Abstract

An analytic theory is proposed that characterizes Q switching in an active mode-locked cavity as the nonlinear interaction of two unstable modes: one symmetric, another antisymmetric. The phase difference between these modes generates a nonlinear beating interaction that gives rise to quasi-periodic behavior in the laser cavity. This quasi-periodic behavior is responsible for the Q-switching phenomenon and is controlled by the interaction and overlap between neighboring pulses. With a linear stability analysis, a simple qualitative description of the Q-switching phenomenon is given that is verified with numerical simulations of the governing active mode-locked equations. This model characterizes the Q switching as a function of the physical parameters of the laser cavity and elucidates the mechanisms for controlling its behavior.

© 2006 Optical Society of America

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    [CrossRef]
  2. F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
    [CrossRef]
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    [CrossRef]
  4. H. A. Haus, E. P. Ippen, and K. Tamura, "Additive-pulse mode-locking in fiber lasers," IEEE J. Quantum Electron. 30, 200-208 (1994).
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  5. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
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  6. J. J. O'Neil, J. N. Kutz, and B. Sandstede, "Theory and simulation of the dynamics and stability of actively mode-locked lasers," IEEE J. Quantum Electron. 38, 1412-1419 (2002).
    [CrossRef]
  7. G. L. Eesley, "Generation of nonequilibrium electron and lattice temperature in copper by picosecond laser pulses," Phys. Rev. B 33, 2144-2151 (1986).
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  12. L. da Silva, M. Perry, M. Feit, and B. Stuart, "The short-pulse laser: a safe, painless surgical tool," Sci. Technol. Rev. September 1995, pp. 29-31.
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  14. K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
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  18. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
    [CrossRef]
  19. M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
    [CrossRef]
  20. F. X. Kärtner and U. Keller, "Stabilization of solitonlike pulses with a slow saturable absorber," Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  21. B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
    [CrossRef]
  22. S. Tsuda, W. H. Knox, E. A. DeSouza, W. J. Jan, and J. E. Cunningham, "Low-loss intracavity AlAs/AlGaAs saturable Bragg reflector for femtosecond mode locking in solid-state lasers," Opt. Lett. 20, 1406-1408 (1995).
    [CrossRef] [PubMed]
  23. F. X. Kärtner, D. Kopf, and U. Keller, "Solitary pulse stabilization and shortening in actively mode-locked lasers," J. Opt. Soc. Am. B 12, 486-496 (1995).
    [CrossRef]
  24. H. A. Haus, "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-330 (1975).
    [CrossRef]
  25. T. Kolokolnikov, T. Erneux, N. Joly, and S. Bielawski, "The Q-switching instability in passively mode-locked lasers," Physica D, submitted for publication.
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    [CrossRef]
  28. P. G. Drazin, Nonlinear Systems (Cambridge u. Press, 1992).
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    [CrossRef]

2004 (1)

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, "Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser," Phys. Rev. E 70, 066612 (2004).
[CrossRef]

2002 (1)

J. J. O'Neil, J. N. Kutz, and B. Sandstede, "Theory and simulation of the dynamics and stability of actively mode-locked lasers," IEEE J. Quantum Electron. 38, 1412-1419 (2002).
[CrossRef]

2000 (1)

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

1999 (2)

D. Y. Tang, W. S. Man, and H. Y. Tam, "Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser," Opt. Commun. 165, 189-194 (1999).
[CrossRef]

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

1998 (2)

1997 (1)

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

1996 (1)

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

1995 (4)

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]

1993 (1)

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser," Opt. Lett. 29, 447-449 (1993).

1992 (2)

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[CrossRef]

1991 (2)

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544-545 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

1986 (1)

G. L. Eesley, "Generation of nonequilibrium electron and lattice temperature in copper by picosecond laser pulses," Phys. Rev. B 33, 2144-2151 (1986).
[CrossRef]

1985 (1)

M. I. Weinstein, "Modulational stability of ground states of the nonlinear Schrödinger equations," SIAM J. Appl. Math. 16, 472-491 (1985).
[CrossRef]

1976 (1)

H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

1975 (1)

H. A. Haus, "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

Akhmediev, N.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, "Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser," Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Andrejco, M. J.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser," Opt. Lett. 29, 447-449 (1993).

Bergman, K.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Bielawski, S.

T. Kolokolnikov, T. Erneux, N. Joly, and S. Bielawski, "The Q-switching instability in passively mode-locked lasers," Physica D, submitted for publication.

