Abstract

A new model is constructed that describes the operation of dual-frequency, pulsed mode-locked laser cavities. The model, which is a combination of dual-channel interactions in the canonical master mode-locking model subject to three different gain models that account for both self- and cross-saturation effects, results in mode-locking dynamics that qualitatively describe the observed experimental dual-frequency laser operation. Specifically, the combination of self- and cross saturation in the gain allows for mode locking at two frequencies simultaneously, which can be of significantly different energies and pulsewidths. The model gives a framework for understanding the operation and stability of the increasingly important and timely technology of dual- and multifrequency mode-locked laser cavities.

© 2006 Optical Society of America

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  1. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley-Interscience, 2002).
    [CrossRef]
  2. Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).
  3. H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).
  4. Z. Ahned and N. Onodera, "High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers," Electron. Lett. 32, 455-455 (1996).
    [CrossRef]
  5. C. Wu and N. K. Dutta, "High repetition-rate optical pulse generation using a rational harmonic mode-locked fiber laser," IEEE J. Quantum Electron. 36, 145-150 (2000).
    [CrossRef]
  6. Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
    [CrossRef]
  7. H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Areas Commun. 6, 1173-1185 (2000).
  8. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, "Self-starting, passively modelocked erbium fiber laser based on the amplifying Sagnac switch," Electron. Lett. 27, 542-544 (1991).
    [CrossRef]
  9. M. L. Dennis and I. N. Duling III, High repetition rate figure eight laser with extracavity feedback," Electron. Lett. 28, 1894-1896 (1992).
    [CrossRef]
  10. F. X. Kartner and U. Keller, "Stabilization of solitonlike pulses with a slow saturable absorber," Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  11. J. N. Kutz, B. C. Collings, K. Bergman, andS. Tsuda, S. Cundiff, W., H. Knox, P. Holmes, and M. Weinstein, "Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector," J. Opt. Soc. Am. B 14, 2681-2690 (1997).
    [CrossRef]
  12. K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse modelocked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
    [CrossRef]
  13. H. A. Haus, E. P. Ippen, and K. Tamura, "Additive-pulse modelocking in fiber lasers," IEEE J. Quantum Electron. 30, 200-208 (1994).
    [CrossRef]
  14. M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).
  15. F. X. KartnerD. Kopf, and U. Keller, "Solitary pulse stabilization and shortening in actively mode-locked lasers," J. Opt. Soc. Am. B 12, 486-4966 (1995).
    [CrossRef]
  16. H. A. Haus , "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-1330 (1975).
    [CrossRef]
  17. 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 equations," Phys. Rev. E 66, 066610 (1992).
    [CrossRef]
  18. N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (2001).
    [CrossRef]
  19. 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]
  20. T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," 53, 1095-1126 (2004).
  21. T. Kapitula, "Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation," Physica D 116, 95-120 (1998).
    [CrossRef]
  22. M. Romagnoli, S. Wabnitz, P. Franco, M. Midrio, L. Bossalini, and F. Fontana, "Role of dispersion in pulse emission from a sliding-frequency fiber laser," J. Opt. Soc. Am. B 12, 938-944 (1995).
    [CrossRef]
  23. 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 28, 055602 (2001).
  24. P. Drazin, Nonlinear Systems (Cambridge, New York, 1992).

2002 (1)

2001 (2)

N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (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 28, 055602 (2001).

2000 (3)

C. Wu and N. K. Dutta, "High repetition-rate optical pulse generation using a rational harmonic mode-locked fiber laser," IEEE J. Quantum Electron. 36, 145-150 (2000).
[CrossRef]

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Areas Commun. 6, 1173-1185 (2000).

1998 (1)

T. Kapitula, "Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation," Physica D 116, 95-120 (1998).
[CrossRef]

1997 (1)

1996 (1)

Z. Ahned and N. Onodera, "High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers," Electron. Lett. 32, 455-455 (1996).
[CrossRef]

1995 (3)

1994 (1)

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

1993 (1)

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

1992 (3)

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

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse modelocked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[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 equations," Phys. Rev. E 66, 066610 (1992).
[CrossRef]

1991 (1)

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

1975 (1)

H. A. Haus , "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-1330 (1975).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley-Interscience, 2002).
[CrossRef]

Ahned, Z.

Z. Ahned and N. Onodera, "High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers," Electron. Lett. 32, 455-455 (1996).
[CrossRef]

Akhmediev, N.

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 28, 055602 (2001).

