Abstract

We develop an iterative (averaging) method to characterize the mode-locking dynamics in a laser cavity mode locked with a combination of wave plates and a passive polarizer. The model explicitly accounts for the effects of self- and cross-phase modulation, an arbitrary alignment of the fast- and slow-axes of the fiber with the wave plates and polarizer, fiber birefringence, saturable gain, and chromatic dispersion. The general averaging scheme results in the cubic-quintic Ginzburg–Landau equation at the leading order and the Swift–Hohenberg equation at the next order. An extensive comparison between the full model and the averaged equations shows a quantitative agreement that allows for characterizing the stability and operating regimes of the laser cavity.

© 2009 Optical Society of America

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    [Crossref]
  2. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
    [Crossref]
  3. 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]
  4. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse mode-locking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
    [Crossref]
  5. M. E. Fermann, M. J. Andrejco, Y. Silberberg, and M. L. Stock, “Passive mode-locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,” Opt. Lett. 29, 447-449 (1993).
  6. M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502-504 (1991).
    [Crossref] [PubMed]
  7. M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
    [Crossref]
  8. 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 lasers,” Phys. Rev. A 65, 063811 (2002).
    [Crossref]
  9. A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
    [Crossref]
  10. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 025604 (2005).
    [Crossref]
  11. 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]
  12. 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]
  13. 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]
  14. 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]
  15. 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]
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  18. W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
    [Crossref]
  19. P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
    [Crossref]
  20. 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]
  21. 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]
  22. E. Ding and J. N. Kutz, “Stability analysis of the mode-locking dynamics in a laser cavity with a passive polarizer,” J. Opt. Soc. Am. B 26, 1400-1411 (2009).
    [Crossref]
  23. B. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for ultrafast mode locking in the normal dispersive regime,” Opt. Lett. 33, 941-943 (2008).
    [Crossref] [PubMed]
  24. 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]
  25. 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]
  26. T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 53, 1095-1126 (2004).
    [Crossref]
  27. G. Strang, “On the construction and comparison of difference schemes,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).
  28. N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
    [Crossref]
  29. 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]
  30. J. M. Soto-Crespo, N. Akhmediev, and K. S. Chiang, “Simultaneous existence of a multiplicity of a stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
    [Crossref]
  31. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).
  32. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674-2682 (1989).
    [Crossref]
  33. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174-176 (1987).
    [Crossref]
  34. E. Farnum and J. N. Kutz, “Multi-frequency mode-locked lasers,” J. Opt. Soc. Am. B 25, 1002-1010 (2008).
    [Crossref]
  35. A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994).
  36. B. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
    [Crossref]
  37. J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
    [Crossref]
  38. D. Anderson, M. Lisak, and A. Berntson, “A variational approach to nonlinear evolution equations in optics,” Pramana, J. Phys. 57, 917-936 (2001).
    [Crossref]

2009 (1)

2008 (7)

2006 (2)

2005 (3)

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

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 Univ. Math. J. 53, 1095-1126 (2004).
[Crossref]

2002 (4)

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]

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. 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]

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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

2001 (4)

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]

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

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (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 (3)

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]

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 (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]

1997 (1)

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

1994 (1)

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

1993 (2)

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

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (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. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[Crossref]

1991 (1)

1989 (1)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674-2682 (1989).
[Crossref]

1987 (1)

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

1968 (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

Afanasjev, V. V.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

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. Chiang, “Simultaneous existence of a multiplicity of a stable and unstable solitons in dissipative systems,” Phys. Lett. A 291, 115-123 (2001).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[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]

Akhmediev, N. N.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

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. Silberberg, and M. L. Stock, “Passive mode-locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,” Opt. Lett. 29, 447-449 (1993).

Bale, B.

Bergman, K.

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]

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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Buckley, J.

Burch, J. M.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994).

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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Chiang, K. S.

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

Chong, A.

Collings, B. C.

Cundiff, S. T.

Ding, E.

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]

Farnum, E.

Fermann, M. E.

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

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[Crossref]

M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502-504 (1991).
[Crossref] [PubMed]

Firth, W. J.

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
[Crossref]

Gerrard, A.

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994).