Brakenhoff, G. J.

Brovelli, L.

F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
[CrossRef]

Calasso, I.

F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
[CrossRef]

Collings, B.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Cundiff, S.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Cunningham, J. E.

da Silva, L.

L. da Silva, M. Perry, M. Feit, and B. Stuart, "The short-pulse laser: a safe, painless surgical tool," Sci. Technol. Rev. September 1995, pp. 29-31.

Dennis, M. L.

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
[CrossRef]

DeSouza, E. A.

Drazin, P. G.

P. G. Drazin, Nonlinear Systems (Cambridge u. Press, 1992).

Duling, I. N.

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
[CrossRef]

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544-545 (1991).
[CrossRef]

Eesley, G. L.

G. L. Eesley, "Generation of nonequilibrium electron and lattice temperature in copper by picosecond laser pulses," Phys. Rev. B 33, 2144-2151 (1986).
[CrossRef]

Erneux, T.

T. Kolokolnikov, T. Erneux, N. Joly, and S. Bielawski, "The Q-switching instability in passively mode-locked lasers," Physica D, submitted for publication.

Feit, M.

L. da Silva, M. Perry, M. Feit, and B. Stuart, "The short-pulse laser: a safe, painless surgical tool," Sci. Technol. Rev. September 1995, pp. 29-31.

Fermann, M. E.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser," Opt. Lett. 29, 447-449 (1993).

Grapinet, M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, "Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser," Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Grelu, P.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, "Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser," Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Haus, H. A.

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

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

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]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[CrossRef]

H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

H. A. Haus, "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

Hönninger, C.

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

Hopkins, J.-M.

J.-M. Hopkins and W. Sibbett, "Utrashort-pulse lasers: big payoffs in a flash," Sci. Am. September 2000, pp. 73-79.

Ippen, E. P.

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]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[CrossRef]

Jan, W. J.

Joly, N.

T. Kolokolnikov, T. Erneux, N. Joly, and S. Bielawski, "The Q-switching instability in passively mode-locked lasers," Physica D, submitted for publication.

Kamp, M.

F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
[CrossRef]

Kärtner, F. X.

Keller, U.

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
[CrossRef]

F. X. Kärtner and U. Keller, "Stabilization of solitonlike pulses with a slow saturable absorber," Opt. Lett. 20, 16-18 (1995).
[CrossRef] [PubMed]

F. X. Kärtner, D. Kopf, and U. Keller, "Solitary pulse stabilization and shortening in actively mode-locked lasers," J. Opt. Soc. Am. B 12, 486-496 (1995).
[CrossRef]

Klimov, V. I.

Knox, W.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Knox, W. H.

Koch, M.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Kolokolnikov, T.

T. Kolokolnikov, T. Erneux, N. Joly, and S. Bielawski, "The Q-switching instability in passively mode-locked lasers," Physica D, submitted for publication.

Kopf, D.

F. X. Kärtner, D. Kopf, and U. Keller, "Solitary pulse stabilization and shortening in actively mode-locked lasers," J. Opt. Soc. Am. B 12, 486-496 (1995).
[CrossRef]

F. X. Kärtner, L. Brovelli, D. Kopf, M. Kamp, I. Calasso, and U. Keller, "Control of solid state laser dynamics of semiconductor devices," Opt. Eng. (Bellingham) 34, 2024-2036 (1995).
[CrossRef]

Kutz, J. N.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, "Theory and simulation of the dynamics and stability of actively mode-locked lasers," IEEE J. Quantum Electron. 38, 1412-1419 (2002).
[CrossRef]

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

Laming, R. I.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

Magnus, W.

W. Magnus, Hill's Equation (Wiley, 1966).

Man, W. S.

D. Y. Tang, W. S. Man, and H. Y. Tam, "Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser," Opt. Commun. 165, 189-194 (1999).
[CrossRef]

Matsas, V. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

McBranch, D. W.

Morier-Genoud, F.

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

Moser, M.

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

Mourou, G. A.

G. A. Mourou and D. Umdstader, "Extreme light," Sci. Am. June 2002, pp. 63-68.

Müller, M.

O'Neil, J. J.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, "Theory and simulation of the dynamics and stability of actively mode-locked lasers," IEEE J. Quantum Electron. 38, 1412-1419 (2002).
[CrossRef]

Paschotta, P.

