Abstract

A purely analytical model treating an average cavity description of a passive self-similar optical resonator is developed. The model introduces an energy area theorem which provides an analytical framework for understanding energy scalability in passive self-similar systems. A qualitative link between self-similar pulses in fiber and solid state systems is explained by the analytical model through the nature of the saturable characteristics of the saturable absorber. The derived expressions for the field amplitude and chirp parameter of the pulse match well with simulation and offer insight into the mechanics of passive self-similar resonator design.

© 2014 Optical Society of America

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  1. W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
    [CrossRef]
  2. F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
    [CrossRef]
  3. 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]
  4. S. Namiki and H. Haus, “Noise of the stretched pulse fiber laser. I. Theory,” IEEE J. Quantum Electron. 33, 649–659 (1997).
    [CrossRef]
  5. F. O. 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]
  6. V. G. Bucklew and C. R. Pollock, “Realizing self-similar pulses in solid-state laser systems,” J. Opt. Soc. Am. B 29, 3027–3033 (2012).
    [CrossRef]
  7. F. Ilday, F. Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express 12, 2731–2738 (2004).
    [CrossRef]
  8. T. Schreiber, B. Ortaç, J. Limpert, and A. Tünnermann, “On the study of pulse evolution in ultra-short pulse mode-locked fiber lasers by numerical simulations,” Opt. Express 15, 8252–8262 (2007).
    [CrossRef]
  9. C. Jirauschek and F. O. Ilday, “Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse,” Phys. Rev. A 83, 063809 (2011).
    [CrossRef]
  10. C. Antonelli, J. Chen, and F. X. Kartner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919–5924 (2007).
    [CrossRef]
  11. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  12. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media (differential equation solution for plane self focusing and one dimensional self modulation of waves interacting in nonlinear media),” Sov. Phys. 34, 62–69 (1972).
  13. W. H. Renninger, A. Chong, and F. W. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  14. 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]
  15. V. Kalashnikov, E. Podivilov, A. Chernykh, and A. Apolonski, “Chirped-pulse oscillators: theory and experiment,” Appl. Phys. B 83, 503–510 (2006).
    [CrossRef]
  16. V. L. Kalashnikov and A. Apolonski, “Chirped-pulse oscillators: a unified standpoint,” Phys. Rev. A 79, 043829 (2009).
    [CrossRef]
  17. V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express 18, 25757–25770 (2010).
    [CrossRef]
  18. V. L. Kalashnikov, Solid State Laser (In Tech, 2012).
  19. B. Oktem, C. Ulgudur, and F. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics 4, 307–311 (2010).
    [CrossRef]
  20. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010).
    [CrossRef]
  21. C. Aguergaray, D. Méchin, V. Kruglov, and J. D. Harvey, “Experimental realization of a mode-locked parabolic Raman fiber oscillator,” Opt. Express 18, 8680–8687 (2010).
    [CrossRef]
  22. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
    [CrossRef]
  23. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).
    [CrossRef]
  24. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
    [CrossRef]
  25. W. H. Renninger, A. Chong, and F. W. Wise, “Area theorem and energy quantization for dissipative optical solitons,” J. Opt. Soc. Am. B 27, 1978–1982 (2010).
    [CrossRef]
  26. F. O. Ilday, J. R. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a fiber laser,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2004), paper. MD8.
  27. J. R. 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]
  28. F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
    [CrossRef]

2012 (2)

W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
[CrossRef]

V. G. Bucklew and C. R. Pollock, “Realizing self-similar pulses in solid-state laser systems,” J. Opt. Soc. Am. B 29, 3027–3033 (2012).
[CrossRef]

2011 (1)

C. Jirauschek and F. O. Ilday, “Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse,” Phys. Rev. A 83, 063809 (2011).
[CrossRef]

2010 (5)

2009 (1)

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

2008 (3)

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (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]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[CrossRef]

2007 (2)

2006 (1)

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

2005 (2)

J. R. 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]

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]

2004 (2)

F. Ilday, F. Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express 12, 2731–2738 (2004).
[CrossRef]

F. O. 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]

2001 (1)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

1997 (1)

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

1993 (1)

1991 (1)

1973 (1)

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

1972 (1)

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

Aguergaray, C.