N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (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 equations," Phys. Rev. E 66, 066610 (1992).
[CrossRef]

Andrejco, M. J.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

Bergman, K.

Bossalini, L.

Chan, K. T.

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

Collings, B. C.

Cundiff, S.

Dennis, M. L.

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

Dong, H.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).

Drazin, P.

P. Drazin, Nonlinear Systems (Cambridge, New York, 1992).

Duling, I. N.

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

Dutta, N. K.

C. Wu and N. K. Dutta, "High repetition-rate optical pulse generation using a rational harmonic mode-locked fiber laser," IEEE J. Quantum Electron. 36, 145-150 (2000).
[CrossRef]

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).

Fermann, M. E.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

Fontana, F.

Franco, P.

Gao, Y.

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

Guiyun, K.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Haus, H. A.

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Areas Commun. 6, 1173-1185 (2000).

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

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse modelocked 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, 323-1330 (1975).
[CrossRef]

Holmes, P.

Ippen, E. P.

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

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

Kapitula, T.

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]

T. Kapitula, "Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation," Physica D 116, 95-120 (1998).
[CrossRef]

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," 53, 1095-1126 (2004).

Kartner, F. X.

Keller, U.

Knox, W. H.

Kopf, D.

Kutz, J. N.

Laming, R. I.

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

Li, Z.

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

Lou, C.

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

Matsas, V. J.

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

Midrio, M.

Onodera, N.

Z. Ahned and N. Onodera, "High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers," Electron. Lett. 32, 455-455 (1996).
[CrossRef]

Payne, D. N.

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

Phillips, M. W.

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

Qida, Z.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Richardson, D. J.

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

Rodrigues, A. S.

N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (2001).
[CrossRef]

Romagnoli, M.

Sandstede, B.

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]

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," 53, 1095-1126 (2004).

Shiquan, Y.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Shuzhong, Y.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Silverberg, Y.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

Soto-Crespo, J. M.

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 28, 055602 (2001).

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 equations," Phys. Rev. E 66, 066610 (1992).
[CrossRef]

Stock, M. L.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

Tamura, K.

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

K. Tamura, H. A. Haus, and E. P. Ippen, "Self-starting additive pulse modelocked erbium fiber ring laser," Electron. Lett. 28, 2226-2228 (1992).
[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 28, 055602 (2001).

Town, G. E.

N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (2001).
[CrossRef]

Tsuda, S.

Wabnitz, S.

Wang, Q.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).

Weinstein, M.

Wu, C.

C. Wu and N. K. Dutta, "High repetition-rate optical pulse generation using a rational harmonic mode-locked fiber laser," IEEE J. Quantum Electron. 36, 145-150 (2000).
[CrossRef]

Xiaoyyi, D.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Zhaohui, L.

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

Zhu, G.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).

Electron. Lett. (4)

Z. Ahned and N. Onodera, "High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers," Electron. Lett. 32, 455-455 (1996).
[CrossRef]

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

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

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

IEEE J. Quantum Electron. (3)

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

C. Wu and N. K. Dutta, "High repetition-rate optical pulse generation using a rational harmonic mode-locked fiber laser," IEEE J. Quantum Electron. 36, 145-150 (2000).
[CrossRef]

H. A. Haus , "A theory of forced mode locking," IEEE J. Quantum Electron. 11, 323-1330 (1975).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

H. A. Haus, "Mode-locking of lasers," IEEE J. Sel. Areas Commun. 6, 1173-1185 (2000).

J. Opt. Soc. Am. B (4)

Opt. Commun. (2)

N. Akhmediev, A. S. Rodrigues, and G. E. Town, "Interaction of dual-frequency pulses in passively mode-locked lasers," Opt. Commun. 187, 419-426 (2001).
[CrossRef]

Z. Li, C. Lou, Y. Gao, and K. T. Chan, "A dual-wavelength and dual-repetition-rate actively mode-locked fiber ring laser," Opt. Commun. 185, 381-385 (2000).
[CrossRef]

Opt. Lett. (2)

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, "Passive modelocking by using nonlinear polarization evolution," Opt. Lett. 29, 447-449 (1993).

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

Phys. Rev. E (2)

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 equations," Phys. Rev. E 66, 066610 (1992).
[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 28, 055602 (2001).