C. Hönninger, P. Paschotta, F. Morier-Genoud, M. Moser, and U. Keller, "Q-switching stability limits of continuous-wave passive mode locking," J. Opt. Soc. Am. B 16, 45-56 (1999).
[CrossRef]

Payne, D. N.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

Perry, M.

L. da Silva, M. Perry, M. Feit, and B. Stuart, "The short-pulse laser: a safe, painless surgical tool," Sci. Technol. Rev. September 1995, pp. 29-31.

Phillips, M. W.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

Richardson, D. J.

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

Sandstede, B.

J. J. O'Neil, J. N. Kutz, and B. Sandstede, "Theory and simulation of the dynamics and stability of actively mode-locked lasers," IEEE J. Quantum Electron. 38, 1412-1419 (2002).
[CrossRef]

Sibbett, W.

J.-M. Hopkins and W. Sibbett, "Utrashort-pulse lasers: big payoffs in a flash," Sci. Am. September 2000, pp. 73-79.

Silverberg, Y.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser," Opt. Lett. 29, 447-449 (1993).

Soto-Crespo, J. M.

J. M. Soto-Crespo, M. Grapinet, P. Grelu, and N. Akhmediev, "Bifurcations and multiple-period soliton pulsations in a passively mode-locked fiber laser," Phys. Rev. E 70, 066612 (2004).
[CrossRef]

Squier, J. A.

Stock, M. L.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser," Opt. Lett. 29, 447-449 (1993).

Stuart, B.

L. da Silva, M. Perry, M. Feit, and B. Stuart, "The short-pulse laser: a safe, painless surgical tool," Sci. Technol. Rev. September 1995, pp. 29-31.

Tam, H. Y.

D. Y. Tang, W. S. Man, and H. Y. Tam, "Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser," Opt. Commun. 165, 189-194 (1999).
[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]

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[CrossRef]

Tang, D. Y.

D. Y. Tang, W. S. Man, and H. Y. Tam, "Stimulated soliton pulse formation and its mechanism in a passively mode-locked fibre soliton laser," Opt. Commun. 165, 189-194 (1999).
[CrossRef]

Tsuda, S.

B. Collings, S. Tsuda, S. Cundiff, J. N. Kutz, M. Koch, W. Knox, and K. Bergman, "Short cavity erbium/ytterbium fiber lasers mode-locked with a saturable Bragg reflector," IEEE J. Sel. Top. Quantum Electron. 3, 1065-1075 (1997).
[CrossRef]

S. Tsuda, W. H. Knox, E. A. DeSouza, W. J. Jan, and J. E. Cunningham, "Low-loss intracavity AlAs/AlGaAs saturable Bragg reflector for femtosecond mode locking in solid-state lasers," Opt. Lett. 20, 1406-1408 (1995).
[CrossRef] [PubMed]

Umdstader, D.

G. A. Mourou and D. Umdstader, "Extreme light," Sci. Am. June 2002, pp. 63-68.

Weinstein, M. I.

M. I. Weinstein, "Modulational stability of ground states of the nonlinear Schrödinger equations," SIAM J. Appl. Math. 16, 472-491 (1985).
[CrossRef]

Wilson, K. R.

Wong, W. S.

H. A. Haus and W. S. Wong, "Solitons in optical communications," Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

Electron. Lett. (4)

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse mode-locked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[CrossRef]

I. N. Duling III, "Subpicosecond all-fiber erbium laser," Electron. Lett. 27, 544-545 (1991).
[CrossRef]

D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively mode-locked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
[CrossRef]

M. L. Dennis and I. N. Duling III, "High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
[CrossRef]

IEEE J. Quantum Electron. (4)

H. A. Haus, "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-330 (1975).
[CrossRef]

H. A. Haus, "Parameter ranges for CW passive mode locking," IEEE J. Quantum Electron. 12, 169-176 (1976).
[CrossRef]

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]

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Figures (13)

Fig. 1
Fig. 1

Qualitative depiction of the Q-switching phenomenon in comparison with a uniform pulse train. The uniform pulse train (top) is the desired configuration for communication and network applications. The modulated Q-switched pulse train (bottom) allows for greater peak intensities, which can be advantageous for certain applications.

