Abstract

It has been recently understood that mode locking of lasers has the signification of a thermodynamic phase transition in a system of many interacting light modes subject to noise. In the same framework, self starting of passive mode locking has the thermodynamic significance of a noise-activated escape process across an entropic barrier. Here we present the first dynamical study of the light mode system. While accordant with the predictions of some earlier theories, it is the first to give precise quantitative predictions for the distribution of self-start times, in closed form expressions, resolving the long standing self starting problem. Numerical simulations corroborate these results, which are also in good agreement with experiments.

© 2006 Optical Society of America

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References

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  1. E. P. Ippen, L. Y. Liu, and H. A. Haus, "Self-starting condition for additive-pulse mode-locked lasers", Opt. Lett. 15, 183 (1990).
    [CrossRef] [PubMed]
  2. C. J. Chen, P. K. A. Wai and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers", Opt. Lett. 20, 350 (1995)
    [CrossRef] [PubMed]
  3. H. A. Haus and E. P. Ippen, "Self-starting of passively mode-locked lasers", Opt. Lett. 16, 1331 (1991)
    [CrossRef] [PubMed]
  4. F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
    [CrossRef] [PubMed]
  5. K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, "Unidirectional ring resonators for selfstarting passively mode-locked lasers", Opt. Lett. 18, 220 (1993)
    [CrossRef] [PubMed]
  6. Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, "Measurements of the self-starting threshold of Kerr-lens modelocking lasers", Opt. Lett. 19, 566 (1994)
    [CrossRef] [PubMed]
  7. Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
    [CrossRef]
  8. F. Krausz and T. Brabec, "Passive mode locking in standing-wave laser resonators", Opt. Lett. 18, 888 (1993)
    [CrossRef] [PubMed]
  9. Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, "Measurements of the self-starting threshold of Kerr-lens modelocking lasers", Opt. Lett. 19, 566 (1994)
    [CrossRef] [PubMed]
  10. J. Hermann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode locked solid-state lasers", Opt. Comm. 98, 111 (1993).
    [CrossRef]
  11. A. K. Komarov, K. P. Komarov and F. M. Mitschke, "Phase-modulation bistability and threshold self-start of laser passive mode locking", Phys. Rev. A. 65, 053803
  12. J. M. Soto-Crespo, N. Akhmediev and G. Town, "Continuous-wave versus pulse regime in a passively modelocked laser with a fast saturable absorber", J. Opt. Soc. Am. B 19, 234 (2002)
    [CrossRef]
  13. H. A. Haus, "Theory of mode locking with a fast saturable absorber", J. Appl. Phys.,  46, 3049 (1975).
    [CrossRef]
  14. H. A. Haus, "Mode-Locking of Lasers", IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000)
    [CrossRef]
  15. A. Gordon and B. Fischer, "Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers", Phys. Rev. Lett. 89, 103901, (2002)
    [CrossRef] [PubMed]
  16. A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity" Opt. Commun. 223, 151 (2003).
    [CrossRef]
  17. A. Gordon and B. Fischer, "Inhibition of modulation instability in lasers by noise", Opt. Lett 18, 1326 (2003).
    [CrossRef]
  18. O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
    [CrossRef]
  19. R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
    [CrossRef] [PubMed]
  20. O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
    [CrossRef]
  21. H. A. Kramers, "Brownian motion in a field of fource and the diffusion model of chemical reactions", Physica (Utrecht) 7, 284 (1940).
    [CrossRef]
  22. P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
    [CrossRef]
  23. B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
    [CrossRef] [PubMed]
  24. M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
    [CrossRef] [PubMed]
  25. R. L. Stratonovich, "Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems", in "Noise in nonlinear dynamical systems, Vol. 1, edited by F. Moss and P. V. E. McClintock, Cambridge University Press (1989)
  26. B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat and Baruch Fischer "Formation and Annihilation of Laser Pulse Quanta in a Thermodynamic-like Pathway", Phys. Rev. Lett. 93,153901 (2004).
    [CrossRef] [PubMed]
  27. H. H. Risken, "The Fokker-Planck Equation", Second edition, Springler-Verlag (1989, 1996).
  28. C. W. GardinerHandbook of Stochastic Methods, 3rd ed., Springer, New York (2004).
  29. P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer New York (2000).
  30. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, p. 281., 2nd edition, Cambridge University Press, New York (1992).

