Abstract

A theoretical model is developed for characterizing mode-locking behavior using the variational method. The variational method is applied to three laser-cavity models, and a reduced master mode-locking model is obtained. Characterizing the mode-locking dynamics is the existence of a stable node, stable spiral, or limit cycle in the reduced equations highlighting the regions of a stable mode-locked operation. Fundamental in driving the laser dynamics is the nontrivial phase profiles generated during the mode-locking process. The variational method provides an excellent theoretical framework for optimizing laser performance for a wide range of mode-locking models.

© 2008 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  39. B. G. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for mode-locking in the normal dispersive regime,” Opt. Lett. 33, 941-943 (2008).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2008 (4)

2007 (2)

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

C. Antonelli, J. Chen, and F. X. Kärtner, “Intracavity pulse dynamics and stability for passively mode-locked lasers,” Opt. Express 15, 5919-5924 (2007).
[CrossRef]

2006 (2)

2005 (2)

2004 (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

2003 (1)

2002 (5)

2001 (2)

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

2000 (1)

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

1999 (1)

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

1998 (3)

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 (1998).
[CrossRef]

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58-103 (1998).
[CrossRef]

J. N. Kutz, P. Holmes, S. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion managed breathers,” J. Opt. Soc. Am. B 15, 87-96 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (3)

1994 (1)

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
[CrossRef]

1993 (3)

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, 894-896 (1993).
[CrossRef]

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331-4341 (1993).
[CrossRef]

1992 (2)

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]

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

1991 (3)

I. N. Duling, “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 modelocked erbium fiber laser based on the amplifying Sagnac switch,” Electron. Lett. 27, 542-544 (1991).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068-2076 (1991).
[CrossRef]

Afanasjev, V. V.

Akhmediev, N.

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

Akhmediev, N. N.

Anderson, D.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[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, 894-896 (1993).
[CrossRef]

Antonelli, C.

Bale, B. G.

Berezhiani, V. I.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

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 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
[CrossRef]

Berntson, A.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Buckley, J.

Chang, K. S.

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

Chen, J.

Chong, A.

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 (1998).
[CrossRef]

Collings, B. C.

Cross, M. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851-1112 (1993).
[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 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
[CrossRef]

Cunningham, J. E.

Davey, R. P.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (IEE, 1992).

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]

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

DeSouza, E. A.

Drazin, P.

P. 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, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544-545 (1991).
[CrossRef]

I. N. Duling and M. L. Dennis, Compact Sources of Ultrafast Lasers (Cambridge U. Press, 1995).
[CrossRef]

Elgin, J. N.

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331-4341 (1993).
[CrossRef]

Evangelides, S.

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, 894-896 (1993).
[CrossRef]

Fujimoto, J. G.

Gordon, J. P.

Greer, E. J.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (IEE, 1992).

Haus, H. A.

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

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking 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, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068-2076 (1991).
[CrossRef]

Hohenberg, P. C.

M. C. Cross and P. C. Hohenberg, “Pattern formation outside of equilibrium,” Rev. Mod. Phys. 65, 851-1112 (1993).
[CrossRef]

Holmes, P.

Ilday, F. O.

Ippen, E. P.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking 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, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068-2076 (1991).
[CrossRef]

Jan, W. J.

Jirauschek, C.

Kapitula, T.

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

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

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58-103 (1998).
[CrossRef]

Kartner, F. X.

Kärtner, F. X.

Kaup, D. J.

B. A. Malomed and D. J. Kaup, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155-159 (1995).
[CrossRef]

Keller, U.

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 (1998).
[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 (1998).
[CrossRef]

Kutz, J. N.

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636-650 (2008).
[CrossRef]

B. G. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for mode-locking in the normal dispersive regime,” Opt. Lett. 33, 941-943 (2008).
[CrossRef]

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 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, “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]

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

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

J. N. Kutz, P. Holmes, S. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion managed breathers,” J. Opt. Soc. Am. B 15, 87-96 (1998).
[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 (1998).
[CrossRef]

J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
[CrossRef]

Laming, R. I.