Fig. 2
Fig. 2

Pulse train solutions of Eq. (1) with constant gain for various values of the elliptic modulus k. The left column depicts the out-of-phase (antisymmetric) solutions given by the cn ( T , k ) of Eq. (3). Note the alternating pattern of positive and negative pulse amplitudes. The right column depicts the in-phase (symmetric) solutions given by the dn ( T , k ) of Eq. (5). Here, there is no nodal separation between neighboring pulses. In the limit k 1 , both the cn ( T , k ) and the dn ( T , k ) solutions converge to a series of well-separated hyperbolic secant pulses: the former with nodal separation, the latter with no nodes. As k 0 , the sinusoidal limit is achieved for which the cn ( T , k ) cos ( T ) and dn ( T , k ) 1 . In this figure and all that follow, a single period P, which is dependent on the elliptic modulus, is given by P = 4 0 π 2 d α ( 1 k 2 sin 2 α ) 1 2 . For k = 0 , P = 2 π , whereas, for k 1 , P .[26]

Fig. 3
Fig. 3

Eigenvalue spectrum of the linearized operator Eq. (11) for the cn ( T , k ) solution Eq. (3) for six elliptic modulus values k. The continuous spectrum does not lead to instability; i.e., it is completely in the left half-plane for k > 0 (at k = 0 , the continuous spectrum starts on the imaginary axis). In contrast, the discrete spectral content exhibits instability with two to four eigenvalues in the right half of the complex plane ( Real { λ } 0 ) . The specific location and number are determined by the value of the elliptic modulus k. The four most important eigenvalues are tracked with different symbols as shown in Fig. 4. Note that the elliptic modulus value for each plot is given at the top of each plot.

Fig. 4
Fig. 4

Real part of the discrete eigenvalues as a function of the elliptic modulus k for the cn ( T , k ) solution Eq. (3). The different symbols used correspond to the eigenvalues shown in Fig. 3. For low k values, four unstable modes, which are composed of two double roots, are present. At a value of k 0.4 , one of the double roots separates, with one eigenvalue increasing and the second decreasing. At a value of k 0.6 , the second double root separates in a similar fashion. Beyond k 0.65 , only two unstable modes remain. In the limit k 1 , the growth rate is identically zero so that neutral stability is established. The solid line denotes the location of the rightmost point of the continuous spectrum for which Real { λ } 0 .

Fig. 5
Fig. 5

Real part of the dominant unstable eigenmodes with nodes (antisymmetric, left column) and without nodes (symmetric, right column) as a function of the elliptic modulus k. Note the qualitative similarity of these modes to the out-of-phase solutions Eq. (3) in Fig. 2 (left) and the in-phase solutions dn ( T , k ) of Eq. (5) in Fig. 2 (right). The nonlinear beating between these unstable modes produces the Q-switching phenomenon. Note that the elliptic modulus value for each row of plots is given between the plots.

Fig. 6
Fig. 6

Eigenvalue spectrum of the linearized operator Eq. (11) for the dn ( T , k ) solution Eq. (5) for six elliptic modulus values k. The discrete spectral content exhibits instability with a number of eigenvalues in the right half of the complex plane. The specific location and number are determined by the value of the elliptic modulus k. The growth rates are typically larger than those found for the cn ( T , k ) solution. The dominant eigenvalues are more clearly illustrated in Fig. 7 as a function of k. Note that the elliptic modulus value for each plot is given at the top of each plot.

Fig. 7
Fig. 7

Real part of the discrete eigenvalues as a function of the elliptic modulus k for the dn ( T , k ) solution Eq. (5). The larger growth rates and numerous unstable modes destabilize the dn ( T , k ) solution, which decays to zero.[6]

Fig. 8
Fig. 8

Pulse dynamics of the Q-switched solution Eq. (18) under the rotating-wave approximation. The cn ( T , k ) , dn ( T , k ) two-mode beating interaction is driven by the phase difference ( k 2 3 ) 6 . Note the beating period as a function of elliptic modulus k.

Fig. 9
Fig. 9

Pulse dynamics demonstrating the onset of the Q-switching instability in a single-period cavity for various elliptic modulus values k. The initial condition is arbitrary in each case. Note the dependence of the beating period on the elliptic modulus as predicted by Eq. (18). Further, as k 1 , the unstable eigenvalues approach zero so that no instability is observed in the numerical computations.[6] To demonstrate the robustness of the Q switching, we performed the simulations to Z = 30 000 . This leads to aliasing in the visualization of the solution. The true dynamics without aliasing is demonstrated in Fig. 10.

