Abstract

A semianalytic approach based on the moment method is used for investigating pulse evolution in mode-locked lasers in which intracavity dispersive and nonlinear effects play a significant role. Its application to an FM mode-locked laser allows us to perform fast parametric studies while predicting the important pulse parameters. When third-order dispersive effects are negligible, a fully analytic treatment is developed that predicts how cavity parameters affect the final steady state. Our analytic approach also allows us to predict relaxation-oscillation behavior as the pulse approaches its steady state. We use this technique to investigate novel aspects specific to FM mode-locked lasers such as stability of and switching between the multiple steady-state solutions. All results obtained are in excellent agreement with numerical simulations.

© 2005 Optical Society of America

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References

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  1. S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
    [CrossRef]
  2. S. E. Harris and R. Targ, "FM oscillation of the He-Ne laser," Appl. Phys. Lett. 5, 202-204 (1964).
    [CrossRef]
  3. G. Geister and R. Ulrich, "Neodymium-fiber laser with integrated-optic mode locker," Opt. Commun. 68, 187-189 (1988).
    [CrossRef]
  4. D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: Theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
    [CrossRef]
  5. H. A. Haus and Y. Silberberg, "Laser mode locking with addition of nonliner index," IEEE J. Quantum Electron. QE-22, 325-331 (1986).
    [CrossRef]
  6. F. X. Kärtner, D. Kopf, and U. Keller, "Solitary-pulse stabilization and shortening in actively mode-locked lasers," J. Opt. Soc. Am. B 12, 486-496 (1995).
    [CrossRef]
  7. J. G. Caputo, C. B. Clausen, M. P. Sørensen, and S. Bischoff, "Amplitude-modulated fiber-ring laser," J. Opt. Soc. Am. B 17, 705-712 (2000).
    [CrossRef]
  8. H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
    [CrossRef]
  9. N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
    [CrossRef]
  10. K. Tamura and M. Nakazawa, "Pulse energy equalization in harmonically FM mode-locked lasers with slow gain," Opt. Lett. 21, 1930-1932 (1996).
    [CrossRef] [PubMed]
  11. M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
    [CrossRef]
  12. A. M. Dunlop, W. J. Firth, and E. M. Wright, "Pulse shapes and stability in Kerr and active mode-locking (KAML)," Opt. Express 2, 204-211 (1998).
    [CrossRef] [PubMed]
  13. S. Longhi and P. Laporta, "Time-domain analysis of frequency modulation laser oscillation," Appl. Phys. Lett. 73, 720-722 (1998).
    [CrossRef]
  14. N. G. Usechak and G. P. Agrawal, "Semi-analytic technique for analyzing mode-locked lasers," Opt. Express 13, 2075-2081 (2005).
    [CrossRef] [PubMed]
  15. D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part II: Experimental results in a Nd:YAG laser with internal FM modulation," IEEE J. Quantum Electron. QE-6, 709-715 (1970).
    [CrossRef]
  16. M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
    [CrossRef]
  17. G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).
  18. N. R. Pereira and L. Stenflo, "Nonlinear Schrödinger equation including growth and damping," Phys. Fluids 20, 1733-1734 (1977).
    [CrossRef]
  19. N.Akhmediev and A.Ankiewicz, eds., Dissipative Solitons (Springer, 2005).
    [CrossRef]
  20. E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).
  21. To remain consistent with Ref. , a Fabry-Perot laser cavity was simulated. In a Fabry-Perot cavity the light passes through each element twice during a single round trip; therefore, DeltaFM→2DeltaFM=0.9 and Psat→Psat/2=12.5 mW.
  22. C. J. McKinstrie, "Effects of filtering on Gordon-Haus timing jitter in dispersion-managed systems," J. Opt. Soc. Am. B 19, 1275-1285 (2002).
    [CrossRef]
  23. J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
    [CrossRef]
  24. J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium-doped fiber laser with soliton pulse shaping," Opt. Lett. 15, 1269-1271 (1989).
    [CrossRef]
  25. T. Brabec, Ch. Spielmann, and F. Krausz, "Mode locking in solitary lasers," Opt. Lett. 16, 1961-1963 (1991).
    [CrossRef] [PubMed]
  26. D. Kopf, F. X. Kärtner, K. J. Weingarten, and U. Keller, "Pulse shortening in a Nd:glass laser by gain reshaping and soliton formation," Opt. Lett. 19, 2146-2148 (1994).
    [CrossRef] [PubMed]
  27. K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

