Abstract

In this paper, the stability of the analytical solutions of the cubic–quintic Ginzburg–Landau equation (CQGLE) in the high-chirp approximation has been studied numerically. The existence domain for the stable solution in the CQGLE parameter set has been found. A temporal and spectral shape of the stable solution as dependent of the cavity parameters has been analyzed. Direct comparison of the spectra with numerical calculations has been performed, demonstrating 102104 accuracy of the analytical solution for chirp parameter f>10. The stable solutions represent the dissipative soliton family with only one composite parameter. Inside this family, the pulse shape in the time domain evolves from the conventional soliton shape, sech2, to a rectangular one in the opposite limit with a parabolic shape as an intermediate one. The obtained theoretical results make it possible to classify experimentally observed highly chirped pulses and to optimize experimental schemes with an all- normal-dispersion cavity.

© 2011 Optical Society of America

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    [CrossRef]
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  23. H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
    [CrossRef]
  24. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
    [CrossRef]
  32. J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, 1964).

2010 (3)

2009 (3)

B. Ortaç, M. Baumgartl, J. Limpert, and A. Tünnermann, “Approaching microjoule-level pulse energy with mode-locked femtosecond fiber laser,” Opt. Lett. 34, 1585–1587 (2009).
[CrossRef] [PubMed]

V. L. Kalashnikov and A. Apolonski “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

2008 (2)

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[CrossRef]

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

2007 (1)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

2006 (1)

2005 (4)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

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

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

2004 (2)

2003 (1)

2002 (3)

1997 (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

1996 (1)

1995 (2)

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]

A. I. Chernykh and S. K. Turitsyn, “Soliton and collapse regimes of pulse generation in passively mode-locking laser systems,” Opt. Lett. 20, 398–400 (1995).
[CrossRef] [PubMed]

1993 (2)

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

M. Nakazawa, E. Yoshida, and Y. Kimura, “Generation of 98 fs optical pulses directly from an erbium-doped fiber ring laser at 1.57 μm,” Electron. Lett. 29, 63–65 (1993).
[CrossRef]

1991 (1)

I. N. Duling III, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

1987 (1)

N. N. Akhmediev, V.  M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 72, 809–818 (1987).
[CrossRef]

1974 (1)

Alan C. Newell, “Nonlinear wave motion,” Lect. Appl. Math. 15, 157–163 (1974).

1972 (1)

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R.  Soc. London Ser. A 326, 289–313 (1972).
[CrossRef]

Afanasjev, V. V.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[CrossRef]

Akhmediev, N.

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[CrossRef]

N. N. Akhmediev, V.  M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 72, 809–818 (1987).
[CrossRef]

Andrejco, M. J.

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Ankiewicz, A.

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

Apolonski, A.

V. L. Kalashnikov and A. Apolonski “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004).
[CrossRef] [PubMed]

Bale, B.

Baumgartl, M.

Brunel, M.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Buckley, J.

Buckley, J. R.

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]

Chartier, T.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Chernykh, A.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Chernykh, A. I.

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]

A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095–10100 (2006).
[CrossRef] [PubMed]

Clark, W. G.

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]

Deng, Y.

Dombi, P.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

Duling, I. N.

I. N. Duling III, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

Eleonskii, V. M.

N. N. Akhmediev, V.  M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 72, 809–818 (1987).
[CrossRef]

Fermann, M. E.

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Fernandez, A.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004).
[CrossRef] [PubMed]

Finot, C.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

Fuji, T.

Fürbach, A.

Graf, R.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

Haboucha, A.

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[CrossRef]

Haus, H. A.

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]

Hideur, A.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Hocking, L. M.

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R.  Soc. London Ser. A 326, 289–313 (1972).
[CrossRef]

Ilday, F. Ö.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton—similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[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]

F. Ö. Ilday and F. W. Wise, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Opt. Lett. 27, 1531–1533 (2002).
[CrossRef]

Ippen, E. P.

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]

Kafka, J. D.

Kalashnikov, V. L.

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

V. L. Kalashnikov and A. Apolonski “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Kieu, K.

Kimura, Y.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Generation of 98 fs optical pulses directly from an erbium-doped fiber ring laser at 1.57 μm,” Electron. Lett. 29, 63–65 (1993).
[CrossRef]

Komarov, A.

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

Krausz, F.

