, 895-904 (1999).
[CrossRef]

Tan, H. H.

Taylor, J. R.

Wang, Ch.-yue

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Xing, Q.

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Yoo, K. M.

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Yu, C. X.

Zhang, W.

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Electron. Lett. (2)

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized Pulse Spacing in Soliton Lasers Due to Gain Depletion and Recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

H. A. Haus, "Mode-Locking of Lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[CrossRef]

B. G. Bale, J. N. Kutz and B. Sandstede, "Optimizing waveguide array mode-locking for high-power fiber lasers," IEEE J. Sel. Top. Quantum Electron. 15, 220-231 (2009).
[CrossRef]

Indiana J. Math. (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

J. Lightwave Technol. (1)

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

Opt. Commun. (4)

Q. Xing, L. Chai, W. Zhang, and Ch.-yue Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Ch.-yue Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. E (1)

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]

SIAM Review (1)

J. N. Kutz, "Mode-locked Soliton Lasers," SIAM Review 48, 629-678 (2006).
[CrossRef]

Supplementary Material (4)

» Media 1: MOV (961 KB)     
» Media 2: MOV (1215 KB)     
» Media 3: MOV (1378 KB)     
» Media 4: MOV (2244 KB)     

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Figures (6)

Fig. 1.
Fig. 1.

(a) Schematic of the experimental setup. The saturable absorber used is single-walled carbon nanotubes. (b) The laser output power versus pump current. Note the three labeled operating regimes: single pulse, double pulse, and an intermediate transition regime from 83 and 87 mA.

Fig. 2.
Fig. 2.

Pulse dynamics in the intermediate regime shown in Fig. 1(b) between one and two pulses for gain pump powers of (a) 83 mA (Media 1), (b) 84 mA (Media 2), (c) 85 mA (Media 3), and (d) 86 mA (Media 4).

Fig. 3.
Fig. 3.

Experimentally measured time series of the separation of two pulses once single pulse operation transitions to multi-pulsing. (a) For pump power 83 mA, the separation of the two pulses is nearly periodic. A best fit sinusoidal function (dashed) is fit to the data (squares) and shows excellent agreement. (b) For pump power 86 mA, the separation of the two pulses is not periodic, and evolves irregularly. The Fourier transform of the time series is shown in (c) for the periodic case and (d) for the irregular case. Note that the spectrum in (d) is a signature that the separation dynamics is chaotic.

Fig. 4.
Fig. 4.

Mode-locked solution branches of the form (3) with (4). Depicted are the 1-pulse branch (N=1), 2-pulse branch (N=2) and 3-pulse branch (N=3) along with their corresponding stability region (shaded gray region). The inset demonstrates the linear (spectral) instability that occurs near the Hopf bifurcation point of the 1-pulse branch of solutions. As the gain increases past the Hopf bifurcation point, bands of oscillatory, unstable modes cross into the right half plane leading to a breather solution and chaotic dynamics.

Fig. 5.
Fig. 5.

Mode-locking dynamics as a function of increasing gain pumping g 0=2.3 (a), 2.52 (b), 2.53 (c), 2.68 (d), 2.72 (e) and 2.75 (f). As the gain is increased, mode-locking is observed to go from 1-pulse per round trip to 2-pulses per round trip via a Hopf bifurcation followed by a chaotic regime of interaction. The parameters for numerical simulation are γ=8, C=5, δ 0=δ 1=0, δ 2=10, τ=0.1 and e 0=1 [18].

Fig. 6.
Fig. 6.

Numerical simulations of the pulse-to-pulse distance and its Fourier transform for two different gain values. For the top panel, g 0=2.68 and a strong periodic signature is observed in the spectrum in (c). As the gain is increased to g 0=2.72 in (b), the periodicity is lost and a stronger chaotic motion is observed as illustrated in the spreading of the spreading of the spectral signatures in (d). Note that the DC signature has been zeroed out in the spectrum. Aside from g 0, the parameters are the same as those used in Fig. 5.

Equations (6)

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

i u z + 1 2 2 u t 2 + γ u 2 u + C v + i δ 0 u ig ( z ) ( 1 + τ 2 t 2 ) u = 0
i v z + C ( w + u ) + i δ 1 v = 0
i w z + Cv + i δ 2 w = 0 ,
g ( z ) = 2 g 0 1 + u 2 / e 0 .
u ( z , t ) = η sech ω t 1 + iA e i θ z ,
g ( z ) = 2 g 0 1 + N u 2 / e 0 .

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