Akhmediev, N.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Anderson, D.

Antonelli, C.

Apolonski, A.

V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express 18, 25757–25770 (2010).
[CrossRef]

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

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

Bucklew, V. G.

Buckley, J. R.

J. R. 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]

F. O. 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]

F. O. Ilday, J. R. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a fiber laser,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2004), paper. MD8.

Chen, J.

Chernykh, A.

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

Chong, A.

W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Area theorem and energy quantization for dissipative optical solitons,” J. Opt. Soc. Am. B 27, 1978–1982 (2010).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (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]

Clark, W. G.

F. O. 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]

Desaix, M.

Dudley, J. M.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

Fernandez, 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]

Fujimoto, J. G.

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]

Harvey, J. D.

C. Aguergaray, D. Méchin, V. Kruglov, and J. D. Harvey, “Experimental realization of a mode-locked parabolic Raman fiber oscillator,” Opt. Express 18, 8680–8687 (2010).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

Hasegawa, A.

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

Haus, H.

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

Haus, H. A.

Ilday, F.

B. Oktem, C. Ulgudur, and F. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics 4, 307–311 (2010).
[CrossRef]

F. Ilday, F. Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express 12, 2731–2738 (2004).
[CrossRef]

Ilday, F. O.

C. Jirauschek and F. O. Ilday, “Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse,” Phys. Rev. A 83, 063809 (2011).
[CrossRef]

J. R. 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]

F. O. 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]

F. O. Ilday, J. R. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a fiber laser,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2004), paper. MD8.

Ippen, E. P.

Jirauschek, C.

C. Jirauschek and F. O. Ilday, “Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse,” Phys. Rev. A 83, 063809 (2011).
[CrossRef]

Kaertner, F.

Kalashnikov, V.

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

Kalashnikov, V. L.

V. L. Kalashnikov and A. Apolonski, “Energy scalability of mode-locked oscillators: a completely analytical approach to analysis,” Opt. Express 18, 25757–25770 (2010).
[CrossRef]

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

V. L. Kalashnikov, Solid State Laser (In Tech, 2012).

Karlsson, M.

Kartner, F. X.

Kruglov, V.

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

Limpert, J.

Lisak, M.

Méchin, D.

Namiki, S.

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

Oktem, B.

B. Oktem, C. Ulgudur, and F. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics 4, 307–311 (2010).
[CrossRef]

Ortaç, B.

Podivilov, E.

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

Pollock, C. R.

Quiroga-Teixeiro, M. L.

Renninger, W.

W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Area theorem and energy quantization for dissipative optical solitons,” J. Opt. Soc. Am. B 27, 1978–1982 (2010).
[CrossRef]

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

Schreiber, T.

Shabat, A. B.

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

Sosnowski, T.

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Tappert, F.

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

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Tünnermann, A.

Ulgudur, C.

B. Oktem, C. Ulgudur, and F. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics 4, 307–311 (2010).
[CrossRef]

Wise, F.

W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[CrossRef]

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

F. Ilday, F. Wise, and F. Kaertner, “Possibility of self-similar pulse evolution in a Ti:sapphire laser,” Opt. Express 12, 2731–2738 (2004).
[CrossRef]

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Area theorem and energy quantization for dissipative optical solitons,” J. Opt. Soc. Am. B 27, 1978–1982 (2010).
[CrossRef]

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

J. R. 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]

F. O. 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]

F. O. Ilday, J. R. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a fiber laser,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2004), paper. MD8.

Zakharov, V. E.