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

A. Fernandez, T. Fuji, A. Poppe, A. Fürbach, F. Krausz, and A. Apolonski, “Chirped-pulse oscillators: a route to high-power femtosecond pulses without external amplification,” Opt. Lett. 29, 1366–1368 (2004).
[CrossRef] [PubMed]

Kulagin, N. E.

N. N. Akhmediev, V.  M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 72, 809–818 (1987).
[CrossRef]

Leblond, H.

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[CrossRef]

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

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Lefrançois, S.

Lim, H.

Limpert, J.

Mathews, J.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, 1964).

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

Nakazawa, M.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Generation of 98 fs optical pulses directly from an erbium-doped fiber ring laser at 1.57 μm,” Electron. Lett. 29, 63–65 (1993).
[CrossRef]

Naumov, S.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

Nelson, L. E.

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]

Newell, Alan C.

Alan C. Newell, “Nonlinear wave motion,” Lect. Appl. Math. 15, 157–163 (1974).

Oktem, B.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton—similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Ortaç, B.

Podivilov, E.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

Poppe, A.

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]

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

Salhi, M.

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[CrossRef]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Sanchez, F.

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[CrossRef]

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

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Silberberg, Y.

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Soto-Crespo, J. M.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[CrossRef]

Soto-Crespo, J.? M.

Stewartson, K.

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R.  Soc. London Ser. A 326, 289–313 (1972).
[CrossRef]

Stock, M. L.

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Tamura, K.

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]

Town, G.

Tünnermann, A.

Turitsyn, S. K.

Ülgüdür, C.

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton—similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Wabnitz, S.

B. Bale and S. Wabnitz, “Strong spectral filtering for a mode-locked similariton fiber laser,” Opt. Lett. 35, 2466–2468 (2010).
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

Walker, R. L.

J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, 1964).

Weiner, M.

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Wise, F.

Wise, F. W.

Yoshida, E.

M. Nakazawa, E. Yoshida, and Y. Kimura, “Generation of 98 fs optical pulses directly from an erbium-doped fiber ring laser at 1.57 μm,” Electron. Lett. 29, 63–65 (1993).
[CrossRef]

Appl. Phys. Lett. (1)

M. E. Fermann, M. J. Andrejco, M. L. Stock, Y. Silberberg, and M. Weiner, “Passive mode locking in erbium fiber laser with negative group delay,” Appl. Phys. Lett. 62, 910–912 (1993).
[CrossRef]

Electron. Lett. (2)

M. Nakazawa, E. Yoshida, and Y. Kimura, “Generation of 98 fs optical pulses directly from an erbium-doped fiber ring laser at 1.57 μm,” Electron. Lett. 29, 63–65 (1993).
[CrossRef]

I. N. Duling III, “Subpicosecond all-fibre erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

IEEE J. Quantum Electron. (1)

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]

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

JETP Lett. (1)

E. Podivilov and V. L. Kalashnikov, “Heavily-chirped solitary pulses in the normal dispersion region: new solutions of the cubic-quintic complex Ginzburg-Landau equation,” JETP Lett. 82, 467–471 (2005).
[CrossRef]

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Nat. Photon. (1)

B. Oktem, C. Ülgüdür, and F. Ö. Ilday, “Soliton—similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Nat. Phys. (1)

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603(2007).
[CrossRef]

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S. Naumov, A. Fernandez, R. Graf, P. Dombi, F. Krausz, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators,” in New J. Phys. 7, 216 (2005).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, S. Naumov, A. Fernandez, R. Graf, and A. Apolonski, “Approaching the microjoule frontier with femtosecond laser oscillators: theory and comparison with experiment,” New J. Phys. 7, 217 (2005).
[CrossRef]

Opt. Express (1)

Opt. Lett. (7)

Phys. Rev. A (4)

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

V. L. Kalashnikov and A. Apolonski “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009).
[CrossRef]

M. Salhi, A. Haboucha, H. Leblond, and F. Sanchez, “Theoretical study of figure-eight all-fiber laser,” Phys. Rev. A 77, 033828(2008).
[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 (3)

V. L. Kalashnikov, “Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation,” Phys. Rev. E 80, 046606 (2009).
[CrossRef]

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

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, pp. 4783–4796 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

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]

Proc. R.? Soc. London Ser. A (1)

L. M. Hocking and K. Stewartson, “On the nonlinear response of a marginally unstable plane parallel flow to a two-dimensional disturbance,” Proc. R.  Soc. London Ser. A 326, 289–313 (1972).
[CrossRef]

Theor. Math. Phys. (1)

N. N. Akhmediev, V.  M. Eleonskii, and N. E. Kulagin, “Exact first order solutions of the nonlinear Schrödinger equation,” Theor. Math. Phys. 72, 809–818 (1987).
[CrossRef]

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J. Mathews and R. L. Walker, Mathematical Methods of Physics (Benjamin, 1964).

