Abstract

A low-dimensional model is constructed via a variational formulation that characterizes the mode-locking dynamics in a laser cavity with a passive polarizer. The theoretical model accounts explicitly for the effects of the passive polarizer with a Jones matrix. In combination with the nonlinear interaction of the orthogonally polarized electromagnetic fields, the evolution of the mode-locked state reduces to the nonlinear interaction of the amplitude, width, and phase chirp. This model allows for an explicit analytic prediction of the steady-state mode-locked state (fixed point) and its corresponding stability. The stability analysis requires a center manifold reduction, which reveals that the solution decays to the mode-locked state on a timescale dependent on the gain bandwidth and the net cavity gain. Quantitative and qualitative agreement is achieved between the full governing model and the low-dimensional model, thus providing for an excellent design tool for characterizing and optimizing mode-locking performance.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  2. D. Spence, P. Kean, and W. Sibbett, “60-femtosecond pulse generation from a self mode-locked Ti:Sapphire laser,” Opt. Lett. 16, 42-44 (1991).
    [CrossRef] [PubMed]
  3. I. N. Duling III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge U. Press, 1995).
    [CrossRef]
  4. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
    [CrossRef]
  5. T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292-1294 (1992).
    [CrossRef] [PubMed]
  6. L. Spinelli, B. Couilland, N. Goldblatt, and D. K. Negus, in Conference on Lasers and Electro-Optics, Vol. 10, OSA Technical Digest Series (Optical Society of America, 1991), paper CPDP7.
  7. F. X. Kärtner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. M. E. Fermann, M. J. Andrejco, Y. Silberberg, and M. L. Stock, “Passive mode locking by using nonlinear polarization evolution in a polarization-maintaining erbium-doped fiber,” Opt. Lett. 18, 894-896 (1993).
    [CrossRef] [PubMed]
  13. I. N. Duling, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544-545 (1991).
    [CrossRef]
  14. 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]
  15. M. L. Dennis and I. N. Duling, “High repetition rate figure eight laser with extracavity feedback,” Electron. Lett. 28, 1894-1896 (1992).
    [CrossRef]
  16. F. O. Ölday, F. W. Wise, and T. Sosnowski, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Opt. Lett. 27, 1531-1533 (2002).
    [CrossRef]
  17. 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 (1994).
    [CrossRef]
  18. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
    [CrossRef]
  19. 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, 2005), pp. 241-265.
    [CrossRef]
  20. J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013-2015 (2005).
    [CrossRef] [PubMed]
  21. J. Proctor and J. N. Kutz, “Nonlinear mode-coupling for passive mode-locking: applications of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933-8950 (2005).
    [CrossRef] [PubMed]
  22. 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]
  23. H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
    [CrossRef]
  24. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
    [CrossRef]
  25. H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
    [CrossRef]
  26. A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
    [CrossRef] [PubMed]
  27. 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]
  28. T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana J. Math. 53, 1095-1126 (2004).
    [CrossRef]
  29. B. 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] [PubMed]
  30. B. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers,” J. Opt. Soc. Am. B 25, 1763-1770 (2008).
    [CrossRef]
  31. B. C. Collings, S. T. Cundiff, N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17, 354-365 (2000).
    [CrossRef]
  32. 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]
  33. D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
    [CrossRef]
  34. D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
    [CrossRef]
  35. A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
    [CrossRef]
  36. K. Spaulding, D. Yong, A. Kim, and J. N. Kutz, “Nonlinear dynamics of mode-locking optical fiber ring lasers,” J. Opt. Soc. Am. B 19, 1045-1054 (2002).
    [CrossRef]
  37. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 26742682 (1989).
    [CrossRef]
  38. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174-176 (1987).
    [CrossRef]
  39. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
    [CrossRef] [PubMed]
  40. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
    [CrossRef]
  41. W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [CrossRef]
  42. S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed soliton in fiber links with in-line filtering presented in the basis of chirped Gauss-Hermite functions,” J. Opt. Soc. Am. B 16, 1321-1331 (1999).
    [CrossRef]
  43. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
    [CrossRef]
  44. I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” Pisma v JETP 63, 814-819 (1996).
  45. 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]
  46. T. Ueda and W. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563-571 (1990).
    [CrossRef] [PubMed]
  47. D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53-64 (1991).
    [CrossRef]
  48. 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] [PubMed]
  49. B. A. Malomed, “Variational methods in nonlinear fiber optics and related fields,” Prog. Opt. 43, 69-191 (2002).
  50. G. Whitham, Linear and Nonlinear Waves (Wiley, 1974).
  51. M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
    [CrossRef]
  52. P. G. Drazin, Nonlinear Systems (Cambridge, 1992).
  53. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).
  54. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, (McGraw-Hill, 1978).
  55. R. P. Davey, N. Langford, and A. I. Ferguson, “Interacting solutions in erbium fibre laser,” Electron. Lett. 27, 1257-1259 (1991).
    [CrossRef]
  56. M. J. Lederer, B. Luther-Davies, 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]
  57. 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]
  58. M. Lai, J. Nicholson, and W. Rudolph, “Multiple pulse operation of a femtosecond Ti:sapphire laser,” Opt. Commun. 142, 45-49 (1997).
    [CrossRef]
  59. 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]
  60. H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128-134 (1998).
    [CrossRef]
  61. B. Sandstede and J. N. Kutz, “Theory of passive harmonic mode-locking using waveguide arrays,” Opt. Express 16, 636-650 (2008).
    [CrossRef] [PubMed]

