Abstract

Soliton area theorems express the pulse energy as a function of the pulse shape and the system parameters. From an analytical solution to the cubic-quintic complex Ginzburg–Landau equation, we derive an area theorem for dissipative optical solitons. In contrast to area theorems for conservative optical solitons, the energy does not scale inversely with the pulse duration, and in addition there is an upper limit to the energy. Energy quantization explains the existence of, and conditions for, multiple-pulse solutions. The theoretical predictions are confirmed with numerical simulations and experiments in the context of dissipative soliton fiber lasers.

© 2010 Optical Society of America

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References

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  1. N. Akhmediev and A. Ankiewics, Dissipative Solitons (Springer, 2005).
    [CrossRef]
  2. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (Differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. JETP 34, 62–69 (1972).
  3. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  4. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-fs pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
    [CrossRef] [PubMed]
  5. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  6. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
    [CrossRef]
  7. B. Proctor, E. Westwig, and F. Wise, “Characterization of a Kerr-lens mode-locked Ti:sapphire laser with positive group-velocity dispersion,” Opt. Lett. 18, 1654–1656 (1993).
    [CrossRef] [PubMed]
  8. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095–10100 (2006).
    [CrossRef] [PubMed]
  9. E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
    [CrossRef] [PubMed]
  10. A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. 32, 2408–2410 (2007).
    [CrossRef] [PubMed]
  11. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
    [CrossRef] [PubMed]
  12. S. Namiki and H. A. Haus, “Noise of the stretched pulse fiber laser I: theory,” IEEE J. Quantum Electron. 33, 649–659 (1997).
    [CrossRef]
  13. W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303–367 (1992).
    [CrossRef]
  14. Some experimental evidence of energy quantization in a normal-dispersion laser was reported previously by J. W. Lou, M. Currie, and F. K. Fatimi, Opt. Express 15, 4960–4965 (2007); however, the observations were analyzed within the CGLE, which fails to account for even qualitative features of the experiments, such as the characteristic shape of the pulse spectrum.
    [CrossRef] [PubMed]

2008 (1)

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

2007 (2)

2006 (1)

2003 (1)

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

2000 (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

1997 (1)

S. Namiki and H. A. Haus, “Noise of the stretched pulse fiber laser I: theory,” IEEE J. Quantum Electron. 33, 649–659 (1997).
[CrossRef]

1993 (2)

1992 (1)

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303–367 (1992).
[CrossRef]

1991 (1)

1973 (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (Differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. JETP 34, 62–69 (1972).

Akhmediev, N.

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

Ankiewics, A.

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

Buckley, J.

Chong, A.

Cundiff, S. T.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Currie, M.

Diddams, S. A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Fatimi, F. K.

Fujimoto, J. G.

Hall, J. L.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Haus, H. A.

Hohenberg, P. C.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303–367 (1992).
[CrossRef]

Ippen, E. P.

Jones, D. J.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Lange, C. H.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Lederer, F.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Lou, J. W.

Michaelis, D.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Namiki, S.

S. Namiki and H. A. Haus, “Noise of the stretched pulse fiber laser I: theory,” IEEE J. Quantum Electron. 33, 649–659 (1997).
[CrossRef]

Nelson, L. E.

Proctor, B.

Ranka, J. K.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Renninger, W.

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]

A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. 32, 2408–2410 (2007).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (Differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. JETP 34, 62–69 (1972).

Stegeman, G. I.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Stentz, A.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Tamura, K.

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Ultanir, E. A.

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

van Saarloos, W.

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303–367 (1992).
[CrossRef]

Westwig, E.

Windeler, R. S.

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Wise, F.

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]

A. Chong, W. H. Renninger, and F. W. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. 32, 2408–2410 (2007).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (Differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. JETP 34, 62–69 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical physics in dispersive dielectric fibers I: Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. Namiki and H. A. Haus, “Noise of the stretched pulse fiber laser I: theory,” IEEE J. Quantum Electron. 33, 649–659 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. A (1)

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. Lett. (1)

E. A. Ultanir, G. I. Stegeman, D. Michaelis, C. H. Lange, and F. Lederer, “Stable dissipative solitons in semiconductor optical amplifiers,” Phys. Rev. Lett. 90, 253903 (2003).
[CrossRef] [PubMed]

Physica D (1)

W. van Saarloos and P. C. Hohenberg, “Fronts, pulses, sources and sinks in generalized complex Ginzburg–Landau equations,” Physica D 56, 303–367 (1992).
[CrossRef]

Science (1)

D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (Differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. JETP 34, 62–69 (1972).

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Variation of the pulse energy as a function of the pulse parameter B. The dotted line separates solutions with | B | < 1 for δ > 0 from those with B > 1 for δ < 0 . Insets: spectral profiles plotted for the respective values of B.

Fig. 2
Fig. 2

Top: theoretical spectra for increasing pulse energy, as B approaches −1; middle: simulated spectra with increasing saturation energy; bottom: measured spectra with increasing pump power. The rightmost spectra correspond to the birth of the second pulse in the cavity.

Fig. 3
Fig. 3

Mode-locked output power versus pump power. The spectra on the right are for the corresponding pump levels.

Fig. 4
Fig. 4

Top: single-pulsing intensity autocorrelation for increasing pulse energies; bottom: output energy versus pulse duration from experiment (left) and simulation (right).

Tables (1)

Tables Icon

Table 1 Area Theorems: (a) Soliton of the NLSE, (b) Dispersion-Managed-Soliton of the CGLE, and (c) Chirped Soliton of the CGLE

Equations (7)

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U z = g U + ( 1 Ω i D 2 ) U t t + ( α + i γ ) | U | 2 U + δ | U | 4 U .
U [ t , z ] = A cosh ( t τ ) + B exp [ i β 2 ln ( cosh ( t τ ) + B ) + i θ z ] .
E = F ( B ) G ( D , Ω , δ ) ,
F ( B ) = { cos 1 ( B ) for   | B | < 1 cosh 1 ( B ) for   B > 1 , }
G ( D , Ω , δ ) = 2 3 ( Δ + 2 ) D 2 ( Δ 8 ) Ω 2 + 12 ( Δ 4 ) D | δ | Ω Ω ( D 2 Ω 2 + 4 ) ,
Δ = 3 D 2 Ω 2 + 16 .
T = ( | B | cosh 1 ( 2 + B ) | B 2 1 | ) D Ω | δ | G ( D , Ω , δ ) 2 ( Δ + 2 ) γ .

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