Abstract

We present an analytical derivation of the distributed model from the experimentally well confirmed lumped approach for the description of light propagation in mode-locked fiber lasers operating in the scalar regime where the dynamics is mainly governed by the propagation of a single field component. As a limiting case of the distributed model we identify the complex cubic-quintic Ginzburg–Landau equation (CQGLE). One important result consists of deriving explicit relations between the coefficients of the distributed models to the realistic laser parameters. We numerically demonstrate that the results obtained by using the general distributed model are in very good agreement with those of the lumped model, whereas results of the CQGLE can significantly deviate for a certain range of parameters. Moreover, we demonstrate that the validity of the CQGLE approach strongly depends on the operation regime of the saturable absorber.

© 2010 Optical Society of America

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References

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  1. A. Chong, W. Renniger, and F. Wise, “All-normal-dispersion femtosecond fiber laser with pulse energy above 20 nJ,” Opt. Lett. 32, 2408–2410 (2007).
    [CrossRef] [PubMed]
  2. F. Ilday, J. Buckley, L. Kuznetsova, and F. Wise, “Generation of 36-femtosecond pulses from a ytterbium fiber laser,” Opt. Express 11, 3550–3554 (2003).
    [CrossRef] [PubMed]
  3. L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal dispersion fiber lasers,” Laser Phys. Lett. 7, 63–67 (2010).
    [CrossRef]
  4. B. Ortaç, A. Zaviyalov, C. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35, 1578–1580 (2010).
    [CrossRef] [PubMed]
  5. P. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 137–173.
    [CrossRef]
  6. B. Ortaç, M. Plötner, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental and numerical study of pulse dynamics in positive net-cavity dispersion modelocked Yb-doped fiber lasers,” Opt. Express 15, 15595–15602 (2007).
    [CrossRef]
  7. H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
    [CrossRef]
  8. L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
    [CrossRef]
  9. N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).
  10. N. Akhmediev and A. Ankiewicz, “Three sources and three component parts of the concept of dissipative solitons” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 1–28.
    [CrossRef]
  11. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604(R) (2005).
    [CrossRef]
  12. E. Ding and N. Kutz, “Operating regimes, split-step modeling, and the Haus master mode-locking model,” J. Opt. Soc. Am. B 26, 2290–2300 (2009).
    [CrossRef]
  13. G. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).
  14. A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
    [CrossRef]
  15. A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
    [CrossRef] [PubMed]
  16. U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
    [CrossRef]
  17. G. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
    [CrossRef] [PubMed]
  18. G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1997).
    [CrossRef]
  19. J. H. B. Nijhof, W. Forysiak, and N. J. Doran, “Dispersion-managed solitons in the normal dispersion regime: a physical interpretation,” Opt. Lett. 23, 1674–1676 (1998).
    [CrossRef]
  20. A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
    [CrossRef]
  21. A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
    [CrossRef]
  22. A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
    [CrossRef] [PubMed]
  23. V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
    [CrossRef]
  24. R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
    [CrossRef]

2010 (2)

2009 (4)

E. Ding and N. Kutz, “Operating regimes, split-step modeling, and the Haus master mode-locking model,” J. Opt. Soc. Am. B 26, 2290–2300 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

2007 (2)

2005 (1)

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

2003 (1)

2001 (1)

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

1999 (1)

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

1998 (1)

1997 (2)

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1997).
[CrossRef]

A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
[CrossRef]

1991 (2)

G. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

1990 (1)

1975 (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Agrawal, G.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1997).
[CrossRef]

G. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

G. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

Akhmediev, N.

N. Akhmediev and A. Ankiewicz, “Three sources and three component parts of the concept of dissipative solitons” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 1–28.
[CrossRef]

Akhmediev, N. N.

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

N. Akhmediev and A. Ankiewicz, “Three sources and three component parts of the concept of dissipative solitons” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 1–28.
[CrossRef]

Bakonyi, Z.

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

Buckley, J.

Chong, A.

Ding, E.

Doran, N. J.

Egorov, O.

B. Ortaç, A. Zaviyalov, C. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35, 1578–1580 (2010).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

Evangelides, S.

L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

Forysiak, W.

Gong, Y. K.

L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal dispersion fiber lasers,” Laser Phys. Lett. 7, 63–67 (2010).
[CrossRef]

Grelu, P.

P. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 137–173.
[CrossRef]

Hasegawa, A.

A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
[CrossRef]

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[CrossRef] [PubMed]

Haus, H.

L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

Haus, H. A.

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Ilday, F.

Iliew, R.

B. Ortaç, A. Zaviyalov, C. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35, 1578–1580 (2010).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

Kalashnikov, V. L.

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

Keller, U.

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

Kodama, Y.

A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
[CrossRef]

A. Hasegawa and Y. Kodama, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[CrossRef] [PubMed]

Komarov, A.

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

Krimer, D. O.

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

Kutz, N.

Kuznetsova, L.

Leblond, H.

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

Lederer, F.

B. Ortaç, A. Zaviyalov, C. Nielsen, O. Egorov, R. Iliew, J. Limpert, F. Lederer, and A. Tünnermann, “Observation of soliton molecules with independently evolving phase in a mode-locked fiber laser,” Opt. Lett. 35, 1578–1580 (2010).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

Limpert, J.

Liu, X. M.

L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal dispersion fiber lasers,” Laser Phys. Lett. 7, 63–67 (2010).
[CrossRef]

Maruto, A.

A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
[CrossRef]

Michaelis, D.

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

Mikhailov, V. P.

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

Mollenauer, L.

L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

Nielsen, C.

Nijhof, J. H. B.

Olsson, N.

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1997).
[CrossRef]

Onishchukov, G.

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

Ortaç, B.

Paschotta, R.

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

Peschel, U.

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

Plötner, M.

Poloyko, I. G.

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

Renniger, W.

Sanchez, F.

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

Schreiber, T.

Soto-Crespo, J. M.

P. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 137–173.
[CrossRef]

Tünnermann, A.

Wang, L. R.

L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal dispersion fiber lasers,” Laser Phys. Lett. 7, 63–67 (2010).
[CrossRef]

Wise, F.

Zaviyalov, A.

Zavyalov, A.

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Hysteresis of dissipative soliton molecules in mode-locked fiber lasers,” Opt. Lett. 34, 3827–3829 (2009).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

Appl. Phys. B (1)

R. Paschotta and U. Keller, “Passive mode locking with slow saturable absorbers,” Appl. Phys. B 73, 653–662 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

G. Agrawal and N. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25, 2297–2306 (1997).
[CrossRef]

J. Appl. Phys. (1)

H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

J. Lightwave Technol. (1)

L. Mollenauer, S. Evangelides, and H. Haus, “Long-distance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[CrossRef]

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

Laser Phys. Lett. (1)

L. R. Wang, X. M. Liu, and Y. K. Gong, “Giant-chirp oscillator for ultra-large net-normal dispersion fiber lasers,” Laser Phys. Lett. 7, 63–67 (2010).
[CrossRef]

Opt. Commun. (1)

V. L. Kalashnikov, D. O. Krimer, I. G. Poloyko, and V. P. Mikhailov, “Ultrashort pulse generation in cw solid-state lasers with semiconductor saturable absorber in the presence of the absorption linewidth enhancement,” Opt. Commun. 159, 237–242 (1999).
[CrossRef]

Opt. Express (2)

Opt. Fiber Technol. (1)

A. Hasegawa, Y. Kodama, and A. Maruto, “Recent progress in dispersion-managed soliton transmission technologies,” Opt. Fiber Technol. 3, 197–213 (1997).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. A (3)

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Discrete family of dissipative soliton pairs in mode-locked fiber lasers,” Phys. Rev. A 79, 053841 (2009).
[CrossRef]

G. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

A. Zavyalov, R. Iliew, O. Egorov, and F. Lederer, “Dissipative molecules with independently evolving or flipping phases in mode-locked fiber lasers,” Phys. Rev. A 80, 043829 (2009).
[CrossRef]

Phys. Rev. E (1)

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

Other (5)

G. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, 1995).

U. Peschel, D. Michaelis, Z. Bakonyi, G. Onishchukov, and F. Lederer, “Dynamics of dissipative temporal solitons,” in Dissipative Solitons, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2005), pp. 161–181.
[CrossRef]

P. Grelu and J. M. Soto-Crespo, “Temporal soliton molecules in mode-locked lasers: collisions, pulsations and vibrations” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 137–173.
[CrossRef]

N. N. Akhmediev and A. Ankiewicz, Solitons, Nonlinear Pulses and Beams (Chapman and Hall, 1997).

N. Akhmediev and A. Ankiewicz, “Three sources and three component parts of the concept of dissipative solitons” in Dissipative Solitons: From Optics to Biology and Medicine, N.Akhmediv and A.Ankiewicz, eds. (Springer, 2008), pp. 1–28.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic of the gain dynamics at a fixed point of the doped fiber over one complete cavity period.

