Abstract

A comprehensive theoretical treatment is given of the phenomenon of harmonic mode-locking in a laser cavity mode-locked by the nonlinear mode-coupling behavior in a waveguide array. The theoretical model completely characterizes oscillatory instabilities and the transition from M to M+1 pulses as a function of increased gain.

© 2008 Optical Society of America

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  1. J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
    [CrossRef]
  2. B. Collings, K. Berman, and W. H. Knox, "Stable multigigahertz pulse train formation in a short cavity passively harmonic modelocked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
    [CrossRef]
  3. M. E. Fermann and J. D. Minelly, "Cladding-pumped passive harmonically mode-locked fiber laser," Opt. Lett. 21, 970-972 (1996).
    [CrossRef] [PubMed]
  4. A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
    [CrossRef]
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    [CrossRef]
  8. M. J. Lederer, B. Luther-Davis, H. H. Tan, C. Jagadish, N. N. Akhmediev, and J. M. Soto-Crespo, "Multipulse operation of a Ti:Sapphire laser mode locked by an ion-implanted semiconductor saturable-absorber mirror," J. Opt. Soc. Am. B 16, 895-904 (1999).
    [CrossRef]
  9. Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
    [CrossRef]
  10. M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
    [CrossRef]
  11. C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
    [CrossRef]
  12. H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
    [CrossRef]
  13. A. N. Pilipetskii, E. A. Golovchenck, and C. R. Menyuk, "Acoustic effect in passively mode-locked fiber ring lasers," Opt. Lett. 20, 907-909 (1995).
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    [CrossRef]
  15. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  16. J. N. Kutz, "Mode-locked soliton lasers," SIAM Rev. 48, 629-678 (2006).
    [CrossRef]
  17. J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
    [CrossRef]
  18. J. N. Kutz, "Mode-locking of fiber lasers via nonlinear mode-coupling," in Dissipative Solitons, Lecture Notes in Physics, N. N. Akhmediev and A. Ankiewicz, eds., (Springer-Verlag, Berlin, 2005) pp. 241-265.
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  19. J. Proctor and J. N. Kutz, "Theory and simulation of passive mode-locking with waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
    [CrossRef]
  20. J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
    [CrossRef] [PubMed]
  21. J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," Math. Comput. Simulation 74, 333-342 (2007).
    [CrossRef]
  22. H. G. Winful and D. T. Walton, "Passive mode locking through nonlinear coupling in a dual-core fiber laser," Opt. Lett. 17, 1688-1690 (1992).
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  23. Y. Oh, S. L. Doty, J.W. Haus, and R. L. Fork, "Robust operation of a dual-core fiber ring laser," J. Opt. Soc. Am. B 12, 2502-2507 (1995).
    [CrossRef]
  24. K. Intrachat and J. N. Kutz, "Theory and simulation of passive mode-locking dynamics using a long period fiber grating," IEEE J. Quantum Electron. 39, 1572-1578 (2003).
    [CrossRef]
  25. D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
    [CrossRef] [PubMed]
  26. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
    [CrossRef]
  27. A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
    [CrossRef]
  28. H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, "Optical discrete solitons in waveguide arrays. 1. Soliton formation," J. Opt. Soc. Am. B 19, 2938-1944 (2002).
    [CrossRef]
  29. U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. 2. Dynamics properties," J. Opt. Soc. Am. B 19, 2637-2644 (2002).
    [CrossRef]
  30. S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
    [CrossRef]
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  33. T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of pulses in the master-modelocking equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
    [CrossRef]

2007

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," Math. Comput. Simulation 74, 333-342 (2007).
[CrossRef]

2006

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

2005

J. Proctor and J. N. Kutz, "Theory and simulation of passive mode-locking with waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
[CrossRef] [PubMed]

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

2003

K. Intrachat and J. N. Kutz, "Theory and simulation of passive mode-locking dynamics using a long period fiber grating," IEEE J. Quantum Electron. 39, 1572-1578 (2003).
[CrossRef]

2002

2000

1999

1998

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

B. Collings, K. Berman, and W. H. Knox, "Stable multigigahertz pulse train formation in a short cavity passively harmonic modelocked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
[CrossRef]

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

1997

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

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

1996

M. E. Fermann and J. D. Minelly, "Cladding-pumped passive harmonically mode-locked fiber laser," Opt. Lett. 21, 970-972 (1996).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

1995

1994

1992

1991

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

1988

1983

J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
[CrossRef]

Aceves, A. B.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Aitchison, J. S.