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

Appl. Phys. B (1)

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

Appl. Phys. Lett. (1)

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

IEEE J. Quantum Electron. (1)

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

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

W. Renninger, A. Chong, and F. Wise, “Pulse shaping and evolution in normal-dispersion mode-locked fiber lasers,” IEEE J. Sel. Top. Quantum Electron. 18, 389–398 (2012).
[CrossRef]

J. Opt. Soc. Am. B (4)

Laser Photon. Rev. (1)

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photon. Rev. 2, 58–73 (2008).
[CrossRef]

Laser Photonics Rev. (1)

F. Wise, A. Chong, and W. Renninger, “High-energy femtosecond fiber lasers based on pulse propagation at normal dispersion,” Laser Photonics Rev. 2, 58–73 (2008).
[CrossRef]

Nat. Photonics (1)

B. Oktem, C. Ulgudur, and F. Ilday, “Soliton-similariton fibre laser,” Nat. Photonics 4, 307–311 (2010).
[CrossRef]

New J. Phys. (1)

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 (5)

Opt. Lett. (1)

Phys. Rev. A (4)

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805 (2010).
[CrossRef]

C. Jirauschek and F. O. Ilday, “Semianalytic theory of self-similar optical propagation and mode locking using a shape-adaptive model pulse,” Phys. Rev. A 83, 063809 (2011).
[CrossRef]

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

Phys. Rev. E (1)

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locked lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[CrossRef]

Phys. Rev. Lett. (2)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010–6013 (2000).
[CrossRef]

F. O. 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]

Sov. Phys. (1)

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

Other (2)

V. L. Kalashnikov, Solid State Laser (In Tech, 2012).

F. O. Ilday, J. R. Buckley, and F. W. Wise, “Self-similar evolution of parabolic pulses in a fiber laser,” in Nonlinear Guided Waves and Their Applications (Optical Society of America, 2004), paper. MD8.

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Figures (4)

Fig. 1.
Fig. 1.

Left: temporal profile of a passive self-similar pulse in a Ti:sapphire resonator. Right: the spectral intensity of the self-similar pulse (black line) is shown alongside the frequency-dependent percent transmission of the Ti:sapphire gain medium (red line).

Fig. 2.
Fig. 2.

Relationship between the energy E and temporal width T of passive self-similar pulses, expressed as E×Tn=C0, is simulated (blue circles), matched to a best-fit energy area theorem with value n (blue line), and compared to the well-known area theorems of the DM soliton (n=3), the NLSE soliton (n=1), and the self-similar model of this paper (n=3). Left: self-similar solid state laser. Right: self-similar fiber laser.

Fig. 3.
Fig. 3.

Left: normalized transmission of the SA as a function of normalized power (P/Ppeak). Middle: logarithmic temporal profile of the pulse. Right: spectral profile of the pulse. Dashed red line, simulated parabolic profile; solid red line, analytic parabolic profile; dashed blue line, simulated chirped-soliton profile; solid blue line, analytical chirped-soliton profile.

Fig. 4.
Fig. 4.

Left: the normalized simulated (dashed line) and analytic (solid line) transmission profiles of the SA required to produce self-similar pulses in solid state (red) and fiber (blue) systems are shown alongside each other. The “loss” states the integrated loss which the simulated, normalized transmission profiles introduce to the pulse. “Ppeak/Psat” states the number of times above saturation the absorber in the numerical simulation were. Right: the analytic chirp parameter α is compared to the chirp parameter from simulation. Solid red line, simulation; solid blue line, analytic.

Tables (1)

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Table 1. Area Theorems

Equations (12)

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A(z,t)z=[g0(z)1+EpEsat(1+1ωc22t2)i2β(z)2t2+iγ(z)|A(z,t)|2Qideal(|A(z,t)|2)l(z)]A(z,t).
iA(z,t)z=[β22t2γ|A(z,t)|2ib1|A(z,t)|2c1|A(z,t)|2+d1+i(gl)]A(z,t),
(2d1t2α2β+d1η)f(t)+(η2t2α2βd1γ)f(t)3γf(t)5+12d1βf(t)+12βf(t)2f(t)=0,
(ib1c1+id1gid1l+id1αβ)f(t)+(igib1il+iαβ)f(t)3+2id1tαβf(t)+2itαβf(t)2f(t)=0,
f(t)=η2t2α2βγ.
b1=gl+3αβ,
c1=2αβη(gl+3αβ)γ,
d1=0.
α=gllnsβ.
b1|A(z,t)|2c1|A(z,t)|2+d1=2(gllns)(ηγ|A(t)|2)γ|A(t)|2+lns,
A(z,t)=A1t2T2exp(iαt2+iηz),
ET3=4(gllns)2βγ.

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