2008 (6)

2007 (1)

2006 (2)

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

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

2005 (4)

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013-2015 (2005).
[CrossRef] [PubMed]

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

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[CrossRef]

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

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)

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 (5)

2001 (2)

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

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

2000 (4)

A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
[CrossRef]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

B. C. Collings, S. T. Cundiff, N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17, 354-365 (2000).
[CrossRef]

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

1999 (4)

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]

S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed soliton in fiber links with in-line filtering presented in the basis of chirped Gauss-Hermite functions,” J. Opt. Soc. Am. B 16, 1321-1331 (1999).
[CrossRef]

M. J. Lederer, B. Luther-Davies, 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]

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]

1998 (3)

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]

H. Kitano and S. Kinoshita, “Stable multipulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Commun. 157, 128-134 (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]

1997 (2)

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 (2)

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” Pisma v JETP 63, 814-819 (1996).

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
[CrossRef]

1995 (2)

1994 (2)

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]

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 (1994).
[CrossRef]

1993 (1)

1992 (3)

T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292-1294 (1992).
[CrossRef] [PubMed]

M. L. Dennis and I. N. Duling, “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 (5)

D. Spence, P. Kean, and W. Sibbett, “60-femtosecond pulse generation from a self mode-locked Ti:Sapphire laser,” Opt. Lett. 16, 42-44 (1991).
[CrossRef] [PubMed]

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

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

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53-64 (1991).
[CrossRef]

1990 (1)

T. Ueda and W. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563-571 (1990).
[CrossRef] [PubMed]

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 26742682 (1989).
[CrossRef]

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174-176 (1987).
[CrossRef]

1975 (1)

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

Akhmediev, N.

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.

Antonelli, C.

Bale, B.

Bale, B. G.

Bender, C. M.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, (McGraw-Hill, 1978).

Bergman, K.

B. C. Collings, S. T. Cundiff, N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17, 354-365 (2000).
[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]

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]

Brabec, T.

Brunel, M.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

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]

Chartier, T.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[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.

Couilland, B.

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

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]

Cundiff, S. T.

Cunningham, J. E.

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]

Dennis, M. L.

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

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

DeSouza, E. A.

Drazin, P. G.

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

Drummond, P.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

Duling, I. N.

M. L. Dennis and I. N. Duling, “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 III and M. L. Dennis, Compact Sources of Ultrashort Pulses (Cambridge U. Press, 1995).
[CrossRef]

Evangelides, 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.

Franco, P.

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
[CrossRef]

Gabitov, I.