Haberl, F.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[Crossref]

M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502-504 (1991).
[Crossref] [PubMed]

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]

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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Hofer, M.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[Crossref]

M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502-504 (1991).
[Crossref] [PubMed]

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]

Kapitula, T.

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 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]

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]

Knox, W. H.

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]

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (2005).
[Crossref]

Kutz, J. N.

E. Ding and J. N. Kutz, “Stability analysis of the mode-locking dynamics in a laser cavity with a passive polarizer,” J. Opt. Soc. Am. B 26, 1400-1411 (2009).
[Crossref]

B. Bale, J. N. Kutz, A. Chong, W. Renninger, and F. Wise, “Spectral filtering for ultrafast 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. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
[Crossref]

E. Farnum and J. N. Kutz, “Multi-frequency mode-locked lasers,” J. Opt. Soc. Am. B 25, 1002-1010 (2008).
[Crossref]

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

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

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]

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]

Langridgea, P. E.

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
[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]

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

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (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 lasers,” Phys. Rev. A 65, 063811 (2002).
[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]

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

McDonald, G. S.

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
[Crossref]

Menyuk, C. R.

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674-2682 (1989).
[Crossref]

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

Ober, M. H.

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[Crossref]

M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502-504 (1991).
[Crossref] [PubMed]

Renninger, W.

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 lasers,” 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]

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

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

Sandstede, B.

T. Kapitula, J. N. Kutz, and B. Sandstede, “The Evans function for nonlocal equations,” Indiana Univ. Math. J. 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]

Schmidt, A. J.

Silberberg, Y.

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

Soto-Crespo, J. M.

J. M. Soto-Crespo 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]

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

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

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

Spaulding, K.

Stock, M. L.

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

Strang, G.

G. Strang, “On the construction and comparison of difference schemes,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

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

Tamura, K.

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

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

Tang, D. Y.

D. Y. Tang, 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 fiber soliton laser,” Opt. Commun. 165, 189-194 (1999).
[Crossref]

Town, G.

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

Wabnitz, S.

J. M. Soto-Crespo, N. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
[Crossref]

Wise, F.

Yong, D.

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. (1)

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]

IEEE J. Quantum Electron. (5)

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]

M. Hofer, M. H. Ober, F. Haberl, and M. E. Fermann, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720-728 (1992).
[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]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674-2682 (1989).
[Crossref]

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

IEEE J. Sel. Top. Quantum Electron. (1)

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

Indiana Univ. Math. J. (1)

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

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

Opt. Commun. (2)

P. E. Langridgea, G. S. McDonald, W. J. Firth, and S. Wabnitz, “Self-sustained mode locking using induced nonlinear birefringence in optical fiber,” Opt. Commun. 97, 178-182 (1993).
[Crossref]

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 (1)

Opt. Lett. (4)

Phys. Lett. A (1)

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

Phys. Rev. A (5)

W. Renninger, A. Chong, and F. Wise, “Dissipative solitons in normal-dispersion fiber lasers,” Phys. Rev. A 77, 023814 (2008).
[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 lasers,” Phys. Rev. A 65, 063811 (2002).
[Crossref]

A. Komarov, H. Leblond, and F. Sanchez, “Multistability and hysteresis phenomena in passively mode-locked fiber lasers,” Phys. Rev. A 71, 053809 (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]

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]

Phys. Rev. E (4)

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

N. Akhmediev, J. M. Soto-Crespo, and G. Town, “Pulsating solitons, chaotic solitons, period doubling, and pulse coexistence in mode-locking lasers: complex Ginzburg-Landau equation approach,” Phys. Rev. E 63, 056602 (2001).
[Crossref]

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. N. Akhmediev, V. V. Afanasjev, and S. Wabnitz, “Pulse solutions of the cubic-quintic complex Ginzburg-Landau equation in the case of normal dispersion,” Phys. Rev. E 55, 4783-4796 (1997).
[Crossref]

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]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

G. Strang, “On the construction and comparison of difference schemes,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 5, 506-517 (1968).

SIAM Rev. (1)

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

Other (2)

A. Gerrard and J. M. Burch, Introduction to Matrix Methods in Optics (Dover, 1994).

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic Press, 2001).