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

Lisak, M.

D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
[CrossRef]

Malomed, B. A.

B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).

B. A. Malomed and D. J. Kaup, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155-159 (1995).
[CrossRef]

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 fiber 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 modelocked erbium fiber laser based on the amplifying Sagnac switch,” Electron. Lett. 27, 542-544 (1991).
[CrossRef]

Miklaszewski, R.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Morgner, U.

Nelson, B. P.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (IEE, 1992).

Payne, D. N.

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

Phillips, M. W.

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

Proctor, J.

J. Proctor and J. N. Kutz, “Averaged models for passive mode-locking using nonlinear mode-coupling,” Math. Comput. Simul. 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]

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

Renninger, W.

Richardson, D. J.

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

Sandstede, B.

J. N. Kutz and B. Sandstede, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636-650 (2008).
[CrossRef]

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

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58-103 (1998).
[CrossRef]

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, 894-896 (1993).
[CrossRef]

Skarka, V.

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Smith, K.

K. Smith, R. P. Davey, B. P. Nelson, and E. J. Greer, Fiber and Solid-State Lasers (IEE, 1992).

Sosnowski, T.

Soto-Crespo, J. M.

J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation,” J. Opt. Soc. Am. B 13, 1439-1449 (1996).
[CrossRef]

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, 894-896 (1993).
[CrossRef]

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 fiber soliton laser,” Opt. Commun. 165, 189-194 (1999).
[CrossRef]

Tamura, K.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking 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 fiber soliton laser,” Opt. Commun. 165, 189-194 (1999).
[CrossRef]

Tsuda, S.

Weinstein, M.

Whitham, G.

G. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974).

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Wise, F. W.

Electron. Lett. (4)

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[CrossRef]

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[CrossRef]

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

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[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
[CrossRef]

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[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 (1998).
[CrossRef]

Indiana J. Math. (1)

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana J. Math. 53, 1095-1126 (2004).
[CrossRef]

J. Opt. Soc. Am. B (8)

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

Opt. Commun. (1)

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

Opt. Express (4)

Opt. Lett. (6)

Phys. Lett. A (1)

J. M. Soto-Crespo, N. Akhmediev, and K. S. Chang, “Simultaneous existence of multiplicity of stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[CrossRef]

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J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331-4341 (1993).
[CrossRef]

W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber laser,” Phys. Rev. A 77, 023814 (2008).
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J. M. Soto-Crespo and N. Akhmediev, “Composite solitons and two-pulse generation in passively mode-locked lasers modeled by the complex quintic Swift-Hohenberg equation,” Phys. Rev. E 66, 066610 (2002).
[CrossRef]

V. Skarka, V. I. Berezhiani, and R. Miklaszewski, “Spatiotemporal soliton propagation in saturating nonlinear optical media,” Phys. Rev. E 56, 1080-1087 (1997).
[CrossRef]

Physica D (2)

T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58-103 (1998).
[CrossRef]

B. A. Malomed and D. J. Kaup, “The variational principle for nonlinear waves in dissipative systems,” Physica D 87, 155-159 (1995).
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[CrossRef]

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[CrossRef]

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P. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).

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Figures (7)

Fig. 1
Fig. 1

(a) Reduced WGAML model with anomalous dispersion showing a stable spiral node ( η 0 , A 0 , ω 0 ) = ( 2.1 , 0.31 , 3.75 ) as the attracting state for the laser. The physical parameters are D = 1 , γ = 8 , τ = 0.1 , δ = 1 , and g 0 = 1.5 . (b) Pulse parameters from the numerical simulation of Eqs. (7a, 7b, 7c) (solid curve) compared with the reduced model of Eqs. (12a, 12b, 12c) (dashed curve). The initial condition for both simulations is u ( 0 , T ) = 1.1 sech ( 0.7 T ) 1 + 0.5 i .