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” Pisma v JETP 63, 814-819 (1996).

Goldblatt, N.

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

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

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]

Guckenheimer, J.

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).

Haboucha, A.

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

Hideur, A.

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Holmes, P.

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]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).

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.

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]

Jagadish, C.

Jan, W. J.

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-modelocking equation,” J. Opt. Soc. Am. B 19, 740-746 (2002).
[CrossRef]

Kärtner, F. X.

Kath, W.

T. Ueda and W. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563-571 (1990).
[CrossRef] [PubMed]

Kath, W. L.

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53-64 (1991).
[CrossRef]

Kean, P.

Keller, U.

Kim, A.

K. Spaulding, D. Yong, A. Kim, and J. N. Kutz, “Nonlinear dynamics of mode-locking optical fiber ring lasers,” J. Opt. Soc. Am. B 19, 1045-1054 (2002).
[CrossRef]

A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
[CrossRef]

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.

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]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Komarov, A.

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
[CrossRef] [PubMed]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[CrossRef]

Kopf, D.

Krausz, F.

Kutz, J. N.

B. 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] [PubMed]

B. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers,” J. Opt. Soc. Am. B 25, 1763-1770 (2008).
[CrossRef]

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
[CrossRef]

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

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: applications of waveguide arrays, dual-core fibers, and/or fiber arrays,” Opt. Express 13, 8933-8950 (2005).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013-2015 (2005).
[CrossRef] [PubMed]

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

K. Spaulding, D. Yong, A. Kim, and J. N. Kutz, “Nonlinear dynamics of mode-locking optical fiber ring lasers,” J. Opt. Soc. Am. B 19, 1045-1054 (2002).
[CrossRef]

A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
[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, “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, 2005), pp. 241-265.
[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]

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]

Langford, N.

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

Leblond, H.

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
[CrossRef] [PubMed]

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[CrossRef]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[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]

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]

Liu, A. Q.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

Luther-Davies, B.

Malomed, B. A.

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

Man, W. S.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

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]

Melnikov, I. V.

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

Menyuk, C. R.

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 26742682 (1989).
[CrossRef]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174-176 (1987).
[CrossRef]

Midrio, M.

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
[CrossRef]

Mihalache, D.

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

Muraki, D.

A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
[CrossRef]

Muraki, D. J.

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53-64 (1991).
[CrossRef]

Negus, D. K.

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

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]

Ölday, F. O.

Orszag, S. A.

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, (McGraw-Hill, 1978).

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]

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]

Proctor, J.

Renninger, W.

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]

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]

Salhi, M.

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
[CrossRef] [PubMed]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Sanchez, F.

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
[CrossRef] [PubMed]

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[CrossRef]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Sandstede, B.

Sazonov, S. V.

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

Shapiro, E. G.

Sibbett, W.

Silberberg, Y.

Sosnowski, T.

Soto-Crespo, J. M.

Spaulding, K.

Spence, D.

Spielmann, Ch.

Spinelli, L.

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

Stock, M. L.

Tam, H. Y.

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

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]

Tan, H. H.

Tang, D. Y.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

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 (1998).
[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]

Turitsyn, S. K.

Ueda, T.

T. Ueda and W. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563-571 (1990).
[CrossRef] [PubMed]

Wabnitz, S.

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
[CrossRef]

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]

Whitham, G.

G. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

Wise, F.

Wise, F. W.

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]

Yong, D.

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]

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]

Zhao, B.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

Zhao, L. M.