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

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Figures (4)

Fig. 1
Fig. 1

Block scheme of possible experiment.

Fig. 2
Fig. 2

Solution existence areas in the plane ( α γ / β κ , σ ζ / κ ): positive ( P m = P m + ) branch of the analytic solution is defined in region IV, negative ( P m = P m ) branch in regions III and IV. Coordinates of the example numerical solutions are marked by points. Dotted curves illustrate the paths with self-similar pulse shape.

Fig. 3
Fig. 3

(a) Pulse spectra. Different stable numerical solutions at point 1 of Fig. 2 are marked by color symbols. (b) Comparison of normalized power spectra calculated analytically (solid curve) and numerically (symbols) for f 23 , 40, 14, and 3 (from top to bottom), the analytics correspond to the P m = P m + solution of Eq. (16).

Fig. 4
Fig. 4

(a) Comparison between numerics (color symbols) and the positive branch of the analytical solution (solid black curve) for the spectral shape. (b) Time-domain shape of the positive branch (solid curves) and fitting (dashed curves) by parabola and sech 2 . Inset shows analytics in the limit R 0 .

Equations (18)

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A z = i ( β 2 t 2 γ | A | 2 ) A + ( σ + α 2 t 2 + κ | A | 2 ( 1 ζ | A | 2 ) ) A ,
A ( z , t ) = a ( τ , z ) e i f Ψ ( τ , z ) i q z ,
a f Ψ z = ( q β Ω 2 γ P ) a + [ β f 2 2 a τ 2 + 2 α Ω f a τ + α f Ω τ a ] ,
a z = ( σ α Ω 2 + κ P ( 1 ζ P ) ) a 2 β Ω f a τ β a f Ω τ + [ α f 2 2 a τ 2 ] .
P ( τ f ) = P m β γ Ω 2 ( τ f ) + O ( 1 / f 2 ) ,
γ d P ( τ f ) d τ = 2 β Ω ( τ f ) d Ω ( τ f ) d τ + O ( 1 / f 2 ) .
( σ α Ω 2 + κ P ( 1 ζ P ) ) P 2 β 2 Ω γ f d Ω d τ β P f d Ω d τ = 0.
β d Ω d t = [ σ α Ω 2 ( t ) + κ ( P m β γ Ω 2 ( t ) ) ( 1 ζ ( P m β γ Ω 2 ( t ) ) ) ] P m 3 β γ Ω 2 ( t ) ( P m β γ Ω 2 ( t ) ) .
P m ± = 3 8 ζ ( 2 α γ β κ ± ( 2 α γ β κ ) 2 16 σ ζ κ ) ,
d Ω dt = κ ζ 3 γ [ Ω 2 ( t ) + γ β ζ ( 1 + α γ β κ ) 5 γ 3 β P m ] ( P m β γ Ω 2 ( t ) ) .
arctanh ( Ω ( t ) Δ ) + 1 R arctan ( Ω ( t ) R Δ ) = t T .
R = 1 + α γ / β κ ζ P m 5 3 .
A ( ω ) = f d τ a ( τ ) e i f ( Ψ ( τ ) ω τ ) .
Ψ ( τ ) τ | τ = τ * = ω ,
A ( ω ) = f a ( τ * ) e i f ( Ψ ( τ * ) ω τ * ) · 2 π i f d 2 Ψ d τ 2 | τ = τ * ( 1 + O ( 1 / f ) ) .
I ( ω ) | A ( ω ) | 2 6 π γ ζ κ ( Δ 2 ω 2 ) ω 2 + R 2 Δ 2 ( 1 + O ( 1 / f ) ) .
I ( ω ) | d z P ( t ) e i φ ( t ) i ω t | 2 6 π γ ζ κ H ( Δ 2 ω 2 ) ω 2 + R 2 Δ 2 ,
f = Δ T = 3 / ( κ γ + α β 2 3 ζ P m κ γ ) ,

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