Fig. 2
Fig. 2

Intra-cavity evolution of the stable dissipative soliton in the mode-locked fiber laser. (Top) intensity pulse profile and (bottom) its peak amplitude and its duration as functions of the intra-cavity position.

Fig. 3
Fig. 3

Numerical comparison between the lumped and the distributed models with differently defined g eff . (a) The stable pulse profiles at g 0 = 0.5 m 1 and (b) the soliton branches.

Fig. 4
Fig. 4

Stable soliton branches versus the dispersion parameter x for both models where g eff L DF = K SA + K OC in the distributed model.

Fig. 5
Fig. 5

Distributed versus lumped model for the case of saturable gain. (a) The evolution of the peak amplitude from the initial condition to the stationary state and (b) the corresponding stable pulse profiles. Laser parameters: D DDL = 0.012 ps 2 , g 0 = 0.7 m 1 , E sat gain = 1   nJ while other parameters are defined in Section 2.

Fig. 6
Fig. 6

Numerical comparison of the soliton branches obtained in the lumped model, the distributed model ( g eff = g 0 ) , and the CQGLE at different SA saturation powers: (a) P sat SA = 83.3   W and (b) P sat SA = 166.7   W .

Equations (47)

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

U ( z , t ) z + i 2 ( β 2 + i g ( z , t ) T 2 2 ) 2 U ( z , t ) t 2 = g ( z , t ) 2 U ( z , t ) + i γ | U ( z , t ) | 2 U ( z , t ) ,
g ( z , t ) t = g 0 g ( z , t ) T relax gain g ( z , t ) | U ( z , t ) | 2 E sat gain ,
g ( z ) = g 0 1 + | U ( z , t ) | 2 / E sat gain ,
g ( z , t ) t = { g ( z , t ) | U ( z , t ) | 2 E sat gain , if   t [ t 1 ; t 2 ] g 0 g ( z , t ) T relax gain , if   t [ 0 ; t 1 ] [ t 2 ; T RT ] . }
g 2 ( z ) = g 1 ( z ) ( 1 pulse | U ( z , t ) | 2 d t E sat gain ) ,
g 1 ( z ) = g 2 ( z ) + ( g 0 g 2 ( z ) ) T RT T relax gain ,
g ( z ) = g 1 ( z ) + g 2 ( z ) 2 = g 0 1 + pulse | U ( z , t ) | 2 d t / E ¯ sat gain ,
E ¯ sat gain = E sat gain T RT T relax gain .
U ( z , t ) z = 1 2 δ ( z , t ) U ( z , t ) ,
δ ( z , t ) t = δ 0 δ ( z , t ) T relax SA δ ( z , t ) | U ( z , t ) | 2 E sat SA ,
U out ( t ) = U in ( t ) exp [ 1 i h 2 δ 0 L SA 1 + | U in ( t ) | 2 / P sat SA ] ,
U ̃ out ( ω ) = U ̃ in ( ω ) exp [ i D DDL ω 2 / 2 ] ,
| U out ( t ) | 2 = T OC | U in ( t ) | 2 .
U ( z , t ) = V ( z ) W ( z , t ) .
V DF ( z ) = exp ( g eff z / 2 )     for   0 z L DF ,
W ( z , t ) z + i 2 ( β 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = g 0 g eff 2 W ( z , t ) + i γ   exp ( g eff z ) | W ( z , t ) | 2 W ( z , t ) ,
W ( z , t ) z + i 2 ( β 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = g 0 g eff 2 W ( z , t ) + i γ exp ( g eff L DF ) 1 g eff L DF | W ( z , t ) | 2 W ( z , t ) .