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

Electron. Lett. (5)

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 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, “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]

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. (6)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 26742682 (1989).
[CrossRef]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174-176 (1987).
[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]

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

A. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465-471 (2000).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
[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 (9)

K. Spaulding, D. Yong, A. Kim, and J. N. Kutz, “Nonlinear dynamics of mode-locking optical fiber ring lasers,” J. Opt. Soc. Am. B 19, 1045-1054 (2002).
[CrossRef]

B. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion fiber lasers,” J. Opt. Soc. Am. B 25, 1763-1770 (2008).
[CrossRef]

B. C. Collings, S. T. Cundiff, N. Akhmediev, J. M. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber laser: experiment,” J. Opt. Soc. Am. B 17, 354-365 (2000).
[CrossRef]

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 (1994).
[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]

B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
[CrossRef]

S. K. Turitsyn and E. G. Shapiro, “Dispersion-managed soliton in fiber links with in-line filtering presented in the basis of chirped Gauss-Hermite functions,” J. Opt. Soc. Am. B 16, 1321-1331 (1999).
[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]

M. J. Lederer, B. Luther-Davies, 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]

Opt. Commun. (5)

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]

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]

Opt. Express (3)

Opt. Lett. (9)

A. Haboucha, H. Leblond, M. Salhi, A. Komarov, and F. Sanchez, “Coherent soliton pattern formation in a fiber laser,” Opt. Lett. 33, 524-526 (2008).
[CrossRef] [PubMed]

J. Proctor and J. N. Kutz, “Passive mode-locking by use of waveguide arrays,” Opt. Lett. 30, 2013-2015 (2005).
[CrossRef] [PubMed]

B. 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] [PubMed]

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]

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

D. Spence, P. Kean, and W. Sibbett, “60-femtosecond pulse generation from a self mode-locked Ti:Sapphire laser,” Opt. Lett. 16, 42-44 (1991).
[CrossRef] [PubMed]

T. Brabec, Ch. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292-1294 (1992).
[CrossRef] [PubMed]

M. E. Fermann, M. J. Andrejco, Y. Silberberg, and M. L. Stock, “Passive mode locking by using nonlinear polarization evolution in a polarization-maintaining erbium-doped fiber,” Opt. Lett. 18, 894-896 (1993).
[CrossRef] [PubMed]

F. O. Ölday, F. W. Wise, and T. Sosnowski, “High-energy femtosecond stretched-pulse fiber laser with a nonlinear optical loop mirror,” Opt. Lett. 27, 1531-1533 (2002).
[CrossRef]

Phys. Rev. A (6)

D. Y. Tang, L. M. Zhao, B. Zhao, and A. Q. Liu, “Mechanism of multi-soliton formation and soliton energy quantization in passively mode-locked fibre lasers,” Phys. Rev. A 72, 043816 (2005).
[CrossRef]

D. Y. Tang, W. S. Man, H. Y. Tam, and P. Drummond, “Observation of bound states of solitons in a passively mode-locked fibre soliton laser,” Phys. Rev. A 64, 033814 (2001).
[CrossRef]

H. Leblond, M. Salhi, A. Hideur, T. Chartier, M. Brunel, and F. Sanchez, “Experimental and theoretical study of the passively mode-locked ytterbium-doped double-clad fiber laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

H. Leblond, S. V. Sazonov, I. V. Melnikov, D. Mihalache, and F. Sanchez, “Few-cycle nonlinear optics of multicomponent media,” Phys. Rev. A 74, 063815 (2006).
[CrossRef]

T. Ueda and W. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563-571 (1990).
[CrossRef] [PubMed]

W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[CrossRef]

Phys. Rev. E (2)

M. Midrio, S. Wabnitz, and P. Franco, “Perturbation theory for coupled nonlinear Schrödinger equations,” Phys. Rev. E 54, 5743-5751 (1996).
[CrossRef]

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
[CrossRef]

Physica D (1)

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53-64 (1991).
[CrossRef]

Pisma v JETP (1)

I. Gabitov and S. K. Turitsyn, “Breathing solitons in optical fiber links,” Pisma v JETP 63, 814-819 (1996).

Pramana, J. Phys. (1)

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

Proc. Natl. Acad. Sci. U.S.A. (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Prog. Opt. (1)

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

SIAM Rev. (1)

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

Other (7)

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

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

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, 2005), pp. 241-265.
[CrossRef]

G. Whitham, Linear and Nonlinear Waves (Wiley, 1974).

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

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, 1983).