Physica D (1)

T. Kapitula, "Stability criterion for bright solitary waves of the perturbed cubic-quintic Schrödinger equation," Physica D 116, 95-120 (1998).
[CrossRef]

Other (5)

P. Drazin, Nonlinear Systems (Cambridge, New York, 1992).

T. Kapitula, J. N. Kutz, and B. Sandstede, "The Evans function for nonlocal equations," 53, 1095-1126 (2004).

G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley-Interscience, 2002).
[CrossRef]

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, "Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs," in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, "Simultaneous mode locked operation of a fiber laser at two wavelengths," in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).

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Figures (11)

Fig. 1
Fig. 1

Growth rate of perturbations to cw solutions as a function of Fourier wavenumber k and gain strength g. For a single cw solution (top) with g > 4.0277, the real part of all eigenvalues are nonpositive for all k. The coupled cw solutions (bottom) have similar eigenvalue structure but require larger gain to achieve stability. Figures 2 and 3 depict the unstable and stable propagations of dual-frequency cw operation for low and high gain values, respectively.

Fig. 2
Fig. 2

For g 0 below threshold, cw solutions are unstable. Here u ( T , 0 ) = v ( T , 0 ) = 1 are perturbed by white noise of amplitude 10–6. Note that the initial conditions first quickly settle to the cw solutions before becoming modulationally unstable. The laser cavity parameters are β = 0.05 , σ = 0.1 , τ = 0.1 , and g 0 = 5.

Fig. 3
Fig. 3

For g 0 above threshold, cw solutions are stabilized. Here u ( T , 0 ) = 0 and v ( T , 0 ) = 1 are perturbed by white noise of amplitude 1.0. Despite the large perturbations, the solutions quickly converge to the cw solution, and stable cw mode locking is achieved. The laser cavity parameters are β = 0.05 , σ = 0.1 , τ = 0.1 , and g 0 = 8.

Fig. 4
Fig. 4

With g 0 sufficiently high, two mode-locked pulsed are formed and stabilized under the cavity-saturation model. Here, the pulses are formed from white noise. The laser cavity parameters are β 1 = β 2 = 0.05 , σ 1 = σ 2 = 0.01 , τ 1 = τ 2 = 0.05 , and the gain parameter g 0 = 0.33.

Fig. 5
Fig. 5

Instability of the dual-frequency mode locking under the cavity-saturation gain model. Here the slightly different initial conditions u ( T , 0 ) = 2 sech ( 6 T ) , and v ( T , 0 ) = 1.6 sech 6 T ) results in a cascade of energy to the energetically dominant v frequency channel. The laser cavity parameters are β 1 = β 2 = 0.05 , σ 1 = σ 2 = 0.01 , τ 1 = τ 2 = 0.05 , and the gain parameter g 0 = 0.31.

Fig. 6
Fig. 6

When the gain is high enough to induce pulse splitting, dual-frequency pulse operation may again become unstable. The initial conditions u ( T , 0 ) = v ( T , 0 ) = 2 sech ( 6 T ) were seeded with a small amount of white noise, and the u field absorbs all system energy. The laser cavity parameters are β 1 = β 2 = 0.05 , σ 1 = σ 2 = 0.01 , τ 1 = τ 2 = 0.05 , and the gain parameter g 0 = 0.45.

Fig. 7
Fig. 7

When the gain is high enough to induce pulse splitting, dual-frequency pulse operation may be feasible, provided the final positions of the pulses in each channel are nonoverlapping. The initial conditions u ( T , 0 ) = v ( T , 0 ) = 2 sech ( 6 T ) were seeded with a small amount of white noise. The laser cavity parameters are β 1 = β 2 = 0.05 , σ 1 = σ 2 = 0.01 , τ 1 = τ 2 = 0.05 , and the gain parameter g 0 = 0.45.

Fig. 8
Fig. 8

Under the cavity-saturable gain model, even a small difference in gain bandwidths τ 1 and τ 2 results in unstable pulse formation. Here u ( T , 0 ) = v ( T , 0 ) = 2 sech ( 6 T ) β 1 = β 2 = 0.05 , τ 1 = 0.05 , τ 2 = 0.051 , σ 1 = σ 2 = 0.01 , and g 0 = 0.30.

Fig. 9
Fig. 9

With different gain bandwidths, the two pulses quickly form and stabilize under gain model 2. Initial conditions are u ( T , 0 ) = v ( T , 0 ) = 0.001 sech ( 0.001 T ) . Here, β 1 = β 2 = 0.05 , g 0 = 0.30, τ 1 = 0.05 , and τ 2 = 0.06 .