2005 (2)

N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
[CrossRef]

N. G. Usechak and G. P. Agrawal, "Semi-analytic technique for analyzing mode-locked lasers," Opt. Express 13, 2075-2081 (2005).
[CrossRef] [PubMed]

2003 (1)

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

2002 (1)

2000 (1)

1998 (5)

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

A. M. Dunlop, W. J. Firth, and E. M. Wright, "Pulse shapes and stability in Kerr and active mode-locking (KAML)," Opt. Express 2, 204-211 (1998).
[CrossRef] [PubMed]

S. Longhi and P. Laporta, "Time-domain analysis of frequency modulation laser oscillation," Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).

K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

1996 (3)

K. Tamura and M. Nakazawa, "Pulse energy equalization in harmonically FM mode-locked lasers with slow gain," Opt. Lett. 21, 1930-1932 (1996).
[CrossRef] [PubMed]

M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
[CrossRef]

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

1995 (1)

1994 (1)

1991 (1)

1989 (1)

J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium-doped fiber laser with soliton pulse shaping," Opt. Lett. 15, 1269-1271 (1989).
[CrossRef]

1988 (1)

G. Geister and R. Ulrich, "Neodymium-fiber laser with integrated-optic mode locker," Opt. Commun. 68, 187-189 (1988).
[CrossRef]

1986 (1)

H. A. Haus and Y. Silberberg, "Laser mode locking with addition of nonliner index," IEEE J. Quantum Electron. QE-22, 325-331 (1986).
[CrossRef]

1977 (1)

N. R. Pereira and L. Stenflo, "Nonlinear Schrödinger equation including growth and damping," Phys. Fluids 20, 1733-1734 (1977).
[CrossRef]

1971 (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
[CrossRef]

1970 (2)

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: Theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part II: Experimental results in a Nd:YAG laser with internal FM modulation," IEEE J. Quantum Electron. QE-6, 709-715 (1970).
[CrossRef]

1964 (1)

S. E. Harris and R. Targ, "FM oscillation of the He-Ne laser," Appl. Phys. Lett. 5, 202-204 (1964).
[CrossRef]

Agrawal, G. P.

N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
[CrossRef]

N. G. Usechak and G. P. Agrawal, "Semi-analytic technique for analyzing mode-locked lasers," Opt. Express 13, 2075-2081 (2005).
[CrossRef] [PubMed]

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).

Baer, T.

J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium-doped fiber laser with soliton pulse shaping," Opt. Lett. 15, 1269-1271 (1989).
[CrossRef]

Bischoff, S.

Brabec, T.

Caputo, J. G.

Clausen, C. B.

Dunlop, A. M.

Firth, W. J.

Geister, G.

G. Geister and R. Ulrich, "Neodymium-fiber laser with integrated-optic mode locker," Opt. Commun. 68, 187-189 (1988).
[CrossRef]

Hall, D. W.

J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium-doped fiber laser with soliton pulse shaping," Opt. Lett. 15, 1269-1271 (1989).
[CrossRef]

Harris, S. E.

S. E. Harris and R. Targ, "FM oscillation of the He-Ne laser," Appl. Phys. Lett. 5, 202-204 (1964).
[CrossRef]

Haus, H. A.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

H. A. Haus and Y. Silberberg, "Laser mode locking with addition of nonliner index," IEEE J. Quantum Electron. QE-22, 325-331 (1986).
[CrossRef]

Ippen, E. P.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

Jones, D. J.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

Kafka, J. D.