W ( z , t ) z + i 2 ( β ¯ 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = g 0 g eff 2 W ( z , t ) + i γ ¯ | W ( z , t ) | 2 W ( z , t ) ,
β ¯ 2 = β 2 + D DDL / L DF ,
γ ¯ = γ exp ( g eff L DF ) 1 g eff L DF .
W ( z , t ) z = ( K SA 2 L SA 1 i h 2 δ 0 1 + | W ( z , t ) | 2 / P ¯ sat SA ) W ( z , t ) ,
P ¯ sat SA = P sat SA   exp ( g eff L DF ) .
W ( z , t ) z + i 2 ( β ¯ 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = ( g 0 K SA / L DF g eff 2 1 i h 2 L DF δ 0 L SA 1 + | W ( z , t ) | 2 / P ¯ sat SA + i γ ¯ | W ( z , t ) | 2 ) W ( z , t ) .
W ( z , t ) z + i 2 ( β ¯ 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = ( g 0 K OC / L DF 2 1 i h 2 δ 0 L SA / L DF 1 + | W ( z , t ) | 2 / P ¯ sat SA + i γ ¯ | W ( z , t ) | 2 ) W ( z , t ) ,
K OC = ln ( T OC ) ,
K SA = { 0 , if   | W ( z , t ) | max 2 P ¯ sat SA δ 0 L SA / 2 , if   | W ( z , t ) | max 2 P ¯ sat SA δ 0 L SA , if   | W ( z , t ) | max 2 P ¯ sat SA . }
W ( z , t ) z + i 2 ( β ¯ 2 + i g 0 T 2 2 ) 2 W ( z , t ) t 2 = ( g 0 K OC / L DF 2 ( 1 i h ) δ ¯ ( z , t ) 2 L DF ) W ( z , t ) ,
δ ¯ ( z , t ) t = δ 0 L SA δ ¯ ( z , t ) T relax SA δ ¯ ( z , t ) | W ( z , t ) | 2 exp ( g eff L DF ) E sat SA ,
W ( z , t ) z + i 2 ( β 2 + i g ( z ) T 2 2 ) 2 W ( z , t ) t 2 = g ( z ) g eff 2 W ( z , t ) + i γ   exp ( g eff z ) | W ( z , t ) | 2 W ( z , t ) ,
g ( z ) = g 0 1 + exp ( g eff z ) Q ( z ) ,
Q ( z ) = pulse | U ( z , t ) | 2 d t / E ¯ sat gain .
W ( z , t ) z + i 2 ( β 2 + i g ( z ) T 2 2 ) 2 W ( z , t ) t 2 = ( g ( z ) g eff 2 + i γ exp ( g eff L DF ) 1 g eff L DF | W ( z , t ) | 2 ) W ( z , t ) ,
g ( z ) = g 0 + g 0 g eff L DF ln [ 1 + Q ( z ) 1 + exp ( g eff L DF ) Q ( z ) ] .
W ( z , t ) z + i 2 ( β ¯ 2 + i g ( z ) T 2 2 ) 2 W ( z , t ) t 2 = ( g ( z ) K OC / L DF 2 1 i h 2 δ 0 L SA / L DF 1 + | W ( z , t ) | 2 / P ¯ sat SA + i γ ¯ | W ( z , t ) | 2 ) W ( z , t ) ,
β ¯ 2 = β 2 + D DDL / L DF ,
γ ¯ = γ exp ( g eff L DF ) 1 g eff L DF ,
P ¯ sat SA = P sat SA   exp ( g eff L DF ) ,
1 | W ( z , t ) | 2 / P ¯ sat SA = 1 | W ( z , t ) | 2 P ¯ sat SA + ( | W ( z , t ) | 2 P ¯ sat SA ) 2 + .
i ψ Z + D 2 2 ψ τ 2 + γ | ψ | 2 ψ = i θ ψ + i ε | ψ | 2 ψ + i β 2 ψ τ 2 i μ | ψ | 4 ψ + η | ψ | 4 ψ ,
ψ = W   exp [ i h δ 0 L SA 2 L DF Z ] ,
D = sgn ( β ¯ 2 ) ,
β = g 0 T 2 2 2 | β ¯ 2 | ,
γ = L D ( γ ¯ h δ 0 L SA 2 L DF P ¯ sat SA ) ,
θ = L D ( g 0 2 K OC + δ 0 L SA 2 L DF ) ,
ε = L D δ 0 L SA 2 L DF P ¯ sat SA ,
μ = L D δ 0 L SA 2 L DF ( P ¯ sat SA ) 2 ,
η = L D h δ 0 L SA 2 L DF ( P ¯ sat SA ) 2 .

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