Abstract

In this contribution we highlight several aspects concerning the numerical simulation of ultra-short pulse mode-locked fiber lasers by a non-distributed model. We show that for fixed system parameters multiple attractors are accessible by different initial conditions especially in the transient region between different mode-locking regimes. The reduction of multiple attractors stabilizing from different quantum noise fields to a single solution by gain ramping is demonstrated. Based on this analysis and model, different regimes of mode-locking obtained by varying the intra-cavity dispersion and saturation energy of the gain fiber are revised and it is shown that a regime producing linearly chirped parabolic pulses known from self-similar evolution is embedded in the wave-breaking free mode-locking regime.

© 2007 Optical Society of America

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References

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  1. N. Akhmediev, A. Ankiewicz, Dissipative Solitons. Lecture Notes in Physics, Band 661 (2005).
  2. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron.,  6, 1173-1185 (2000).
    [CrossRef]
  3. A. K. Komarov and K. P. Komarov, "Multistability and hysteresis phenomena in passive mode-locked lasers," Phys. Rev. E 62, 7607 - 7610 (2000).
    [CrossRef]
  4. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. Zhou, and C. -P. Huang, "Fourth-order dispersion-limited solitary pulses," Opt. Lett. 19, 1465 (1994).
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    [CrossRef]
  6. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, "Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment," IEEE J. Quantum Electron. 31, 591-598 (1995).
    [CrossRef]
  7. F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  8. F. Ilday, F. Wise, and F. Kaertner, "Possibility of self-similar pulse evolution in a Ti:sapphire laser," Opt. Express 12, 2731-2738 (2004).
    [CrossRef] [PubMed]
  9. G. P. Agrawal, Nonlinear Fiber Optics, (3rd edition, Academic, New York 2001).
  10. N. Akhmediev, J. M. Soto-Crespo, and G. Town, "Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: Complex Ginzburg-Landau equation approach," Phys. Rev. E 63, 056602 (2001).
    [CrossRef]
  11. U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, "Dynamics of Dissipative Temporal Solitons," Lect. Notes Phys. 661, 161-181 (2005).
    [CrossRef]
  12. A. Ruehl, O. Prochnow, D. Wandt, D. Kracht, B. Burgoyne, N. Godbout, and S. Lacroix, "Dynamics of parabolic pulses in an ultrafast fiber laser," Opt. Lett. 31, 2734-2736 (2006).
    [CrossRef] [PubMed]
  13. M. E. Ferman, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey. "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
    [CrossRef]
  14. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B,  19, 461 (2002).
    [CrossRef]
  15. A. C. Peacock, R. J. Kruhlak, J. D. Harvey, J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Comm. 206, 171-177 (2002).
    [CrossRef]
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  17. C. Nielsen, B. Ortaç, T. Schreiber, J. Limpert, R. Hohmuth, W. Richter, and A. Tünnermann, "Self-starting self-similar all-polarization maintaining Yb-doped fiber laser," Opt. Express 13, 9346-9351 (2005).
    [CrossRef] [PubMed]

2006

2005

2004

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

F. Ilday, F. Wise, and F. Kaertner, "Possibility of self-similar pulse evolution in a Ti:sapphire laser," Opt. Express 12, 2731-2738 (2004).
[CrossRef] [PubMed]

2002

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B,  19, 461 (2002).
[CrossRef]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Comm. 206, 171-177 (2002).
[CrossRef]

2001

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

2000

M. E. Ferman, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey. "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef]

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

A. K. Komarov and K. P. Komarov, "Multistability and hysteresis phenomena in passive mode-locked lasers," Phys. Rev. E 62, 7607 - 7610 (2000).
[CrossRef]

1995

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, "Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment," IEEE J. Quantum Electron. 31, 591-598 (1995).
[CrossRef]

1994

IEEE J. Quantum Electron.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, "Stretched-pulse additive pulse mode-locking in fiber ring lasers: Theory and experiment," IEEE J. Quantum Electron. 31, 591-598 (1995).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

J. Opt. Soc. Am. B

Lect. Notes Phys.

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, "Dynamics of Dissipative Temporal Solitons," Lect. Notes Phys. 661, 161-181 (2005).
[CrossRef]

Opt. Comm.

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Comm. 206, 171-177 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. V. Tognetti, M. N. Miranda, and H. M. Crespo, "Dispersion-managed mode-locking dynamics in a Ti:sapphire laser," Phys. Rev. A 74, 033809 (2006).
[CrossRef]

Phys. Rev. E

A. K. Komarov and K. P. Komarov, "Multistability and hysteresis phenomena in passive mode-locked lasers," Phys. Rev. E 62, 7607 - 7610 (2000).
[CrossRef]

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

Phys. Rev. Lett.