Fig. 2
Fig. 2

(a) Reduced QMML model with normal dispersion showing a stable node ( η 0 , A 0 , ω 0 ) = ( 0.9 , 10.3 , 0.25 ) as the attracting state for the laser. The physical parameters are D = 1 , γ = 1 , τ = 0.1 , δ = 1 , g 0 = 4 , β = 0.5 , and μ = 0.1 . (b) Pulse parameters from numerical simulation of Eq. (5) (solid curve) compared with the reduced model of Eqs. (12a, 12b, 12c) (dashed curve). The initial condition for both simulations is u ( 0 , T ) = 0.1 sech ( 0.1 T ) 1 + 3 i .

Fig. 3
Fig. 3

(a) Time- and (b) spectral-domain profiles for the mode-locked solutions (FPs) in Figs. 1, 2 for both anomalous (solid curve) and normal (dotted curve) dispersion regimes. Two key features are exemplified: the narrow time-domain mode-locked pulse in the anomalous regime and the broad time-domain squared-off spectrum of the normal regime.

Fig. 4
Fig. 4

(a) Breathing mode-locked solution in the WGAML model with anomalous dispersion with the parameters D = 1 , γ = 5 , C = 5 , τ = 0.1 , δ = 1 , and g 0 = 2.4 . The bifurcation from a stable pulse (stable spiral point) with g 0 = 1.5 to a breathing solution (limit cycle) is the first step toward multipulse mode locking [33]. (b) Solution parameter values from the numerical simulation of Eqs. (7a, 7b, 7c) (solid curve) compared with those from the reduced model of Eqs. (12a, 12b, 12c) (dashed curve).

Fig. 5
Fig. 5

Mode-locking dynamics in the WGAML model with a constant gain [ g ( z ) = g 0 ] for (a) anomalous and (b) normal dispersions. The globally attracting FPs present in this case are spiral nodes. The constant gain allows for an explicit analytic prediction of the stability dynamics. Here τ = 0.1 and δ = 1 with (a) g 0 = 3 and (b) g 0 = 2 .

Fig. 6
Fig. 6

(a) Reduced QMML model with the QMML ansatz and normal dispersion showing a stable node ( η f , B f , A f ) = ( 1.6 , 1.6 , 2.2 ) as the attracting state for the laser. The physical parameters are D = 0.4 , γ = 1 , τ = 0.2 , δ = 1 , g 0 = 1.5 , β = 0.7 , and μ = 0.2 . (b) Pulse parameters from the full numerical simulation of Eq. (5) (solid curve) compared with those obtained from the reduced model of Eqs. (20a, 20b, 20c) (dashed curve). The initial condition for both the full model and reduced model (gray square) is u ( 0 , T ) = 0.5 ( 1 + cosh ( T 2 ) ) 1 2 exp [ i 2 log ( 1 + cosh ( T 2 ) ) ] .

Fig. 7
Fig. 7

Spectral domain ( Ω ) profiles of the QMML solutions of Eq. (6) as a function of the parameter B. The distinct spectral shapes have all been experimentally observed [22, 23, 24], suggesting that the ansatz of Eq. (6) and its dynamics of Eqs. (20a, 20b, 20c) are critical in understanding the dynamics of certain laser systems. The values of B in the subfigures are (a) B = 4 , (b) B = 1, (c) B = 0 , and (d) B = 0.9 .

Tables (1)

Tables Icon

Table 1 Complete Characterization of the Mode-Locking Dynamics in the MML, QMML, and WGAML Models as a Function of the Gain g 0 with δ = 1 a

Equations (35)