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Figures (11)

Fig. 1
Fig. 1

Experimental configuration of a ring laser cavity that includes quarter-wave plates (QWP), passive polarizer, half-wave plate (HWP), ytterbium-doped amplification, and output coupler. The Yb-doped section of the fiber is fused with standard single-mode fiber and treated in a distributed fashion. The birefringence of the system can be adjusted easily by a pair of polarization controllers (not shown in the figure). The angles α 1 , α 2 , α 3 , and α 4 can all be measured with reasonable accuracy.

Fig. 2
Fig. 2

Spectrum of the linearized CQGLE about the numerical solution of the form in Eq. (23), where ω = 0.4275 , B = 1.6603 , A = 1.3819 , and φ = 0.0491 z . The parameters are α 1 = 0.1 π , α 2 = 0.554 π , α 3 = 0.23 π , α p = 0.43 π , g 0 = 0.7 , τ = 0.2 K = 0.1 , and D = 0.3 . Top: constant gain showing an unstable eigenvalue with R { λ } > 0 . Bottom: saturable gain showing its stabilizing influence since all R { λ } 0 .

Fig. 3
Fig. 3

The evolutions of the peak amplitude of | q | governed by the CQGLE (dotted curves) and CNLS (solid curves) for different values of α j . Top row (left to right): ( α 2 , α 3 , α p ) = ( 0.34 π , 0.23 π , 0.3 π ) and α 1 = 0.05 π , 0.15 π , 0.35 π , and 0.5 π . Second row (left to right): ( α 1 , α 3 , α p ) = ( 0.4 π , 0.23 π , 0.3 π ) and α 2 = 0 , 0.3 π , 0.35 π , and 0.5 π . Third row (left to right): ( α 1 , α 2 , α p ) = ( 0.4 π , 0.34 π , 0.3 π ) and α 3 = 0 , 0.1 π , 0.4 π , and 0.5 π . Bottom row (left to right): ( α 1 , α 2 , α 3 ) = ( 0.4 π , 0.34 π , 0.23 π ) and α p = 0 , 0.15 π , 0.35 π , and 0.5 π .

Fig. 4
Fig. 4

Variations of c 1 and the relative error between the CQGLE and full dynamics. It is observed that the error is smaller when c 1 is less negative.

Fig. 5
Fig. 5

The evolutions of the peak amplitude of | q | governed by the CQGLE (dotted curves) and CNLS (solid curves) for different values of D, g 0 and K. Top row (left to right): D = 0.09 , 0.1 , 0.2 , 1 . Middle row (left to right): g 0 = 0.2 , 0.6 , 0.8 , 0.9 . Bottom row (left to right): K = 0 , 0.2 , 0.9 , 1.1 .

Fig. 6
Fig. 6

Induced multi-pulsing instability generated as the average cavity dispersion becomes too small, i.e., D = D c = 0.1 . The same phenomenon occurs for large energies if the gain g 0 is increased significantly, reflecting a critical breakdown of the balance between chromatic dispersion and self-phase modulation.

Fig. 7
Fig. 7

Expected blowup of the CQGLE when c 2 , 3 > 0 . The full simulation is shown on the left, and the CQGLE is shown on the right. Here B = 1 3 , K = 0.1 , α 1 = 0.35 π , α 2 = 0.1 π , α 3 = 0.2 π , and α p = 0.45 π . This suggests that the results of the CQGLE for c 2 , 3 > 0 are strongly dependent upon the initial conditions and highly suspect when compared to the full cavity evolution dynamics.

Fig. 8
Fig. 8

Possible operating regions of mode locking in the context of CQGLE. Only the gray regions satisfy conditions of Eq. (22) so that they are considered the possible operating regimes of the cavity. Here B = 1 3 , K = 0.1 , and α p = 0.45 π .

Fig. 9
Fig. 9

Actual mode-locking regions of the CQGLE in Eq. (21). Stable mode-locked pulses are found in the white regions, which also satisfy conditions of Eq. (22). The parameters used are D = 0.3 , τ = 0.2 , Γ = 0.1 , and g 0 = Γ c 1 .