2006 (1)

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

2005 (3)

O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
[CrossRef] [PubMed]

2004 (1)

O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
[CrossRef]

2003 (2)

A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity" Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, "Inhibition of modulation instability in lasers by noise", Opt. Lett 18, 1326 (2003).
[CrossRef]

2002 (2)

A. Gordon and B. Fischer, "Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers", Phys. Rev. Lett. 89, 103901, (2002)
[CrossRef] [PubMed]

J. M. Soto-Crespo, N. Akhmediev and G. Town, "Continuous-wave versus pulse regime in a passively modelocked laser with a fast saturable absorber", J. Opt. Soc. Am. B 19, 234 (2002)
[CrossRef]

2000 (1)

H. A. Haus, "Mode-Locking of Lasers", IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000)
[CrossRef]

1995 (1)

1994 (2)

1993 (3)

1991 (3)

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
[CrossRef] [PubMed]

H. A. Haus and E. P. Ippen, "Self-starting of passively mode-locked lasers", Opt. Lett. 16, 1331 (1991)
[CrossRef] [PubMed]

1990 (2)

E. P. Ippen, L. Y. Liu, and H. A. Haus, "Self-starting condition for additive-pulse mode-locked lasers", Opt. Lett. 15, 183 (1990).
[CrossRef] [PubMed]

P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
[CrossRef]

1975 (1)

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

1940 (1)

H. A. Kramers, "Brownian motion in a field of fource and the diffusion model of chemical reactions", Physica (Utrecht) 7, 284 (1940).
[CrossRef]

Borkovec, M.

P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
[CrossRef]

Hänggi, P.

P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
[CrossRef]

Talkner, , P.

P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
[CrossRef]

Akhmediev, N.

Bekker, A.

Berger, N. K.

Brabec, T.

F. Krausz and T. Brabec, "Passive mode locking in standing-wave laser resonators", Opt. Lett. 18, 888 (1993)
[CrossRef] [PubMed]

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
[CrossRef] [PubMed]

Chen, C. J.

Chou, Y.-F.

Fischer, B.

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
[CrossRef]

O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
[CrossRef]

A. Gordon and B. Fischer, "Inhibition of modulation instability in lasers by noise", Opt. Lett 18, 1326 (2003).
[CrossRef]

A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity" Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, "Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers", Phys. Rev. Lett. 89, 103901, (2002)
[CrossRef] [PubMed]

Fujimoto, J. G.

Gat, O.

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
[CrossRef] [PubMed]

O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
[CrossRef]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
[CrossRef]

Gordon, A.

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
[CrossRef]

O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
[CrossRef]

A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity" Opt. Commun. 223, 151 (2003).
[CrossRef]

A. Gordon and B. Fischer, "Inhibition of modulation instability in lasers by noise", Opt. Lett 18, 1326 (2003).
[CrossRef]

A. Gordon and B. Fischer, "Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers", Phys. Rev. Lett. 89, 103901, (2002)
[CrossRef] [PubMed]

Haus, H. A.

Hermann, J.

J. Hermann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode locked solid-state lasers", Opt. Comm. 98, 111 (1993).
[CrossRef]

Ippen, E. P.

Jacobson, J.

Katz, M.

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

Komarov, A. K.

A. K. Komarov, K. P. Komarov and F. M. Mitschke, "Phase-modulation bistability and threshold self-start of laser passive mode locking", Phys. Rev. A. 65, 053803

Komarov, K. P.

A. K. Komarov, K. P. Komarov and F. M. Mitschke, "Phase-modulation bistability and threshold self-start of laser passive mode locking", Phys. Rev. A. 65, 053803

Kramers, H. A.