C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, (McGraw-Hill, 1978).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Experimental configuration of a ring laser cavity that includes a passive polarizer, erbium-doped amplification, output coupler, and polarization controllers. The erbium-doped section of fiber is fused with standard single-mode fiber and is treated in a distributed fashion. The polarization controllers can be used to tune the cavity birefringence. For a short cavity, the polarizer suppresses the polarization rotation and phase-slip generated from the birefringence so that stable mode locking is achieved.

Fig. 2
Fig. 2

Ideal mode-locking behavior in the laser cavity with a passive polarizer. Here, the white-noise initial condition settles to the attracting mode-locked on the order of thousands of round trips. For this computation, g 0 = 0.1 , τ = 0.1 , γ = 0.1 , K = 0.1 , and α = 0.01 with the polarizer set so that θ = 0.28 π .

Fig. 3
Fig. 3

Depiction of the characteristic oscillatory attenuation of the maximum of the U and V fields to the mode-locked state along with the energy and gain equilibration dynamics corresponding to Fig. 2. In what follows, the characteristic timescale of the oscillatory decay of the fields will be explicitly calculated and shown to be on a slow scale compared to the round-trip cavity time. The initial condition is given by (5) with ψ = 0.7 , P = 0.27 π , β = 0 and η = ω = 1 .

Fig. 4
Fig. 4

Low-dimensional mode-locking behavior [Eqs. (9, 10)] in the laser cavity with a passive polarizer. Here, the parameters chosen are identical to the full evolution shown in Figs. 2, 3 with g 0 = 0.1 , τ = 0.1 , γ = 0.1 , K = 0.1 , L = 1 , and α = 0.01 with the polarizer set so that θ = 0.28 π . Note that the low-dimensional mode-locking dynamics agree quantitatively with the governing evolution. Specifically, the fast oscillatory behavior and slow attenuation to the mode-locked state are commensurate with the full evolution equations. The initial conditions are the same as in Fig. 3. The steady-state mode-locked solution is achieved for ( η , ω , β ) = ( 0.1975 , 0.4444 , 6.3 × 10 4 ) , which should be compared to the full evolution values of ( η , ω , β ) = ( 0.1982 , 0.4461 , 14 × 10 4 ) . Thus the 5 × 5 system reproduces the pulse width and height to within 1 % .

Fig. 5
Fig. 5

Depiction of the low-dimensional intracavity dynamics once mode locking has been achieved. The parameters considered are identical to those in Fig. 4. The evolution dynamics shows the explicit jump in the amplitude, polarization, and phase-slip variables induced by the polarizer and its jump condition (10).

Fig. 6
Fig. 6

Comparison of the full governing evolution model, Eqs. (1, 4), (dotted lines) with the three-degree of freedom low-dimensional reduction, Eq. (12) (solid lines). The mode-locking dynamics in the laser cavity is accurately captured with the three-degree of freedom system with the primary difference being in a phase-slip that develops due to the slightly different periods of oscillation in the two systems. Regardless, all the critical features of the mode-locking process, including the oscillatory decay and its associated timescales, are captured very well with the low-dimensional model considered. The steady-state mode-locked solution is achieved (after several thousand round trips) for ( η , ω , β ) = ( 0.2421 , 0.4921 , 90 × 10 4 ) , which should be compared to the full evolution values of ( η , ω , β ) = ( 0.1982 , 0.4461 , 14 × 10 4 ) . Thus the 3 × 3 system reproduces the pulse width and height to within 10 % and 20 % , respectively. The 5 × 5 system, Eq. (9), has not been included in the plot, since it reproduces the pulse width and height of the full governing equations to within 1 % , thus producing a line that is directly on top of the dotted curve.

Fig. 7
Fig. 7

Three-dimensional phase-space evolution for the low-dimensional model (12) (top) and the full governing equations (1) with Eq. (4) (bottom). The three-dimensional approximation to the evolution dynamics is remarkably good as the fixed point and oscillation period of the decay is quantitatively close to the full evolution equation aside from an initial transient behavior in the full equations.