J. D. Kafka, T. Baer, and D. W. Hall, "Mode-locked erbium-doped fiber laser with soliton pulse shaping," Opt. Lett. 15, 1269-1271 (1989).
[CrossRef]

Kärtner, F. X.

Keller, U.

Kopf, D.

Krausz, F.

Kubota, H.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Kuizenga, D. J.

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: Theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part II: Experimental results in a Nd:YAG laser with internal FM modulation," IEEE J. Quantum Electron. QE-6, 709-715 (1970).
[CrossRef]

Laporta, P.

S. Longhi and P. Laporta, "Time-domain analysis of frequency modulation laser oscillation," Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

Longhi, S.

S. Longhi and P. Laporta, "Time-domain analysis of frequency modulation laser oscillation," Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

McKinstrie, C. J.

Nakazawa, M.

E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).

K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

K. Tamura and M. Nakazawa, "Pulse energy equalization in harmonically FM mode-locked lasers with slow gain," Opt. Lett. 21, 1930-1932 (1996).
[CrossRef] [PubMed]

M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
[CrossRef]

Pereira, N. R.

N. R. Pereira and L. Stenflo, "Nonlinear Schrödinger equation including growth and damping," Phys. Fluids 20, 1733-1734 (1977).
[CrossRef]

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
[CrossRef]

Sahara, A.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

Santhanam, J.

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

Siegman, A. E.

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part II: Experimental results in a Nd:YAG laser with internal FM modulation," IEEE J. Quantum Electron. QE-6, 709-715 (1970).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: Theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

Silberberg, Y.

H. A. Haus and Y. Silberberg, "Laser mode locking with addition of nonliner index," IEEE J. Quantum Electron. QE-22, 325-331 (1986).
[CrossRef]

Sørensen, M. P.

Spielmann, Ch.

Stenflo, L.

N. R. Pereira and L. Stenflo, "Nonlinear Schrödinger equation including growth and damping," Phys. Fluids 20, 1733-1734 (1977).
[CrossRef]

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
[CrossRef]

Tamura, K.

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).

K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
[CrossRef]

K. Tamura and M. Nakazawa, "Pulse energy equalization in harmonically FM mode-locked lasers with slow gain," Opt. Lett. 21, 1930-1932 (1996).
[CrossRef] [PubMed]

Targ, R.

S. E. Harris and R. Targ, "FM oscillation of the He-Ne laser," Appl. Phys. Lett. 5, 202-204 (1964).
[CrossRef]

Ulrich, R.

G. Geister and R. Ulrich, "Neodymium-fiber laser with integrated-optic mode locker," Opt. Commun. 68, 187-189 (1988).
[CrossRef]

Usechak, N. G.

N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
[CrossRef]

N. G. Usechak and G. P. Agrawal, "Semi-analytic technique for analyzing mode-locked lasers," Opt. Express 13, 2075-2081 (2005).
[CrossRef] [PubMed]

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
[CrossRef]

Weingarten, K. J.

Wong, W. S.

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

Wright, E. M.

Yoshida, E.

K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).

M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
[CrossRef]

Zuegel, J. D.

N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

S. E. Harris and R. Targ, "FM oscillation of the He-Ne laser," Appl. Phys. Lett. 5, 202-204 (1964).
[CrossRef]

S. Longhi and P. Laporta, "Time-domain analysis of frequency modulation laser oscillation," Appl. Phys. Lett. 73, 720-722 (1998).
[CrossRef]

Electron. Lett. (1)

M. Nakazawa, E. Yoshida, and K. Tamura, "10 GHz, 2 ps regeneratively and harmonically FM mode-locked erbium fibre ring laser," Electron. Lett. 32, 1285-1287 (1996).
[CrossRef]

IEEE J. Quantum Electron. (5)

N. G. Usechak, J. D. Zuegel, and G. P. Agrawal, "FM mode-locked fiber lasers operating in the autosoliton regime," IEEE J. Quantum Electron. 41, 753-761 (2005).
[CrossRef]