Akhmediev, N. N.

Arnold, J. M.

Bergman, K.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Berman, K.

Boskovic, A.

Boyd, A. R.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Brabec, T.

Carruthers, T. F.

Chai, L.

Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Christodoulides, D. N.

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, "Discrete self-focusing in nonlinear arrays of coupled waveguides," Opt. Lett. 13, 794-796 (1988).
[CrossRef] [PubMed]

Collings, B.

Collings, B. C.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Curley, P. F.

Davey, R. P.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

De Angelis, C.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Doty, S. L.

Droulias, S.

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

Duling, I. N.

Eisenberg, H. S.

Ferguson, A. I.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

Fermann, M. E.

Fork, R. L.

Golovchenck, E. A.

Gordon, J. P.

J. P. Gordon, "Interaction forces among solitons in optical fibers," Opt. Lett. 8, 396-398 (1983).
[CrossRef]

Grudinin, A. B.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Guy, M. J.

Haus, H.

Haus, H. A.

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

Haus, J.W.

Hizanidis, K.

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

Horowitz, M.

Intrachat, K.

K. Intrachat and J. N. Kutz, "Theory and simulation of passive mode-locking dynamics using a long period fiber grating," IEEE J. Quantum Electron. 39, 1572-1578 (2003).
[CrossRef]

Ippen, E. P.

Jagadish, C.

Joseph, R. I.

Kapitula, T.

Kinoshita, S.

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Kitano, H.

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Knox, W. H.

B. Collings, K. Berman, and W. H. Knox, "Stable multigigahertz pulse train formation in a short cavity passively harmonic modelocked Er/Yb fiber laser," Opt. Lett. 23, 123-125 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Krausz, F.

Kutz, J. N.

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," Math. Comput. Simulation 74, 333-342 (2007).
[CrossRef]

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

J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, "Theory and simulation of passive mode-locking with waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

K. Intrachat and J. N. Kutz, "Theory and simulation of passive mode-locking dynamics using a long period fiber grating," IEEE J. Quantum Electron. 39, 1572-1578 (2003).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "Stability of pulses in the master-modelocking equation," J. Opt. Soc. Am. B 19, 740-746 (2002).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

Lai, M.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Langford, N.

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

Lederer, F.

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. 2. Dynamics properties," J. Opt. Soc. Am. B 19, 2637-2644 (2002).
[CrossRef]

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Lederer, M. J.

Lee, K. F.

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Luther-Davis, B.

Menyuk, C. R.

Minelly, J. D.

Morandotti, R.

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

U. Peschel, R. Morandotti, J. M. Arnold, J. S. Aitchison, H. S. Eisenberg, Y. Silberberg, T. Pertsch, and F. Lederer, "Optical discrete solitons in waveguide arrays. 2. Dynamics properties," J. Opt. Soc. Am. B 19, 2637-2644 (2002).
[CrossRef]

H. S. Eisenberg, R. Morandotti, Y. Silberberg, J. M. Arnold, G. Pennelli, and J. S. Aitchison, "Optical discrete solitons in waveguide arrays. 1. Soliton formation," J. Opt. Soc. Am. B 19, 2938-1944 (2002).
[CrossRef]

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

Muschall, R.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Namiki, S.

Nicholson, J.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Noske, P. U.

Oh, Y.

Payne, D. N.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Pennelli, G.

Pertsch, T.

Peschel, T.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Peschel, U.

Pilipetskii, A. N.

Proctor, J.

