Abstract

We present a theoretical description of the generation of ultrashort, high-energy pulses in an all-normal dispersion laser cavity with spectral filtering. A reduced variational model based upon the Haus master mode-locking equations with quintic saturation is shown to characterize the experimentally observed dynamics. Critical in driving the intracavity dynamics is the nontrivial phase profiles generated and their periodic modification from the spectral filter. The theory gives a simple geometrical description of the intracavity dynamics and possible operation modes of the laser cavity. Further, it provides a simple and efficient method for optimizing the laser cavity performance.

© 2008 Optical Society of America

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References

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  1. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  2. I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
    [CrossRef]
  3. F. O. Ilday, F. W. Wise, and T. Sosnowski, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Opt. Lett. 27, 1531-1533 (2002).
    [CrossRef]
  4. F. O. Ilday, J. Buckley, F. W. Wise, and W. G. Clark, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  5. J. Buckley, F. W. Wise, F. O. Ilday, and T. Sosnowski, “Femtosecond fiber lasers with pulse energies above 10 nJ,” Opt. Lett. 30, 1888-1890 (2005).
    [CrossRef] [PubMed]
  6. A. Chong, J. Buckley, W. Renninger, and F. Wise, “All-normal-dispersion femtosecond fiber laser,” Opt. Express 14, 10095-10100 (2006).
    [CrossRef] [PubMed]
  7. A. Chong, W. 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]
  8. A. Chong, W. Renninger, and F. Wise, “Properties of normal-dispersion femtosecond fiber lasers,” J. Opt. Soc. Am. B 25, 140-148 (2008).
    [CrossRef]
  9. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  10. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
    [CrossRef]
  11. A. Fernandez, T. Fuji, A. Poppe, A. Frbach, 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]
  12. 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]
  13. 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, 4783-4796 (1997).
    [CrossRef]
  14. J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
    [CrossRef]
  15. J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
    [CrossRef]
  16. 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]
  17. V. L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007).
    [CrossRef]
  18. V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  21. N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124-3128 (2008).
    [CrossRef]
  22. B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for mode-locking in the normal dispersive regime,” Opt. Lett. 33, 911-913 (2008).
    [CrossRef] [PubMed]
  23. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
    [CrossRef]
  24. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).
  25. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
    [CrossRef]
  26. C. Jirauschek, U. Morgner, and F. X. Krtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716-1721 (2002).
    [CrossRef]
  27. C. Antonelli, J. Chen, and F. X. Kärtner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919-5924 (2007).
    [CrossRef] [PubMed]
  28. N. G. Usechak and G. P. Agrawal, “Semi-analytic technique for analyzing mode-locked lasers,” Opt. Express 13, 2075-2081 (2005).
    [CrossRef] [PubMed]
  29. N. G. Usechak and G. P. Agrawal, “Rate-equation approach for frequency-modulation mode locking using the moment method,” J. Opt. Soc. Am. B 22, 2570-2580 (2005).
    [CrossRef]
  30. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099-2111 (1997).
    [CrossRef]
  31. S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099-2111 (1997).
    [CrossRef]
  32. E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
    [CrossRef]
  33. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
    [CrossRef]
  34. P. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).

2008 (5)

2007 (4)

2006 (5)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
[CrossRef]

P. A. Belanger, “On the profile of pulses generated by fiber lasers:the highly-chirped positive dispersion regime (similariton),” Opt. Express 14, 12174-12182 (2006).
[CrossRef] [PubMed]

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

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
[CrossRef]

E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
[CrossRef]

2005 (3)

2004 (2)

2002 (4)

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

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).

C. Jirauschek, U. Morgner, and F. X. Krtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716-1721 (2002).
[CrossRef]

2001 (2)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

2000 (1)

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

1997 (3)

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, 4783-4796 (1997).
[CrossRef]

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099-2111 (1997).
[CrossRef]

S. Namiki, E. P. Ippen, H. A. Haus, and C. X. Yu, “Energy rate equations for mode-locked lasers,” J. Opt. Soc. Am. B 14, 2099-2111 (1997).
[CrossRef]

1996 (1)

1995 (1)

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

1994 (1)

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
[CrossRef]

1992 (1)

P. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).

1991 (1)

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, 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]

Agrawal, G. P.

Akhmediev, N.

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124-3128 (2008).
[CrossRef]

E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

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, 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]

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Ankiewicz, A. A.

E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
[CrossRef]

Antonelli, C.

Apolonski, A.

Bale, B. G.

Belanger, P. A.

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Buckley, J.

Chang, K. S.

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

Chen, J.

Chernykh, A.

V. L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
[CrossRef]

Chong, A.

Clark, W. G.

F. O. Ilday, J. Buckley, F. W. Wise, and W. G. Clark, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Dennis, M. L.

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

Drazin, P.

P. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).

Duling, I. N.

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

Fernandez, A.