Fig. 10
Fig. 10

With the different gain bandwidths, the two pulses quickly form and stabilize under the hybrid gain mode. Here, the gain is strongly weighted toward cavity saturation (α = 0.95). Initial conditions are u ( T , 0 ) = v ( T , 0 ) = 0.001 sech ( 0.001 T ) . Here, β 1 = β 2 = 0.05 , g 0 = 0.30, τ 1 = 0.05 , and τ 2 = 0.06 .

Fig. 11
Fig. 11

Under gain model 2, the energies in the two channels are nearly equal, despite differences in gain bandwidth parameters τ 1 and τ 2 . When the hybrid gain model 3 is weighted toward the saturable cavity limit, with 95% cavity saturation and 5% self-saturation, the energy difference in the channels is evident. In both cases, g 0 = 0.30, β 1 = β 2 = 0.05 , τ 1 = 0.05 , and τ 2 = 0.06 .

Equations (27)

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i u Z + [ 1 2 i g 1 ( Z ) τ 1 ] 2 u T 2 + ( 1 i β 1 ) u 2 u + 2 v 2 u + i σ 1 u 4 u + i [ γ 1 g 1 ( Z ) ] u = 0 ,
i v Z + [ 1 2 i g 2 ( Z ) τ 2 ] 2 v T 2 + ( 1 i β 2 ) v 2 v + 2 u 2 v + i σ 2 v 4 v + i [ γ 2 g 2 ( Z ) ] v = 0 ,
g 1 ( Z ) = g 2 ( Z ) = g u v ( z ) = 2 g 0 1 + ( u 2 + v 2 ) ( 2 e 0 ) ,
g 1 ( Z ) = g u ( Z ) = 2 g 0 1 + u 2 e 0 ,
g 2 ( Z ) = g v ( Z ) = 2 g 0 1 + v 2 e 0 ,
g 1 ( Z ) = α g u v + ( 1 α ) g u ,
g 2 ( Z ) = α g u v + ( 1 α ) g v ,
Ω = A 2 ,
σ A 4 β A 2 + ( γ g ) = 0 .
g > γ β 2 4 σ ,
A 2 = { β + [ β 2 4 σ ( γ g ) ] 1 2 } ( 2 σ ) .
i Z u ̃ + ( 1 2 i g τ ) 2 T 2 u ̃ + [ ( 1 i β ) A 2 + i 2 σ A 4 ] u ̃ * + [ ( 1 i 2 β ) A 2 + i 3 σ A 4 + i ( γ g ) ] u ̃ = 0 .
Z R + 2 T 2 ( g τ R + 1 2 I ) + ( 4 σ A 4 2 β A 2 ) R = 0 ,
Z I + 2 T 2 ( g τ I 1 2 R ) 2 A 2 R = 0 .
d d Z [ R ̂ I ̂ ] = [ g τ k 2 4 σ A 4 + 2 β A 2 1 2 k 2 1 2 k 2 + 2 A 2 g τ k 2 ] [ R ̂ I ̂ ] ,
λ ( k ) = { g τ k 2 + A 2 [ β 2 4 σ ( γ g ) ] 1 2 } ± 1 2 D ( k ) ,
D ( k ) = 4 A 4 [ β 2 4 σ ( γ g ) ] k 4 + 4 A 2 k 2 .
λ k = k [ 2 g τ ± 2 A 2 k 2 D ( k ) ] = 0 .
k 4 4 A 2 k 2 + 4 A 4 { 1 4 g τ 2 [ β 4 σ ( γ g ) ] } 1 + 4 g 2 τ 2 = 0 ,
k = ± 2 A { 1 2 g τ [ 1 + β 4 σ ( γ g ) 1 + 4 g 2 τ 2 ] 1 2 } 1 2 = 0 .
g 3 + 1 4 σ ( β 4 σ γ ) g 2 1 16 σ τ 2 > 0
Ω 1 = A 2 + 2 B 2 ,
Ω 2 = B 2 + 2 A 2 ,
σ A 4 β A 2 + ( γ g ) = 0 ,
σ B 4 β B 2 + ( γ g ) = 0 .
d d z [ R ̂ u I ̂ u R ̂ v I ̂ v ] = [ M 2 [ 0 0 2 A 2 0 ] [ 0 0 2 A 2 0 ] M 2 ] [ R ̂ u I ̂ u R ̂ v I ̂ v ] ,
λ ( k ) = { g τ k 2 + A 2 [ β 2 4 σ ( γ g ) ] 1 2 } ± 1 2 [ D ( k ) + 4 A 2 k 2 ] 1 2 ,

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