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," Math. Comput. Simulation 74, 333-342 (2007).
[CrossRef]

J. Proctor and J. N. Kutz, "Nonlinear mode-coupling for passive mode-locking: application of waveguide arrays, dual-core fibers, and/or fiber arrays," Opt. Express 13, 8933-8950 (2005).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, "Theory and simulation of passive mode-locking with waveguide arrays," Opt. Lett. 13, 2013-2015 (2005).
[CrossRef]

Richardson, D. J.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

Rudolph, W.

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

Sandstede, B.

Silberberg, Y.

Soto-Crespo, J. M.

Spielmann, Ch.

Tan, H. H.

Taylor, J. R.

Trillo, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Wabnitz, S.

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Walton, D. T.

Wang, C.

Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Winful, H. G.

Xing, Q.

Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

Yoo, K. M.

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

Yu, C. X.

Zhang, W.

Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

App. Phys. Lett.

S. Droulias, K. Hizanidis, D. N. Christodoulides, and R. Morandotti, "Waveguide array-grating compressors," App. Phys. Lett. 87, 131104 (2005).
[CrossRef]

Electron. Lett.

A. B. Grudinin, D. J. Richardson, and D. N. Payne, "Energy quantization in figure eight fibre laser," Electron. Lett. 28, 1391-1393 (1992).
[CrossRef]

R. P. Davey, N. Langford, and A. I. Ferguson, "Interacting solutions in erbium fibre laser," Electron. Lett. 27, 1257-1259 (1991).
[CrossRef]

IEEE J. Quantum Electron.

J. N. Kutz, B. C. Collings, K. Bergman, and W. H. Knox, "Stabilized pulse spacing in soliton lasers due to gain depletion and recovery," IEEE J. Quantum Electron. 34, 1749-1757 (1998).
[CrossRef]

K. Intrachat and J. N. Kutz, "Theory and simulation of passive mode-locking dynamics using a long period fiber grating," IEEE J. Quantum Electron. 39, 1572-1578 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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

J. Lightwave Technol.

J. Opt. Soc. Am. B

Math. Comput. Simulation

J. Proctor and J. N. Kutz, "Averaged models for passive mode-locking using nonlinear mode-coupling," Math. Comput. Simulation 74, 333-342 (2007).
[CrossRef]

Opt. Commun.

Q. Xing, L. Chai, W. Zhang, and C. Wang, "Regular, period-doubling, quasi-periodic, and chaotic behavior in a self-mode-locked Ti:sapphire laser," Opt. Commun. 162, 71-74 (1999).
[CrossRef]

M. Lai, J. Nicholson, and W. Rudolph, "Multiple pulse operation of a femtosecond Ti:sapphire laser," Opt. Commun. 142, 45-49 (1997).
[CrossRef]

C. Wang, W. Zhang, K. F. Lee, and K. M. Yoo, "Pulse splitting in a self-mode-locking Ti:sapphire laser," Opt. Commun. 137, 89-92 (1997).
[CrossRef]

H. Kitano and S. Kinoshita, "Stable multipulse generation from a self-mode-locked Ti:sapphire laser," Opt. Commun. 157, 128-134 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

A. B. Aceves, C. De Angelis, T. Peschel, R. Muschall, F. Lederer, S. Trillo, and S. Wabnitz, "Discrete selftrapping soliton interactions, and beam steering in nonlinear waveguide arrays," Phys. Rev. E 53, 1172-1189 (1996).
[CrossRef]

Phys. Rev. Lett.

H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd and J. S. Aitchison, "Discrete spatial optical solitons in waveguide arrays," Phys. Rev. Lett. 81, 3383-3386 (1998).
[CrossRef]

SIAM Rev.

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

Other

J. N. Kutz, "Mode-locking of fiber lasers via nonlinear mode-coupling," in Dissipative Solitons, Lecture Notes in Physics, N. N. Akhmediev and A. Ankiewicz, eds., (Springer-Verlag, Berlin, 2005) pp. 241-265.
[CrossRef]

L. Spinelli, B. Couilland, N. Goldblatt, and D. K. Negus, in Conference on Lasers and Electro-Optics 10, (1991) OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1991), paper CPDP7.

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Figures (13)

Fig. 1.
Fig. 1.