H. A. Kramers, "Brownian motion in a field of fource and the diffusion model of chemical reactions", Physica (Utrecht) 7, 284 (1940).
[CrossRef]

Krausz, F.

F. Krausz and T. Brabec, "Passive mode locking in standing-wave laser resonators", Opt. Lett. 18, 888 (1993)
[CrossRef] [PubMed]

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
[CrossRef] [PubMed]

Kuo, N.-P.

Liu, H.-H.

Liu, L. Y.

Menyuk, C. R.

Mitschke, F. M.

A. K. Komarov, K. P. Komarov and F. M. Mitschke, "Phase-modulation bistability and threshold self-start of laser passive mode locking", Phys. Rev. A. 65, 053803

Rosen, A.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Schmidt, A. J.

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

Smulakovsky, V.

Soto-Crespo, J. M.

Spielman, Ch.

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

Spielmann, Ch.

Tamura, K.

Town, G.

Vodonos, B.

Wai, P. K. A.

Wang, J.

Weill, R.

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Wintner, E.

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

IEEE J. Quantum Electron. (1)

Ch. Spielman, F. Krausz, T. Brabec, E. Wintner and A. J. Schmidt, "Experimental study of additive-pulse mode locking in an Nd:Glass laser", IEEE J. Quantum Electron. 27, 1207 (1991)
[CrossRef]

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

H. A. Haus, "Mode-Locking of Lasers", IEEE J. Sel. Top. Quantum Electron. 6, 1173 (2000)
[CrossRef]

J. Appl. Phys. (1)

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

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

New J. Phys. (1)

O. Gat, A. Gordon and B. Fischer, "Light-mode locking - A new class of solvable statistical physics systems", New J. Phys. 7, 151 (2005)
[CrossRef]

Opt. Comm. (1)

J. Hermann, "Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode locked solid-state lasers", Opt. Comm. 98, 111 (1993).
[CrossRef]

Opt. Commun. (1)

A. Gordon and B. Fischer, Phase transition theory of pulse formation in passively mode-locked lasers with dispersion and Kerr nonlinearity" Opt. Commun. 223, 151 (2003).
[CrossRef]

Opt. Lett (1)

A. Gordon and B. Fischer, "Inhibition of modulation instability in lasers by noise", Opt. Lett 18, 1326 (2003).
[CrossRef]

Opt. Lett. (9)

E. P. Ippen, L. Y. Liu, and H. A. Haus, "Self-starting condition for additive-pulse mode-locked lasers", Opt. Lett. 15, 183 (1990).
[CrossRef] [PubMed]

F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
[CrossRef] [PubMed]

H. A. Haus and E. P. Ippen, "Self-starting of passively mode-locked lasers", Opt. Lett. 16, 1331 (1991)
[CrossRef] [PubMed]

K. Tamura, J. Jacobson, E. P. Ippen, H. A. Haus, and J. G. Fujimoto, "Unidirectional ring resonators for selfstarting passively mode-locked lasers", Opt. Lett. 18, 220 (1993)
[CrossRef] [PubMed]

F. Krausz and T. Brabec, "Passive mode locking in standing-wave laser resonators", Opt. Lett. 18, 888 (1993)
[CrossRef] [PubMed]

Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, "Measurements of the self-starting threshold of Kerr-lens modelocking lasers", Opt. Lett. 19, 566 (1994)
[CrossRef] [PubMed]

Y.-F. Chou, J. Wang, H.-H. Liu, and N.-P. Kuo, "Measurements of the self-starting threshold of Kerr-lens modelocking lasers", Opt. Lett. 19, 566 (1994)
[CrossRef] [PubMed]

C. J. Chen, P. K. A. Wai and C. R. Menyuk, "Self-starting of passively mode-locked lasers with fast saturable absorbers", Opt. Lett. 20, 350 (1995)
[CrossRef] [PubMed]