M. E. Ferman, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey. "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef]

F. Ö. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-Similar Evolution of Parabolic Pulses in a Laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Other

G. P. Agrawal, Nonlinear Fiber Optics, (3rd edition, Academic, New York 2001).

N. Akhmediev, A. Ankiewicz, Dissipative Solitons. Lecture Notes in Physics, Band 661 (2005).

Supplementary Material (1)

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Figures (12)

Fig. 1.
Fig. 1.

Illustration of the fiber laser cavity elements used for the simulations.

Fig. 2.
Fig. 2.

Basin of attractors presented in the space of temporal and spectral width (RMS values) for the parameters of Esat=100 pJ and β2=0.004 ps2. The lines connect the measured values after each roundtrip (after the SMF). The arrows indicate the evolution from the initial condition to the attractor marked with yellow circles. (gray: below transform-limited condition). A description of the attractors A1 to A3 is given in the text.

Fig. 3.
Fig. 3.

Examples of the convergence to the different attractors in the temporal domain: (a) attractor A1, (b) attractor A3, (c)+(d) attractor A2. (linear scale: 0oe-15-13-8252-i001≥max)

Fig. 4.
Fig. 4.

(a) Poincare map of the transient evolution from quantum noise with (black line) and without (red line) gain ramping. (b) Transient evolution in the temporal domain from quantum noise to the steady state solution using gain ramping, where attractor A1 is obtained instead of attractor A2 (compare to Fig. 3(d)). (linear scale: 0 oe-15-13-8252-i002 ≥max)

Fig. 5.
Fig. 5.

(a) Spectrogram of the transient evolution of a Gaussian initial pulse to the unstable attractor of a parabolic pulse turning into noise [movie: 1.5MB] and (b) the corresponding transient temporal evolution. (linear scale: 0 oe-15-13-8252-i002 -15-13-8252-i003" xmlns:xlink="http://www.w3.org/1999/xlink"/> ≥max) [Media 1]

Fig. 6.
Fig. 6.

Number of roundtrips until convergence starting from noise (a) and from a Gaussian pulse (b). (linear scale: 0 oe-15-13-8252-i004 3000 [white: unstable])

Fig. 7.
Fig. 7.

Pulse duration (FWHM) after the SMF of the converged solution that started from noise (a) and from a Gaussian pulse (b). (linear scale: 0 oe-15-13-8252-i005 10 ps [white: unstable])

Fig. 8.
Fig. 8.

Spectral bandwidth (FWHM) after the SMF of the converged solution that started from noise (a) and from a Gaussian pulse (b). (linear scale: 5 nm oe-15-13-8252-i006 35 nm [white: unstable])

Fig. 9.
Fig. 9.

Intra-cavity pulse evolution for (a) Esat=100 pJ, β2=-0.002ps2, (b) Esat=200 pJ, β2=+0.003 ps2, and (c) Esat=2000 pJ, β2=+0.03 ps2. (logarithmic scale: -30 dB oe-15-13-8252-i007 0 dB)

Fig. 10.
Fig. 10.

Temporal and spectral kurtosis of the converged solution after the SMF. (linear scale: -1.11 oe-15-13-8252-i008 -0.2, oe-15-13-8252-i009-0.86)

Fig. 11.
Fig. 11.

Temporal pulse profiles normalized in time to their FWHM for β2 net=0.0045 ps2, Esat=400 pJ (red), β2 net=0.01 ps2, Esat=400 pJ (blue) and β2 net=0.01 ps2, Esat=100 pJ (green) shown in linear (a) and logarithmic scale (b) in comparison with a perfect parabolic shape (black). (Spectral profiles are not shown but exhibit similar features.)

Fig. 12.
Fig. 12.

Spectrograms of the steady state solution for (a) β2 net=0.0045 ps2, Esat=400 pJ, (b) β2 net=0.01 ps2, Esat=400 pJ and (c) β2 net=0.01 ps2, Esat=100 pJ. (Spectrogram resolution: 600 fs). (linear scale: 0 oe-15-13-8252-i010 max)

Tables (1)

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Tab. 1. Parameters of the simulation

Equations (5)

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A z = g 2 A + n 2 β n i n + 1 n ! n T n A + i γ A 2
R = R unsat + R sat ( 1 1 1 + P P sat )
g = g 0 1 + E E sat
ε = i E i N 2 E i N 1 2 i E i N 1 2 with E i as the complex field amplitude
k = μ 4 σ 4 3

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