Fig. 10
Fig. 10

Pulse dynamics demonstrating the Q-switching phenomenon in a single-period cavity for various elliptic modulus values k. To visualize the Q switching without aliasing, the simulations are demonstrated for Z [ 30 000 , 30 030 ] . Note the clear dependence of the beating period on the elliptic modulus as predicted by Eq. (18). As k 1 , the unstable eigenvalues approach zero so that no instability is observed in the numerical computations.[6]

Fig. 11
Fig. 11

Amplitude fluctuations in the center of the laser cavity ( T = 0 ) for Z [ 30 000 , 30 100 ] . Oscillations occur for the Q-switched solutions that are sufficiently far away from the k 1 limit. Note that the period of the oscillations decreases as a function of k. The k 1 limit essentially has an infinite period and does not Q switch.

Fig. 12
Fig. 12

Pulse train dynamics generated from the laser cavity corresponding to Fig. 10. It is assumed that the output coupler is located at each Z = n , where n = 1 , 2, …. By retrieving the amplitude slices of Fig. 10 at the various output coupler locations Z = n and concatenating them, one can construct the mode-locked pulse train. This behavior is the hallmark feature of Q switching, i.e., the slow modulation of a uniform pulse train. Q switching is not observed in the k 1 , since the unstable eigenvalues go to zero.

Fig. 13
Fig. 13

Pulse dynamics demonstrating the onset of the Q-switching instability in a multipulse per round-trip cavity configuration (four periods) for various elliptic modulus values k. The initial condition is arbitrary in each case. For this case, the longer cavity allows for long-wavelength instabilities to grow, which drives the lower-k-value solutions to a weakly turbelent regime. As found previously, for k 1 the unstable eigenvalues approach zero so that no instability is observed in the numerical computations.[6]

Equations (26)

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i Q Z + 1 2 2 Q T 2 + Q 2 Q i { g ( Z ) ( 1 + τ 2 T 2 ) M [ Γ cn 2 ( ω T , k ) ] } Q = 0 ,
g ( Z ) = 2 g 0 1 + Q 2 e 0 .
Q ( Z , T ) = k cn ( T , k ) exp [ i ( 1 2 k 2 ) Z ] ,
M = 2 k 2 g τ ,
Γ = 1 2 k 2 ( 2 k 2 + 1 τ 1 ) .
Q ( Z , T ) = dn ( T , k ) exp [ i ( k 2 2 1 ) Z ] ,
M = 2 k 2 g τ ,
Γ = 1 2 k 2 ( k 2 + 1 τ ) .
Q ( Z , T ) = ( Q 0 + Q ̃ ) exp ( i Ω Z ) ,
Q ̃ ( Z , T ) = ( R + i I ) .
W Z = [ L 1 L 2 L 2 + 2 Q 0 2 L 1 ] W = LW ,
L 1 = g τ 2 T 2 + g M Γ + M cn 2 ( ω T , k ) ,
L 2 = 1 2 2 T 2 Q 0 2 + Ω .
Lv = λ v .
Q ( Z , T ) = A ( Z ) cn ( T , k ) + B ( Z ) dn ( T , k )
i d A d Z + 2 k 2 1 2 A + ( 1 k 2 ) ( 2 A B 2 + A * B 2 ) i [ g + g τ ( 2 k 2 1 ) M Γ ] A = 0 ,
i d B d Z + 2 k 2 2 B + ( 1 1 k 2 ) ( 2 A 2 B + A 2 B * ) i [ g + g τ ( 2 k 2 ) M Γ + M ( 1 1 k 2 ) ] B = 0 ,
( A 2 k 2 ) A + k 2 ( 2 A B 2 + A * B 2 ) + i ( 2 k 2 g τ M ) A = 0 ,
( B 2 1 ) B + 1 k 2 ( 2 A 2 B + A 2 B * ) + i k 2 ( 2 k 2 g τ M ) B = 0 .
i d A d Z + 2 k 2 1 2 A + 2 ( 1 k 2 ) B 2 A = 0 ,
i d B d Z + 2 k 2 2 B + 2 ( 1 1 k 2 ) A 2 B = 0 ,
A ( Z ) = A 0 exp { i [ ( k 2 1 2 ) + 2 ( 1 k 2 ) B 0 2 ] Z } ,
B ( Z ) = B 0 exp { i [ ( 1 k 2 2 ) + 2 ( 1 1 k 2 ) A 0 2 ] Z } .
A 0 2 = k 2 3 ,
B 0 2 = 1 3 ,
Q ( Z , T ) = 1 3 [ k cn ( T , k ) exp ( i 2 k 2 1 6 Z ) + dn ( T , k ) exp ( i 2 + k 2 6 Z ) ] ,

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