Possible experimental laser configuration for a mode-locked laser cavity with a waveguide array responsible for the NLMC.

Fig. 2.
Fig. 2.

Spatial diffraction of a CW beam propagating in a waveguide array governed by Eq. (1). The left figure shows the low-intensity evolution and associated discrete spatial diffration with initial condition A 0=1, while the right figure shows the high-intensity evolution and associated self-focusing for A 0=3.

Fig. 3.
Fig. 3.

Plotted are the amplitude η versus the fixed-gain parameter gf [left curve], from solving Eq. (14), and the corresponding solutions [right curve] for variable gain with g 0 computed via Eq. (15) from gf .

Fig. 4.
Fig. 4.

The left plot shows the existence curve of secant solutions for variable gain Eq. (10). The modelocking solution is stable along the solid curve; it destabilizes on the lower branch at g 0=1.4 due to unstable counter-propagating radiation modes as shown in the lower-right panel, while the instability on the upper branch at g 0=2.3 is due to a Hopf bifurcation which will be discussed later. The upper-right panel contains the linear dispersion relation λ(k) associated with radiation modes. The instability at g 0=1.4 for low-amplitude solitons is due to the upper branch of radiation modes crossing the imaginary axis, resulting in the instability caused by the counter-propagating waves shown in the lower-right panel.

Fig. 5.
Fig. 5.

The upper-right panel contains the complete spectrum for the mode-locking solution at the instability point g 0=2.3 shown in the left panel. Eigenvalues are plotted for the linearization on the interval (-4,4) [blue bullets] and (-10,10) [red crosses] with 300 equidistant mesh points in T. The instability is caused by a localized Hopf eigenmode, whose real and imaginary part we plot in the lower-right figure, with temporal frequency 12.06.

Fig. 6.
Fig. 6.

Plotted are the amplitude η versus variable gain g 0 for the harmonic modelocking solutions with M=1,…,4 and the fixed-gain modelocking solutions [left]. Solutions are stable for amplitudes η in the range indicated to the right; they are unstable to oscillatory modes above the upper horizontal dotted line and to counter-propagating waves below the lower horizontal dotted line.

Fig. 7.
Fig. 7.

Demonstration of mode-locking (A 0, A 1 and A 2) in the variable gain model with parameters as in Eq. (16) and initial white-noise. The L 2 norm is also depicted for A 0, A 1 and A 2. Note that the dynamics here and the settling to a steady-state solution is tremendously robust with what appears to be an infinite basin of attraction.

Fig. 8.
Fig. 8.

Dynamic evolution and associated bifurcation structure of the transition from one pulse per round trip to two pulses per round trip. The corresponding values of gain are g 0=2.3,2.35,2.5,2.55,2.7, and 2.75. For the lowest gain value only a single pulse is present. The pulse then becomes a periodic breather before undergoing a “chaotic” transition between a breather and a two-pulse solution. Above a critical value (g 0≈2.75), the two-pulse solution is stabilized. The corresponding gain dynamics is given in Fig. 9.

Fig. 9.
Fig. 9.

Gain dynamics associated with the transition from one pulse per round trip to two pulses per round trip for the temporal dynamics given in Fig. 8. The left column is the full gain dynamics for Z∊[0,4000], while the right column is a detail over Z=10 or Z=50 units, for values of gain equal to g 0=2.3,2.35,2.5,2.55,2.7, and 2.75. Initially a single pulse is present (top panel), which becomes a periodic breather (following two panels) before undergoing a “chaotic” transition between a breather and a two-pulse solution (following two panels) until the two-pulse is stabilized (bottom panel) at g 0≈2.75.

Fig. 10.
Fig. 10.

Fourier spectrum of gain dynamics oscillations as a function of wavenumber. The last Z=51.15 units (which gives 1024 data points) are selected to construct the time series of the gain dynamics and its Fourier transform minus its average. This is done for the data in Figs. 8 and 9 for g 0=2.35,2.5 and 2.55.

Fig. 11.
Fig. 11.