M. Nakazawa, H. Kubota, A. Sahara, and K. Tamura, "Time-domain ABCD matrix formalism for laser mode-locking and optical pulse transmission," IEEE J. Quantum Electron. 34, 1075-1081 (1998).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part I: Theory," IEEE J. Quantum Electron. QE-6, 694-708 (1970).
[CrossRef]

H. A. Haus and Y. Silberberg, "Laser mode locking with addition of nonliner index," IEEE J. Quantum Electron. QE-22, 325-331 (1986).
[CrossRef]

D. J. Kuizenga and A. E. Siegman, "FM and AM mode locking of the homogeneous laser--Part II: Experimental results in a Nd:YAG laser with internal FM modulation," IEEE J. Quantum Electron. QE-6, 709-715 (1970).
[CrossRef]

IEICE Trans. Electron. (2)

K. Tamura, E. Yoshida, and M. Nakazawa, "Forced phase modulation and self phase modulation effects in dispersion-tuned mode-locked fiber lasers," IEICE Trans. Electron. E81-C, 195-200 (1998).

E. Yoshida, K. Tamura, and M. Nakazawa, "Intracavity dispersion effects of a regeneratively and harmonically FM mode-locked erbium-doped fiber laser," IEICE Trans. Electron. E81-C, 189-194 (1998).

J. Lightwave Technol. (1)

H. A. Haus, D. J. Jones, E. P. Ippen, and W. S. Wong, "Theory of soliton stability in asynchronous modelocking," J. Lightwave Technol. 14, 622-627 (1996).
[CrossRef]

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

Opt. Commun. (2)

J. Santhanam and G. P. Agrawal, "Raman-induced spectral shifts in optical fibers: general theory based on the moment method," Opt. Commun. 222, 413-420 (2003).
[CrossRef]

G. Geister and R. Ulrich, "Neodymium-fiber laser with integrated-optic mode locker," Opt. Commun. 68, 187-189 (1988).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Phys. Fluids (1)

N. R. Pereira and L. Stenflo, "Nonlinear Schrödinger equation including growth and damping," Phys. Fluids 20, 1733-1734 (1977).
[CrossRef]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, and V. I. Talanov, "Averaged description of wave beams in linear and nonlinear media (the method of moments)," Radiophys. Quantum Electron. 14, 1062-1070 (1971).
[CrossRef]

Other (3)

G. P. Agrawal, Applications of Nonlinear Fiber Optics (Academic, 2001).

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

To remain consistent with Ref. , a Fabry-Perot laser cavity was simulated. In a Fabry-Perot cavity the light passes through each element twice during a single round trip; therefore, DeltaFM→2DeltaFM=0.9 and Psat→Psat/2=12.5 mW.

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Figures (8)

Fig. 1
Fig. 1

(Color online) Steady-state pulse shapes in (a) the anomalous dispersion regime and (b) the normal dispersion regime. Plot (a) is fit with the expected autosoliton, and plot (b) is fit with the expected Gaussian.

Fig. 2
Fig. 2

Evolution of pulse energy E, width τ, and chirp q over multiple round trips in the case of anomalous (top row) and normal (bottom row) dispersion.

Fig. 3
Fig. 3

Pulse width predicted by our theory as a function of modulation depth in the (a) anomalous and (b) normal dispersion regimes. In each case, we compare our predictions with numerical simulations and with the theories given in Refs. [4, 5] (when β ¯ 2 < 0 ), and 10.

Fig. 4
Fig. 4

Steady-state pulse width τ, chirp q (left column), temporal shift ξ, and frequency shift Ω (right column) in the anomalous (top row) and normal (bottom row) dispersion regimes.

Fig. 5
Fig. 5

Same as Fig. 4 except the pulse parameters are plotted as a function of the nonlinear parameter.

Fig. 6
Fig. 6

Chirp as a function of average dispersion β ¯ 2 from Eq. (28).