Fig. 10
Fig. 10

(a) Left-most mode-locking region shown in Fig. 9 where α 3 = 0.1 π . (b)–(d) The corresponding values of c 1 , c 2 , and c 3 with B = 1 3 , K = 0.1 , and α p = 0.45 π .

Fig. 11
Fig. 11

Failure of the CQGLE for large | c 1 | . The parameters are α 1 = 0.3 π , α 2 = 0.244 π , α 3 = 0.5 π , and α p = 0.23 π such that c 1 = 1.73 . Other parameters are g 0 = 1.56 , τ = 0.2 , and D = 0.3 . Left: The CNLS showing a failure to mode lock. Right: The averaged equation (CQGLE) showing an erroneous mode-locking behavior.

Equations (41)

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

i u z + D 2 2 u t 2 K u + ( | u | 2 + A | v | 2 ) u + B v 2 u * = i R u ,
i v z + D 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 t 2 ) σ Γ σ ,
g ( z ) = 2 g 0 1 + 1 E 0 ( | u | 2 + | v | 2 ) d t .
W λ 4 = ( e i π 4 0 0 e i π 4 ) ,
W λ 2 = ( i 0 0 i ) ,
W p = ( 1 0 0 0 ) .
J j = R ( α j ) W R ( α j ) ,
R ( α j ) = ( cos ( α j ) sin ( α j ) sin ( α j ) cos ( α j ) ) .
u n = q n cos α p , v n = q n sin α p ,
( u ( 0 , t ) v ( 0 , t ) ) = J 1 ( u n v n ) = ( cos α p i cos ( 2 α 1 α p ) sin α p i sin ( 2 α 1 α p ) ) q n 2 ,
( u ( z , t ) v ( z , t ) ) = e ( g Γ ) z J K J L ( u ( 0 , t ) v ( 0 , t ) ) ,
J K = ( e i K z 0 0 e i K z ) and J L = ( L 0 0 L ) .
L = 1 + z ( i D 2 + g τ ) t 2 .
( u ( 1 , t ) v ( 1 , t ) ) = e ( g Γ ) J K J NL J K J 1 ( cos α p sin α p ) L 2 q n ,
q n + 1 = 1 2 e i | q n | 2 e ( g Γ ) Q ̃ ( | q n | 2 ) L 2 q n ,
Q ̃ = e i K [ i ( cos ( 2 α 2 2 α 3 α p ) + i cos ( 2 α 3 α p ) ) ( cos ( 2 α 1 α p w ) + i cos ( α p w ) ) ] [ + e 2 i K ( sin ( 2 α 2 2 α 3 α p ) i sin ( 2 α 3 α p ) ) ( i sin ( 2 α 1 α p w ) + sin ( [ α p w ] ) ) ] .
q z = ( i b + c ) q + ε d 1 2 q t 2 + ε 2 d 2 4 q t 4 ,
q ( 1 , t ) = e i b + c [ 1 + ε d 1 2 t 2 + ε 2 ( d 1 2 2 + d 2 ) 4 t 4 ] q ( 0 , t ) .
c = g Γ + log | Q ̃ | 2 ,
b = | q | 2 + arg Q ̃ ,
ε d 1 = i D 2 + g τ ,
ε 2 d 2 = 1 4 ( i D 2 + g τ ) 2 .
c = g Γ + c 1 + c 2 | q | 2 + c 3 | q | 4 ,
b = | q | 2 + b 1 + b 2 | q | 2 + b 3 | q | 4 ,