Fig. 8
Fig. 8

Comparison of the steady-state mode-locked solution for the full governing evolution model (1, 4) (dotted lines) with the three-degree of freedom low-dimensional reduction, Eq. (12) (solid lines). The mode-locking dynamics in the laser cavity is accurately captured with the three-degree-of-freedom system. The steady-state mode-locked solution is achieved for ( η , ω , β ) = ( 0.2421 , 0.4921 , 90 × 10 4 ) , which should be compared to the full evolution values of ( η , ω , β ) = ( 0.1982 , 0.4461 , 14 × 10 4 ) . Thus the 3 × 3 system reproduces the pulse width and height to within 10 % and 20 % , respectively. The comparison between the transient responses of both models is given in Fig. 6.

Fig. 9
Fig. 9

Laser cavity dynamics, Eq. (1) and Eq. (4), with the same parameters as those in Fig. 2 with the exception that the birefringence strength has been increased to K = 0.5 . The increased birefringence strength and the corresponding phase-slip generated between the orthogonally polarized components prevents mode-locking from occurring.

Fig. 10
Fig. 10

Onset of stable multipulsing mode locking with increasing amplifier pump strength. Here, the parameters are those in Fig. 2 with the exception that the gain strength has been increased to g 0 = 0.2 . The increased gain strength leads to the formation of two identical mode-locked pulses. Further increasing the gain can lead to more identical pulses in the laser cavity. The low-dimensional model remains an accurate theoretical description provided that the gain is appropriately modified [61].

Fig. 11
Fig. 11

Mode-locking dynamics in a laser cavity with a long cavity period. Here, the parameters are those in Fig. 2 with the exception that the polarizer is now applied at Z = 5 , 10 , 15 , instead of at Z = 1 , 2 , 3 , . The increased cavity length makes more pronounced the discrete effect of the polarizer. In this case the low-dimensional model, Eq. (9), would need to be considered as the reduction to Eq. (12) is no longer valid.

Equations (45)

Equations on this page are rendered with MathJax. Learn more.