Fig. 7
Fig. 7

(Color online) Temporal shift per round trip as a function of pulse-modulator detuning using the dominant pulse parameters ( τ = 0.787 ps , q = 0.0179 ). These figures identify the stable (squares) and unstable (circles) operating locations as well as the strength and direction of the pulse velocity for the following cases: (a) β ¯ 2 < 0 and β ¯ 3 = 0 , (b) β ¯ 2 < 0 and β ¯ 3 > 0 , (c) β ¯ 2 < 0 and β ¯ 3 < 0 , and (d) β ¯ 2 > 0 and β ¯ 3 > 0 . The imaginary part of the modulator’s signal is plotted by the dashed curve to aid in location identification for a fixed modulation depth of Δ FM = 0.45 .

Fig. 8
Fig. 8

(Color online) Temporal shift per round trip as a function of pulse-modulator detuning using the secondary pulse parameters ( τ 0 = 16.2 ps , q 0 = 111.73 ). These figures identify the stable (squares) and unstable (circles) operating locations as well as the strength and direction of the pulse velocity for the following cases: (a) β ¯ 2 < 0 and β ¯ 3 = 0 , (b) β ¯ 2 < 0 and β ¯ 3 > 0 , and (c) result of large TOD on stable pulses. The imaginary part of the modulator’s signal is plotted by the dashed curve to aid in location identification for a fixed modulation depth of Δ FM = 0.45 .

Tables (1)

Tables Icon

Table 1 Parameter Values Used in This Paper

Equations (36)