B. Vodonos, A. Bekker, V. Smulakovsky, A. Gordon, O. Gat, N. K. Berger and B. Fischer "Experimental study of the stochastic nature of the pulsation self-starting process in passive mode-locking," Opt. Lett. 30, 2787 (2005).
[CrossRef] [PubMed]

Phys. Rev. A. (1)

A. K. Komarov, K. P. Komarov and F. M. Mitschke, "Phase-modulation bistability and threshold self-start of laser passive mode locking", Phys. Rev. A. 65, 053803

Phys. Rev. E. (1)

O. Gat, A. Gordon and B. Fischer, "Solution of a statistical mechanics model for pulse formation in lasers", Phys. Rev. E. 70, 046108 (2004)
[CrossRef]

Phys. Rev. Lett. (3)

M. Katz, A. Gordon, O. Gat, and B. Fischer, "Non-Gibbsian Stochastic Light-Mode Dynamics of Passive Mode Locking", Phys. Rev. Lett. 97, 113902 (2006)
[CrossRef] [PubMed]

A. Gordon and B. Fischer, "Phase Transition Theory of Many-Mode Ordering and Pulse Formation in Lasers", Phys. Rev. Lett. 89, 103901, (2002)
[CrossRef] [PubMed]

R. Weill, A. Rosen, A. Gordon, O. Gat, and B. Fischer, "Critical Behavior of Light in Mode-Locked Lasers," Phys. Rev. Lett. 95, 013903 (2005).
[CrossRef] [PubMed]

Physica (Utrecht) (1)

H. A. Kramers, "Brownian motion in a field of fource and the diffusion model of chemical reactions", Physica (Utrecht) 7, 284 (1940).
[CrossRef]

Rev. Mod. Phys. (1)

P.  Hänggi, P.  Talkner and M.  Borkovec, "Reaction-rate theory: fifty years after Kramers," Rev. Mod. Phys. 62, 251 (1990).
[CrossRef]

Other (6)

R. L. Stratonovich, "Some Markov methods in the theory of stochastic processes in nonlinear dynamical systems", in "Noise in nonlinear dynamical systems, Vol. 1, edited by F. Moss and P. V. E. McClintock, Cambridge University Press (1989)

B. Vodonos, R. Weill, A. Gordon, A. Bekker, V. Smulakovsky, O. Gat and Baruch Fischer "Formation and Annihilation of Laser Pulse Quanta in a Thermodynamic-like Pathway", Phys. Rev. Lett. 93,153901 (2004).
[CrossRef] [PubMed]

H. H. Risken, "The Fokker-Planck Equation", Second edition, Springler-Verlag (1989, 1996).

C. W. GardinerHandbook of Stochastic Methods, 3rd ed., Springer, New York (2004).

P. E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer New York (2000).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical recipes in C: The Art of Scientific Computing, p. 281., 2nd edition, Cambridge University Press, New York (1992).

Supplementary Material (1)

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Figures (6)

Fig. 1.
Fig. 1.

Right: A recording of the time evolution of a laser waveform envelope under the action of a saturable absorber, obtained from a numerical simulation as described before. Most of the time is spent in a quiescent quasi-cw configuation of the laser, until a rare noise fluctuation quickly drives the system across the entropic barrier. Left: The horizontal position of the red dot shows the time-dependent pulse power in the simulation, and the curve shows the free energy function, which is the potential in the effective one-degree of freedom dynamics described below (s9.avi–1.9MB).

Fig. 2.
Fig. 2.

The mean field free energy function (Eq. 6) and its extrema, which are also approximately the fixed points of the deterministic part of the self-start dynamics Eq. (12). The minimum at ξc =0 corresponds to cw, the one at ξp corresponds to the pulsed state, and the maximum at ξb corresponds to the saddle of the barrier separating the cw and pulsed states.

Fig. 3.
Fig. 3.

A graphical comparison between the mean self-start times calculated by the asymoptotic formulas Eq. (13) (full line) and Eq. (14) (dashed line). The latter approaches the former in the lower left part of the figure. Note that rather large values of N are required to reach the region of validity of Eq. (14).