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i u z + D 2 2 u t 2 + γ u 2 u + i δ u i g ( z ) ( 1 + τ 2 t 2 ) u = 0 ,
g ( Z ) = 2 g 0 1 + u 2 e 0 .
i u z + D 2 2 u t 2 + ( γ i β ) u 2 u + i δ u i g ( z ) ( 1 + τ 2 t 2 ) u = 0 ,
u ( z , t ) = η sech 1 + i A ω t exp ( i ϕ ) ,
i u z + D 2 2 u t 2 + ( γ i β ) u 2 u + i μ u 4 u + i δ u i g ( z ) ( 1 + τ 2 t 2 ) u = 0 ,
u ( z , t ) = η B + cosh ω t exp [ i A ln ( B + cosh ω t ) + i ϕ ] ,
i u z + D 2 2 u t 2 + γ u 2 u + C v + i δ 0 u i g ( z ) ( 1 + τ 2 t 2 ) u = 0 ,
i v z + C ( w + u ) + i δ 1 v = 0 ,
i w z + C v + i δ 2 w = 0 ,
L = i 2 ( u u Z * u * u Z ) D 2 u T 2 + γ 2 u 4 ,
L = 1 ω 2 [ Γ 0 η 2 ω A Z + 2 η 2 ω ϕ Z η 2 A ω Z D 3 η 2 ω 3 ( 1 + A 2 ) + 2 3 γ η 4 ω ] ,
R [ u ] = i { τ g ( Z ) 2 u T 2 + ( g ( Z ) δ ) u + β u 2 u μ u 4 u } .
δ L δ p j = 2 R { R [ u s ] u s * p j d T } ,
η z = η ω 2 ( D A τ g ( A 2 + 7 ) 3 ) + 3 η ( g δ + β η 2 + α 2 μ η 4 ) ,
ω z = ω 3 ( D A + τ g ( A 2 2 ) ) η 2 ( β ω α 1 μ ω η 2 ) ,
A z = η 2 ( γ β A α 3 μ A η 2 ) ω 2 ( 1 + A 2 ) ( D + τ g A ) ,
g ( z ) = 2 g 0 1 + 2 η 2 ω e 0 .
A 0 = 1 + 1 + 8 τ 2 g 2 2 τ g > 0 .
A 0 { 2 τ g + O ( τ 3 ) anomalous 1 τ g 0 + O ( τ ) normal .
η 0 = ω 0 = 3 ( g 0 δ ) τ g
λ 1 2 ( g 0 δ ) ( τ g 0 ) 2 ,
λ 2 , 3 3 ( g 0 δ ) τ g 0 ± i 7 ( g 0 δ ) τ g 0 ,
η 0 = 3 ( g 0 δ ) 2 ,
ω 0 = 3 2 τ g 0 ( g 0 δ ) ,
λ 1 4 ( g 0 δ ) τ g 0 ,
λ 2 , 3 1 2 ( g 0 δ ) ( τ g 0 ) 2 ± i 7 ( g 0 δ ) τ g 0 3 .
η z = f ( η , A , B ) α 1 ( B ) ,
B z = g ( η , A , B ) α 2 ( B ) ,
A z = h ( η , A , B ) α 2 ( B ) ,
f = 2 η ( g δ ) [ F S F + G ] η 3 A R 2 + τ g η 3 2 [ 2 Y 3 W R F ( G + F ) + ( W R ( G + F ) F ) A 2 ] + μ η 5 [ F F ( G + F ) 2 Z ] ,
g = 2 η ( g δ ) F + 2 ϵ η 3 [ S F F ( F G ) ] + η 2 A R 2 + τ g η 3 2 [ [ 2 Y 3 W R F ( F G ) ] [ R ( F G ) F + W ] A 2 ] + μ η 5 [ F F ( F G ) + 2 Z ] ,
h = 4 ( g δ ) A F ϵ η 2 A [ F 2 F + F ] 2 A F f η α 1 μ η 4 A 3 [ F + F F F ] τ g η 2 A ( A 2 + 1 ) 6 [ R + 3 Q F F ] + η 2 ( 1 + A 2 ) 4 [ 3 R F F R ] γ η 2 [ 3 F 2 F F ] .
F = d t Θ , G = ln Θ d t Θ , Q = t sinh 3 t d t Θ 4 ,
R = sinh 2 t d t Θ 2 , S = ln Θ d t Θ 2 , W = sinh 2 t ln Θ d t Θ 3 ,
Y = cosh t ln Θ d t Θ 2 , Z = ln Θ d t Θ 3

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