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

T R A T + i 2 ( β ¯ 2 + i g ¯ T 2 2 ) L R 2 A t 2 1 6 β ¯ 3 L R 3 A t 3 = i γ ¯ L R A 2 A + 1 2 ( g ¯ α ¯ ) L R A + M ( A , t ) ,
P ave = 1 T m T m 2 T m 2 A ( t , z ) 2 d t .
A ( T , t ) = a [ sech ( t τ ) ] 1 + i q exp ( i κ T ) ,
τ 2 = g ¯ T 2 2 + β ¯ 2 q g ¯ α ¯ ,
a 2 = 1 2 γ ¯ τ 2 [ ( q 2 2 ) β ¯ 2 + 3 g ¯ T 2 2 q ] ,
κ = L R 2 T R τ 2 [ ( 1 q 2 ) β ¯ 2 2 g ¯ T 2 2 q ] ,
q = 3 β ¯ 2 2 g ¯ T 2 2 ± [ ( 3 β ¯ 2 2 g ¯ T 2 2 ) 2 + 2 ] 1 2 .
A ( T , t ) = a { sech [ ( t ξ ) τ ] } 1 + i q exp [ i Ω ( t ξ ) + i k T + i ϕ 0 ] .
A ( T , t ) = a { exp [ ( t ξ ) 2 2 τ 2 ] } 1 + i q exp [ i Ω ( t ξ ) + i k T + i ϕ 0 ] .
E ( T ) = A ( T , t ) 2 d t ,
ξ ( T ) = 1 E t A ( T , t ) 2 d t ,
Ω ( T ) = i 2 E [ A * A t A A * t ] d t ,
q ( T ) = i E ( t ξ ) [ A * A t A A * t ] d t ,
τ 2 ( T ) = 2 C 3 E ( t ξ ) 2 A ( T , t ) 2 d t ,
d E d T = [ A * A t + A A * T ] d t ,
d ξ d T = 1 E d E d T ξ + 1 E t [ A * A t + A A * t ] d t ,
d Ω d T = 1 E d E d T Ω + i 2 E [ T ( A * A T ) T ( A A * t ) ] d t ,
d q d T = 1 E d E d T q + i E ( t ξ ) [ T ( A * A t ) T ( A A * t ) ] d t ,
τ d τ d T = 1 2 E d E d T τ 2 + C 3 E ( t ξ ) 2 [ A * A T + A A * T ] d t .
T R L R d E d T = ( g ¯ α ¯ ) E g ¯ T 2 2 2 τ 2 [ C 0 ( 1 + q 2 ) + 2 Ω 2 τ 2 ] E ,
T R L R d ξ d T = β ¯ 2 Ω g ¯ T 2 2 q Ω + β ¯ 3 4 τ 2 [ C 0 ( 1 + q 2 ) + 2 Ω 2 τ 2 ] ,
T R L R d Ω d T = C 0 g ¯ T 2 2 τ 2 ( 1 + q 2 ) Ω + Δ FM ω m L R Ψ 0 sin [ ω m ( ξ t m ) ] ,
T R L R d q d T = β ¯ 2 τ 2 [ C 0 ( 1 + q 2 ) + 2 Ω 2 τ 2 ] g ¯ T 2 2 τ 2 q [ C 1 ( 1 + q 2 ) + 2 Ω 2 τ 2 ] + C 2 γ ¯ E 2 π τ + β ¯ 3 Ω τ 2 [ 3 2 C 0 ( 1 + q 2 ) + Ω 2 τ 2 ] + Δ FM ω m τ L R Ψ 1 cos [ ω m ( ξ t m ) ] ,
T R L R d τ d T = C 3 β ¯ 2 τ q + C 3 β ¯ 3 τ q Ω + C 0 C 3 g ¯ T 2 2 2 τ ( C 4 q 2 ) ,
T R L R d E d T = g ¯ ss α ¯ g ¯ ss T 2 2 2 τ ss 2 C 0 ( 1 + q ss 2 ) = 0 ,
T R L R d q d T = β ¯ 2 τ ss 2 C 0 ( 1 + q ss 2 ) g ¯ ss T 2 2 τ ss 2 q ss C 1 ( 1 + q ss 2 ) + C 2 γ ¯ E ss 2 π τ ss + Δ FM ω m τ ss L R Ψ 1 = 0 ,
T R L R d τ d T = C 3 β ¯ 2 q ss + C 0 C 3 g ¯ ss T 2 2 2 ( C 4 q ss 2 ) = 0 .
q ss = d ± d 2 + C 4 ,
g ¯ ss = α ¯ [ 1 C 0 T 2 2 2 τ ss 2 ( 1 + q ss 2 ) ] 1 α ¯ + C 0 T 2 2 α ¯ 2 τ ss 2 ( 1 + q ss 2 ) .
τ FWHM = 2 [ 2 ln ( 2 ) ] 1 2 ( g ¯ ss L R Δ FM ) 1 4 ( T 2 ω m ) 1 2 .
ω r = L R τ 0 T R B C D 2 ,
B = C 3 τ ss ( β ¯ 2 C 0 g ¯ T 2 2 q ss ) ,
C = C 2 γ ¯ E 2 π + 4 Δ FM ω m 2 τ ss 3 C 3 L R ,
D = 1 2 τ ss [ 2 β ¯ 2 C 0 q ss g ¯ T 2 2 C 1 ( 1 + 3 q ss 2 ) ] .
Ω ( T ) = Δ FM ω m τ 2 C 0 L R g ¯ T 2 2 ( 1 + q 2 ) Ψ 0 sin [ ω m ( ξ t m ) ] + C exp [ C 0 L R g ¯ T 2 2 ( 1 + q 2 ) τ 2 T R T ] ,
T R d ξ d T = L R ( β ¯ 2 g ¯ T 2 2 q ) Δ FM ω m τ 2 C 0 L R g ¯ T 2 2 ( 1 + q 2 ) Ψ 0 sin [ ω m ( ξ t m ) ] + L R β ¯ 3 4 τ 2 { C 0 ( 1 + q 2 ) + 2 Δ FM 2 ω m 2 τ 6 C 0 2 L R 0 g ¯ 2 T 2 4 ( 1 + q 2 ) 2 Ψ 0 2 sin 2 [ ω m ( ξ t m ) ] } .

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