i U Z + 1 2 2 U T 2 K U + ( U 2 + A V 2 ) U + B V 2 U * = i R U ,
i V Z + 1 2 2 V T 2 + K V + ( A U 2 + V 2 ) V + B U 2 V * = i R V ,
R σ = G ( Z ) ( 1 + τ 2 T 2 ) σ γ σ
G ( Z ) = 2 g 0 1 + ( U 2 + V 2 ) d T .
( U + V + ) = ( c 2 + α s 2 ( 1 α ) s c ( 1 α ) s c α c 2 + s 2 ) ( U V ) .
( U V ) = ( exp ( i ψ 2 ) cos P exp ( + i ψ 2 ) sin P ) η sech ω T exp [ i ( β T 2 + φ 2 ) ] ,
L = L ( U , U * , V , V * ) d T ,
L = i 2 ( U * U Z U U Z * ) + i 2 ( V * V Z V V Z * ) 1 2 U T 2 1 2 V T 2 + 1 2 U 4 + 1 2 V 4 + A U 2 V 2 K ( U 2 V 2 ) + B 2 ( V 2 U * 2 + U 2 V * 2 ) .
δ L δ ρ i = 2 R { ( i R U U * ρ i + i R V V * ρ i ) d T } ,
d η d Z = 1 15 π 2 ω ( 2 η + ω ) × [ 2 η ( 6 g 0 τ π 4 β 2 30 π 2 ( γ + β ) η ω + 15 π 2 ( 2 g 0 γ β ) ω 2 10 g 0 τ ( 12 + π 2 ) ω 4 ) ] ,
d ω d Z = 32 π 2 g 0 τ β 2 240 g 0 τ ω 4 30 π 2 β ω ( 2 η + ω ) 15 π 2 ( 2 η + ω ) ,
d β d Z = 1 6 π 2 ( 2 η + ω ) × [ 16 ( 3 + π 2 ) g 0 τ β ω 3 + 12 π 2 β 2 ( 2 η + ω ) 12 ω 2 ( 2 η + ω ) ( ω 2 ( 1 B sin 2 ( 2 P ) sin 2 ψ ) η ) ] ,
d P d Z = 1 3 B η sin ( 2 P ) sin ( 2 ψ ) ,
d ψ d Z = 2 K 4 B η 3 cos ( 2 P ) sin 2 ψ .
η + = f 1 2 + f 2 2 ,
P + = tan 1 f 1 f 2 ,
ψ + = arg [ f 1 f 2 tan P + ] ,
f 1 = η [ ( 1 α ) s c cos P exp ( i ψ 2 ) + ( α c 2 + s 2 ) sin P exp ( i ψ 2 ) ] ,
f 2 = η [ ( α c 2 + s 2 ) cos P exp ( i ψ 2 ) + ( 1 α ) s c sin P exp ( i ψ 2 ) ] .
d η d Z = 1 15 π 2 ω ( 2 η + ω ) × [ 2 η ( 6 g 0 τ π 4 β 2 30 π 2 ( γ + β ) η ω + 15 π 2 ( 2 g 0 γ β ) ω 2 10 g 0 τ ( 12 + π 2 ) ω 4 ) ] ,
d ω d Z = 32 π 2 g 0 τ β 2 240 g 0 τ ω 4 30 π 2 β ω ( 2 η + ω ) 15 π 2 ( 2 η + ω ) ,
d β d Z = 6 ( 2 η + ω ) [ ω 2 ( ω 2 η ) π 2 β 2 ] 8 ( 3 + π 2 ) g 0 τ β ω 3 3 π 2 ( 2 η + ω ) .
d η 0 d Z = 2 β 0 η 0 ,
d ω 0 d Z = 2 β 0 ω 0 ,
d β 0 d Z = 2 π 2 ( β 0 2 π 2 + η 0 ω 0 2 ω 0 4 ) ,
J 0 = ( 0 0 2 a 2 0 0 2 a 2 a 2 π 2 4 a 3 π 2 0 ) ,
η ¯ = a 2 + ε η 1 + ε 2 η 2 + ε 3 η 3 + ,
ω ¯ = a + ε ω 1 + ε 2 ω 2 + ε 3 ω 3 + ,
β ¯ = 0 + ε β 1 + ε 2 β 2 + ε 3 β 3 + .
β 1 = 0 ,
a = 1 2 r ( 2 g r ) ,
η 1 = 2 a ω 1 .
β 2 = ( 2 g r ) 2 b π 2 r ,
ω 1 = g ( 2 g r ) 2 b 12 r 3 .
η 1 = g ( 2 g r ) 3 b 12 r 4 .
η 2 = 5 g 2 ( 2 g r ) 4 b 2 144 r 6 ,
ω 2 = g 2 ( 2 g r ) 3 b 2 36 r 5 .
J = J 0 + ε J 1 + ε 2 J 2 + ,
λ 3 + c 2 λ 2 + c 1 λ + c 0 + O ( ε 3 ) = 0 ,
c 0 = ε ( 2 g r ) 5 ( 8 ε g 2 b 2 ε g r b r 2 ( 12 + ε b ) ) 48 g π 2 r 5 ,
c 1 = 1 72 π 2 r 8 × [ ( 2 g r ) 4 ( 18 r 4 + 12 ε g r 3 b + 28 ε 2 g 4 b 2 28 ε 2 g 3 r b 2 + ε g 2 r 2 b ( 7 ε b 24 ) ) ] ,
c 2 = 1 12 g π 2 r × [ ε ( 2 g r ) ( 12 ε g ( 2 g r ) b + π 2 ( 8 ε g 2 b 6 ε g r b + r 2 ( 12 + ε b ) ) ) ] .
λ = λ ( 0 ) + ε λ ( 1 ) + ε 2 λ ( 2 ) + ,
λ 1 = 0 ε r ( 2 g r ) g + O ( ε 2 ) ,
λ 2 , 3 = ± i 2 a 2 π ε i g ( 2 g r ) 3 b 6 π r 4 + ε 2 h + O ( ε 3 ) ,

Metrics