Demonstration of bi-stability of the one and two pulse per round trip configurations in the harmonic mode-locking process. The gain is first increased from g 0=0.9 to g 0=2.75 and then decreased back. Shown are stable single pulse per round trip (g 0=0.9), one and two pulse per round trip (g 0=2.3 in middle panels), and breathing one pulse and two pulse per round trip (g 0=2.55 in bottom panels) configurations. The top right panel shows the one- and two-pulse branch of solutions along with the stable solutions from the remaining panels indicated by circles.

Fig. 12.
Fig. 12.

Dynamic evolution and associated bifurcation structure of the transition from two pulses per round trip to three pulses per round trip. The corresponding values of gain are g 0=3.7,3.9,4.1, and 4.3. For the lowest gain, two pulses are present. One pulse then becomes a periodic breather before undergoing a “chaotic” transition between a breather and a two-pulse solution. Above a critical value, three pulses are stabilized.

Fig. 13.
Fig. 13.

Dynamic evolution to four and five pulses per round trip for values of gain equal to g 0=5.7 and 7.1, respectively. The dynamical bifurcations to these steady-state configurations follow patterns similar to the transitions from one to two and two to three pulses.

Equations (25)

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i dA n + C ˜ ( A n 1 + A n + 1 ) + β A n 2 A n = 0 ,
i A n Z + 1 2 2 A n T 2 + β n A n 2 A n + C ¯ ( A n + 1 + A n 1 ) + n A n iG n ( Z ) ( 1 + τ n 2 T 2 ) A n = 0
G n ( Z ) = 2 g 0 1 + A 0 2 e 0 ,
i A 0 Z + 1 2 2 A 0 T 2 + β 0 A 0 2 A 0 + CA 1 + 0 A 0 ig ( Z ) ( 1 + τ 2 T 2 ) A 0 = 0
i A 1 Z + 1 2 2 A 1 T 2 + β 1 A 1 2 A 1 + C ( A 2 + A 0 ) + 1 A 1 = 0
i A 2 Z + 1 2 2 A 2 T 2 + β 2 A 2 2 A 2 + CA 1 + 2 A 2 = 0 .
i A 0 Z + 1 2 2 A 0 T 2 + β 0 A 0 2 A 0 + CA 1 + 0 A 0 ig ( Z ) ( 1 + τ 2 T 2 ) A 0 = 0
i A 1 Z + C ( A 2 + A 0 ) + 1 A 1 = 0
i A 2 Z + CA 1 + 2 A 2 = 0
A i = Q i ( T ) exp ( i Θ 0 Z )
Q 2 = C Θ 0 + 2 Q 1 .
Q 1 = C Θ 0 + 1 C 2 ( Θ 0 + 1 ) ( Θ 0 + 2 ) Q 0 = : P Q 0 ,
Θ 0 Q 0 + 1 2 2 Q 0 T 2 + β 0 Q 0 2 Q 0 + C P Q 0 + 0 Q 0 ig ( Q 0 ) ( 1 + τ 2 T 2 ) Q 0 = 0 .
g ( Z ) = 2 g 0 1 + Q 0 2 e 0 .
Q 0 = η sech ( ωT ) 1 + iA .
g f = g 0 1 + 2 η 2 ωe 0 ,
g ( Z ) = 2 g f .
ω 2 A + CP i + γ 0 2 g f ( 1 + τω 2 ( 1 A 2 ) ) = 0
ω 2 ( A 2 2 ) + 2 β 0 η 2 12 g f τω 2 A = 0
3 A + 4 g f τ ( 2 A 2 ) = 0
g 0 = g f ( 1 + 2 η 2 ωe 0 ) .
( e 0 , τ , C , β , γ 0 , γ 1 , γ 2 , g 0 ) = ( 1 , 0.1 , 5 , 8 , 0 , 0 , 10 , 2.3 ) ,
e λZ+ikT ( B 0 , B 1 , B 2 ),
k 2 2 + 0 Θ 0 2 ig f ( 1 τk 2 ) C 0 C + 1 + Θ 0 C 0 C + 2 + Θ 0 = 0 .
g ( Z ) = 2 g 0 1 + M A 0 2 e 0 ,

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