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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  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).
  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)
  3. H. A. Haus and E. P. Ippen, "Self-starting of passively mode-locked lasers", Opt. Lett. 16, 1331 (1991)
  4. F. Krausz, T. Brabec and Ch. Spielmann, "Self-starting passive mode locking", Opt. Lett. 16, 235 (1991)
  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)
  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)
  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)
  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)
  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
    [CrossRef]
  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)
  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]
  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).
  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]
  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]
  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]
  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).
  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]

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]

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]

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).

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]

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)

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)

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

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).

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)

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

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]

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]

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]

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]

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]

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).

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]

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]

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]

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]

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]

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]

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]

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]

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).

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]

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]

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
[CrossRef]

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
[CrossRef]

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)

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

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]

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
[CrossRef]

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]

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]

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).

Opt. Lett. (9)

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
[CrossRef]

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]

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

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).

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).

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