Frbach, A.

Fuji, T.

Fujimoto, J. G.

Grelu, Ph.

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124-3128 (2008).
[CrossRef]

Haus, H. A.

Ilday, F. O.

Ippen, E. P.

Jirauschek, C.

Kalashnikov, V. L.

V. L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007).
[CrossRef]

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
[CrossRef]

Kärtner, F. X.

Krausz, F.

Krtner, F. X.

Kutz, J. N.

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Malomed, B. A.

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).

Morgner, U.

Namiki, S.

Podivilov, E.

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
[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]

B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for mode-locking in the normal dispersive regime,” Opt. Lett. 33, 911-913 (2008).
[CrossRef] [PubMed]

Sosnowski, T.

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124-3128 (2008).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[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, 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]

Tamura, K.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
[CrossRef]

Tsoy, E. N.

E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
[CrossRef]

Usechak, N. G.

Wabnitz, S.

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, 4783-4796 (1997).
[CrossRef]

Wise, F.

Wise, F. W.

Yu, C. X.

Appl. Phys. B (1)

V. L. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503-510 (2006).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

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

J. Opt. Soc. Am. B (8)

Opt. Express (5)

Opt. Lett. (5)

Phys. Lett. A (2)

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

N. Akhmediev, J. M. Soto-Crespo, and Ph. Grelu, “Roadmap to ultra-short record high-energy pulses out of laser oscillators,” Phys. Lett. A 372, 3124-3128 (2008).
[CrossRef]

Phys. Rev. A (2)

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

V. L. Kalashnikov and A. Chernykh, “Spectral anomalies and stability of chirped-pulse oscillators,” Phys. Rev. A 75, 033820 (2007).
[CrossRef]

Phys. Rev. E (3)

E. N. Tsoy, A. A. Ankiewicz, and N. Akhmediev, “Dynamical models for dissipative localized waves of the complex Ginzburg-Landau equation,” Phys. Rev. E 73, 036621 (2006).
[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, 4783-4796 (1997).
[CrossRef]

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

Phys. Rev. Lett. (1)

F. O. Ilday, J. Buckley, F. W. Wise, and W. G. Clark, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Pramana, J. Phys. (1)

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Prog. Opt. (1)

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).

SIAM Rev. (1)

J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
[CrossRef]

Other (2)

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

P. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).

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Figures (11)

Fig. 1
Fig. 1

Top: spectral profiles of Eq. (3) with large chirp parameter A = 10 and (a) B = 0.97 , (b) B = 0.5 , (c) B = 1.1 , (d) B = 35 . Bottom: representative experimental spectra from the ANDi laser corresponding to the theoretical pulses. Here the data is from Renninger et al. [9].

Fig. 2
Fig. 2

(a) Reduced CQGLE model with the ansatz Eq. (6) showing a stable node ( η 0 , B 0 , A 0 ) = ( 1.6 , 1.6 , 2.2 ) as the attracting state for the laser. The physical parameters are D = 0.4 , τ = 0.2 , δ = 1 , g 0 = 1.5 , e 0 = 1 , β = 0.7 , and μ = 0.2 . (b) Pulse parameters from full numerical simulation of Eq. (1) (solid curves) compared with those obtained from the reduced model Eq. (7) (dashed curves). The initial condition for both the full model and reduced model (gray square) is u ( 0 , T ) = 0.5 ( 1 + cosh ( T 2 ) ) 1 2 exp [ i 2 log ( 1 + cosh ( T 2 ) ) ] .

Fig. 3
Fig. 3

Experimental laser cavity configuration includes an amplifier with parabolic gain bandwidth Ω g and a Gaussian spectral filter with bandwidth Ω f . A typical pulse solution with spectral bandwidth Ω p is also shown. Note that the key parameter Γ = Ω f Ω p .

Fig. 4
Fig. 4

For three different ratios of the filter-to-pulse bandwidth, Γ = ( a ) , (b) 0.87, (c), (d) 0.62, and (e), (f) 0.5, the resultant spectral and temporal profiles is displayed. In each row of subfigures, the postfiltered pulse (solid curves) profile and spectrum is shown along with a fitted pulse solution Eq. (6) with ( η f , B f , A f ) (dashed curves). The inset shows the Gaussian filter (shaded) relative to the prefiltered pulse spectrum. The values of the fixed point and resultant filtered point are shown in Table 1.

Fig. 5
Fig. 5

(a) Postfiltered values of η, B, and A versus Γ. The horizontal dotted lines correspond to the pulse parameter values at the fixed point. (b) Phase–plane dynamics of a laser configuration with the parameters D = 0.4 , τ = 0.2 , δ = 1 , g 0 = 3 , e 0 = 1 , β = 0.5 , and μ = 0.1 . Depicted is the jump condition from the fixed-point (solid circle) to the postfiltered pulse (gray square, circle, and triangle) followed by the evolution back to the fixed point. The gray circle, square, and triangle in (b) correspond to Γ = 0.87 , 0.62, and 0.5 in (a).