H 1 ( | q | 2 ) = log { 1 4 ( [ cos ( 2 α 1 + 2 α 2 2 α 3 2 α p K S ) + cos ( 2 α 1 + 2 α 2 2 α 3 2 α p + K S ) + 4 sin K cos ( α 1 + α 2 2 α p ) cos ( α 1 α 2 + 2 α 3 S ) cos ( 2 α 3 2 α p K + S ) cos ( 2 α 3 2 α p + K + S ) ] 2 + [ 4 sin K sin ( α 1 α 2 ) sin ( α 1 α 2 + 2 α 3 S ) + 2 cos ( α 1 α 2 ) cos ( α 1 + α 2 2 α 3 + K S ) + cos ( 2 α 1 2 α 3 K S ) + cos ( 2 α 2 2 α 3 K S ) ] 2 ) 1 2 }
h 21 = cos ( 2 α 1 + 2 α 2 2 α 3 2 α p K S ) cos ( 2 α 1 + 2 α 2 2 α 3 2 α p + K S ) 4 sin K cos ( α 1 + α 2 2 α p ) cos ( α 1 α 2 + 2 α 3 S ) + cos ( 2 α 3 2 α p K + S ) + cos ( 2 α 3 2 α p + K + S ) ,
h 22 = 4 sin K sin ( α 1 α 2 ) sin ( α 1 α 2 + 2 α 3 S ) + 2 cos ( α 1 α 2 ) cos ( α 1 + α 2 2 α 3 + K S ) + cos ( 2 α 1 2 α 3 K S ) + cos ( 2 α 2 2 α 3 K S ) ,
q z = ( τ g + i D 2 ) 2 q t 2 [ ( g 2 τ 2 4 D 2 16 ) + i D g τ 4 ] 4 q t 4 + ( g + c 1 Γ ) q + [ c 2 + i ( 1 + b 2 ) ] | q | 2 q + ( c 3 + i b 3 ) | q | 4 q .
c 2 > 0 > c 3 .
q ( z , t ) = ω B + cosh ω t exp i { A 2 log ( B + cosh ω t ) + φ } ,
d ω d z = 1 2 ( I 1 I 11 I 2 I 5 + I 2 I 6 ) × [ 4 ( c 1 + g Γ ) ( I 1 I 11 I 2 I 5 ) ω + ( 2 g τ I 12 I 2 + ( A 2 3 ) g I 13 I 2 τ 2 g τ I 11 I 7 I 4 ( A D I 2 + ( A 2 3 ) g I 11 τ ) ) ω 3 + 4 c 3 ( I 11 I 3 I 14 I 2 ) ω 5 ] ,
d A d z = 1 12 ( I 1 I 11 I 2 I 5 + I 2 I 6 ) × [ 24 A ( c 1 + g Γ ) ( I 2 I 6 2 I 1 I 8 ) 3 ( 4 ( 3 + 3 b 2 2 A c 2 ) I 2 2 4 A g τ I 8 ( ( A 2 3 ) I 4 + 2 I 7 ) + I 1 ( 8 ( 1 + b 2 A c 2 ) I 3 + ( 1 + A 2 ) I 15 ( 3 D + 2 A g τ ) ) + I 2 ( 2 ( A + A 3 ) g τ I 9 I 4 ( 3 D 3 A 2 D + 10 A g τ + 2 A 3 g τ ) + 4 A ( 2 c 2 I 8 + g τ I 7 ) ) ) ω 2 + 8 ( 3 ( b 3 A c 3 ) I 1 I 16 + 5 b 3 I 2 I 3 + 3 A c 3 ( I 10 I 2 I 3 I 2 + 2 I 3 I 8 ) ) ω 4 ] ,
d B d z = 1 2 ( I 1 I 11 I 2 I 5 + I 2 I 6 ) × [ ( I 5 I 6 ) ω 2 ( 4 c 2 I 2 g τ ( ( A 2 3 ) I 4 + 2 I 7 ) + 4 c 3 I 3 ω 2 ) + I 1 ( 4 ( c 1 + g Γ ) I 6 4 c 3 I 14 ω 4 ( 4 c 2 I 11 + A D I 4 + g τ ( 2 I 12 + ( A 2 3 ) I 13 ) ) ω 2 ) ] .
I 1 = d t θ , I 2 = d t θ 2 , I 3 = d t θ 3 , I 4 = s 2 θ 3 d t ,
I 5 = log θ θ d t , I 6 = t s θ 2 d t , I 7 = c θ 2 d t ,
I 8 = t s θ 3 d t ,
I 9 = t s 3 θ 4 d t , I 10 = t s θ 4 d t , I 11 = log θ θ 2 d t ,
I 12 = c log θ θ 2 d t , I 13 = s 2 log θ θ 3 d t ,
I 14 = log θ θ 3 d t ,
I 15 = s 2 θ 4 d t , I 16 = 1 θ 4 d t ,

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