Fig. 4.
Fig. 4.

Left: An experimental histogram of the measured self-start times in an additive-pulse passively mode locked fiber laser [23], shown on a logarithmic scale. An exponential tail is clearly observed, as expected for a noise-activated escape process. Right: The mean cw lifetime dependence on the intracavity power P in a set of self-starting experiments. The results are for two sets of measurements (shown in circles and triangles). The estimated mean cw lifetime is shown on logarithmic scale versus 1/P 2. The results are compatible with the predictions following from the Arrhenius formula as discussed in the text.

Fig. 5.
Fig. 5.

A histogram of the numerically measured self-start times in logarithmic (top) and linear (bottom) scales, in a simulation performed with N=600, γ=20. The distribution is very well described by an exponential, as expected for a noise activated barrier crossing.

Fig. 6.
Fig. 6.

A graphical comparison between the mean self-start times calculated by the asymoptotic formula Eq. (13) shown in black lines labeled by the number N of degrees of freedom, and full numerical simulation, shown with red crosses. The discrepancy between the analytically and numerically calculated lifetimes is less than 10%, and is within the statistical error.

Equations (19)

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ψ ˙ m = g ( t ) ψ m + γ s ψ m 2 ψ m + Γ m ( t ) ,
Γ m ( t ) | Γ n * ( t ) = 2 W δ nm δ ( t t )
Γ m ( t ) Γ n ( t ) = 0 .
𝒫 = 1 N j = 1 N ψ j 2 .
g = 1 N 𝒫 { γ s j = 1 N ψ j 4 + R e [ j = 1 N ψ j * Γ j ] } ,
ρ ( ψ 1 , . . . , ψ N ) δ ( 𝒫 P 0 ) exp ( γ s W j ψ j 4 ) .
F ( ξ ) = 1 2 ξ 2 γ ln ( γ ξ ) ,
t esc ~ t dyn e Δ E T
1 2 p . m = γ s p m 2 ( 1 p m NP 0 ) W P 0 p m + W γ s NP 0 ( j m p j 2 ) p m + η m ( t ) .
η m ( t ) η m ( t ) = Wp m ( 1 p m NP 0 ) δ ( t t ) ,
ξ ( τ ) = ξ 2 ξ ξ 3 γ + ε ( 1 1 NP 0 2 ( j m p j 2 ) ξ ) + εξ ( 1 ξ γ ) η ( τ ) ,
Q = j m p m 2 ,
1 2 ξ ( τ ) = ξ 2 ξ ξ 3 γ + ε ( 1 2 ( 1 ξ γ ) 2 ξ ) + ε ξ ( 1 ξ γ ) η ( τ ) ,
t cw = 1 γ s P 0 e 2 ξ b ξ b 2 γ γ ξ b γ 2 ξ b π 2 F ( ξ b ) ε 5 2 e F ( ξ b ) ε ,
t cw ~ 1 γ s P 0 e 2 π 2 1 γ ε 5 2 e 1 2 ε .
t cw ~ 1 γ s P 0 e 2 π 2 ( γ P 0 2 T ) 3 2 1 N 5 2 e NT 2 γ s P 0 2
[ ξ 2 ξ ξ 3 γ + ε ( 1 2 ( 1 ξ γ ) 2 ξ ) ] τ ξ ¯ ( ξ ) ξ + ε ξ ( 1 ξ γ ) 2 τ ξ ¯ ( ξ ) ξ 2 = 1 2 ,
τ esc = 1 2 ε 0 ξ d ξ 0 ξ γ ( γ ξ ) ξ ( γ ξ ) 2 exp ( 1 ε F ( ξ ) F ( ξ ) + 2 ( ξ ξ ) ξ 2 ( ξ ) 2 γ ) d ξ ,
τ esc = e 2 ξ b ξ b 2 γ γ ξ b γ ξ b π 2 F ( ξ b ) ε 1 2 e 1 ε F ( ξ ) ( 1 + ο ( 1 ) ) .

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