Fig. 6
Fig. 6

(a) Illustration of the laser cavity with discrete filter and four labeled positions in the cavity. (b) The intracavity mode-locked evolution in the experimentally relevant variables along with the action of the spectral filtering (dotted line). (c) The output spectral profiles at the labeled intracavity positions of (a) and (b). This prototypical pulse evolution is characteristic of the all-normal dispersion fiber laser [8]. The Gaussian spectral filter (shaded) is shown in the prefiltered and postfiltered positions 4 and 1, respectively. Note that high-energy, high peak intensity pulses can be obtained if the output coupler is placed at position 3.

Fig. 7
Fig. 7

Geometric representation of the mode-locking operation. The jump condition associated with spectral filtering on the parameters is demonstrated with the gray circles [prefiltering (−) and postfiltering (+)]. The output coupler (OC) is placed at the optimal location where the energy and peak intensity is maximized. The inset shows the temporal profile at the output coupler (dashed curve) and the fixed point (solid curve). Note the significant increase in intensity and pulse energy.

Fig. 8
Fig. 8

Intracavity mode-locking dynamics with Γ = 0.5 and g 0 = 1.5 , 1.9, 2.5, and 3.0 (circle, square, triangle, star). The evolution lines are strongly effected by the gain level and can be used to understand the laser performance as a function of the gain parameter g 0 .

Fig. 9
Fig. 9

Laser performance versus gain parameter g 0 . (a) Pulse duration, (b) breathing ratio, (c) pulse energy at fixed point, and (d) energy ratio increase (maximum energy divided by the fixed point energy).

Fig. 10
Fig. 10

Intracavity mode-locking dynamics with g 0 = 3 . The evolution lines are strongly effected by the filter-to-pulse bandwidth ratio: Γ = 0.87 , 0.62, and 0.5 (circle, square, triangle).

Fig. 11
Fig. 11

Laser performance versus the filter-to-pulse bandwidth ratio parameter Γ: (a) breathing ratio and (b) relative energy increase.

Tables (1)

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Table 1 Resultant Filtered Solution Parameters Corresponding to the Solution Ansatz Eq. (6) for Specified Bandwidth Ratios Γ and Fixed Point ( η 0 , B 0 , A 0 ) = ( 1 , 0.5 , 3.3 ) .

Equations (18)

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i u Z + D 2 2 u T 2 + ( 1 i β ) u 2 u + i μ u 4 u + i δ u i g ( Z ) ( 1 + τ 2 T 2 ) u = 0 ,
g ( Z ) = 2 g 0 1 + u 2 e 0 ,
u ( Z , T ) = η B + cosh ( ω T ) e i [ A ln ( B + cosh ( ω T ) ) + φ Z ] .
δ δ p j ( L [ u s ] d T ) = 2 R { i R [ u s ] u s * p j d T } .
L = i 2 ( u u Z * u * u Z ) D 2 u T 2 + 1 2 u 4 ,
R = τ g ( Z ) 2 u T 2 + ( g ( Z ) δ ) u + β u 2 u μ u 4 u .
u ( Z , T ) = η ( Z ) B ( Z ) + cosh ( η ( Z ) T ) e i α ( Z , T ) ,
D x = g ,
D = [ H η H 0 0 ( H G ) η ( G + H ) 0 0 0 A H 1 2 ( G H ) H A H 0 1 2 η ( G + H ) η H ] .
g 1 = 2 η ( g δ ) H 2 β η 3 H + μ η 5 H 1 2 τ g η 3 ( A 2 + 1 ) R ,
g 2 = D 2 η 3 A R 2 η ( g δ ) G 3 β η 3 S 2 μ η 5 Z + 1 2 τ g η 3 [ 3 W 2 Y W A 2 ] ,
g 3 = 3 D 4 η 2 ( 1 + A 2 ) R + 3 2 η 2 H 2 A ( g δ ) H + β η 2 A H 1 3 μ η 4 A H + 1 2 τ g η 2 A ( 1 + A 2 ) Q ,
g 4 = D 4 η 3 ( 1 + A 2 ) R + 1 2 η 3 H 2 η A ( g δ ) H β η 3 A H + 1 3 μ η 5 A H 1 12 τ g η 3 A ( 1 + A 2 ) R .
H = d t ϴ , G = ln ϴ d t ϴ , Q = t sinh 3 t d t ϴ 4 ,
R = sinh 2 t d t ϴ 2 , S = ln ϴ d t ϴ 2 ,
W = sinh 2 t ln ϴ d t ϴ 3 ,
Y = cosh t ln ϴ d t ϴ 2 , Z = ln ϴ d t ϴ 3 ,
η 2 u 0 2 = 0.36 W [ D ] [ T 0 ] 2 40 W ,

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