Abstract

A theoretical model is constructed that describes the operation of a pulsed mode-locked laser simultaneously operating at N frequency channels. The model, which is a combination of standard WDM interactions in the canonical master mode-locking model subject to both self- and cavity-saturating gain effects, results in mode-locking dynamics that qualitatively describe the N-frequency channel operation. It is further in agreement with the observed experimental dual-frequency (N=2) laser operation. In the model, it is the combination of self- and cavity-gain saturation that simultaneously allows for mode-locking at N frequencies, which can be of significantly different energies and pulse widths. The model provides a framework for understanding the operation and stability of identically mode-locked pulses at multiple frequencies, thus contributing to the characterization of the increasingly important and timely technology of dual- and multifrequency mode-locked laser cavities.

© 2008 Optical Society of America

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References

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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,” Proc. SPIE 4974, 43-49 (2003).
    [CrossRef]
  3. H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
    [CrossRef]
  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. E. Farnum, L. Butson, and J. N. Kutz, “Theory and simulation of dual-frequency mode-locked lasers,” J. Opt. Soc. Am. B 23, 257-264 (2006).
    [CrossRef]
  8. H. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. 6, 1173-1185 (2000).
    [CrossRef]
  9. J. N. Kutz, “Mode-locked soliton lasers,” SIAM Rev. 48, 629-678 (2006).
    [CrossRef]
  10. D. J. Richardson, R. I. Laming, D. N. Payne, V. J. Matsas, and M. W. Phillips, “Self-starting, passively modelocked erbium fiber laser based on the amplifying Sagnac switch,” Electron. Lett. 27, 542-544 (1991).
    [CrossRef]
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    [CrossRef]
  12. F. X. Kartner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16-18 (1995).
    [CrossRef] [PubMed]
  13. J. N. Kutz, B. C. Collings, K. Bergman, S. Tsuda, S. Cundiff, W. H. Knox, P. Holmes, and M. Weinstein, “Mode-locking pulse dynamics in a fiber laser with a saturable Bragg reflector,” J. Opt. Soc. Am. B 14, 2681-2690 (1997).
    [CrossRef]
  14. 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]
  15. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200-208 (1994).
    [CrossRef]
  16. M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, “Passive modelocking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,” Opt. Lett. 29, 447-449 (1993).
  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 (1995).
    [CrossRef]
  18. H. A. Haus, “A theory of forced mode locking,” IEEE J. Quantum Electron. 11, 323-330 (1975).
    [CrossRef]
  19. G. P. Agrawal, Fiber-Optic Communication Systems, 3rd ed. (Wiley-Interscience, 2002).
    [CrossRef]
  20. 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]
  21. L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
    [CrossRef]
  22. A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1995), Chap. 10.
  23. P. G. Drazin, Nonlinear Systems (Cambridge U. Press, 1992).
  24. T. Kapitula and B. Sandstede, “Instability mechanism for bright solitary-wave solutions to the cubic-quintic Ginzburg-Landau equation,” J. Opt. Soc. Am. B 15, 2757-2762 (1998).
    [CrossRef]
  25. T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58-103 (1998).
    [CrossRef]
  26. R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks-Cole, 2004).

2006 (2)

2004 (1)

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

2003 (1)

Y. Shiquan, L. Zhaohui, Y. Shuzhong, D. Xiaoyyi, K. Guiyun, and Z. Qida, “Dual-wavelength actively mode-locked erbium dobed fiber laser using FBGs,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

2002 (1)

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

1998 (2)

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

T. Kapitula and B. Sandstede, “Instability mechanism for bright solitary-wave solutions to the cubic-quintic Ginzburg-Landau equation,” J. Opt. Soc. Am. B 15, 2757-2762 (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 (2)

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 in a polarizing-maintaining erbium-doped fiber laser,” Opt. Lett. 29, 447-449 (1993).

1992 (2)

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]

1991 (2)

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]

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
[CrossRef]

1975 (1)

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

Agrawal, G. P.

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

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]

Andrejco, M. J.

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

Bergman, K.

Burden, R. L.

R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks-Cole, 2004).

Butson, 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,” Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

Drazin, P. G.

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

Duling, I. N.

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

Dutta, N. K.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
[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]

Evangelides, S. G.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
[CrossRef]

Faires, J. D.

R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks-Cole, 2004).

Farnum, E.

Fermann, M. E.

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

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]

Gordon, J. P.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
[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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Hasegawa, A.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1995), Chap. 10.

Haus, H.

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

Haus, H. A.

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, 323-330 (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.

Kartner, F. X.

Kärtner, F. X.

Keller, U.

Knox, W. H.

Kodama, Y.

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1995), Chap. 10.

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]

Mollenauer, L. F.

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
[CrossRef]

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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Richardson, D. J.

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

Sandstede, B.

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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Silverberg, Y.

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

Stock, M. L.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, “Passive modelocking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,” 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]

Tsuda, S.

Wang, Q.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Zhu, G.

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

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, 323-330 (1975).
[CrossRef]

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

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

J. Lightwave Technol. (1)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-long distance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991) (see appendix).
[CrossRef]

J. Opt. Soc. Am. B (5)

Opt. Commun. (1)

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 in a polarizing-maintaining erbium-doped fiber laser,” 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]

Physica D (1)

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

Proc. SPIE (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,” Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

H. Dong, G. Zhu, Q. Wang, and N. K. Dutta, “Simultaneous mode locked operation of a fiber laser at two wavelengths,” Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

SIAM Rev. (1)

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

Other (5)

A. Hasegawa and Y. Kodama, Solitons in Optical Communications (Oxford U. Press, 1995), Chap. 10.

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

R. L. Burden and J. D. Faires, Numerical Analysis, 8th ed. (Brooks-Cole, 2004).

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

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

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Figures (10)

Fig. 1
Fig. 1

When saturable gain is included as a perturbation to the constant gain model, the positive eigenvalue is shifted to zero. Thus a saturable gain can stabilize sech-type solutions to the master mode-locking case. Shown are eigenvalues for the linearization around the chirpless solutions u ( t , z ) = 2 sech ( 2 t ) e i z . The gain saturates at g sat = 0.25 . System parameters are τ = 0.1 , γ = 0.30 , and σ = 0.01 .

Fig. 2
Fig. 2

Saturable gain can stabilize the two-pulse solution provided that there is sufficient self-saturation. When the saturation parameter exceeds 0.25, the two positive eigenvalues are pushed to zero corresponding to phase and translation invariances. System parameters are g 0 = 0.25 , τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 3
Fig. 3

When each channel’s gain independently saturates, the dual-pulse operation is especially robust. With initial conditions of white noise, hyperbolic secantlike pulses quickly form and stabilize. The system parameters are g 0 = 0.25 , τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 4
Fig. 4

When both channels’ gain only saturates with the cavity’s total energy, the dual-pulse operation is unstable. With known steady-state solutions that are perturbed with white noise of an amplitude of 0.001, one channel quickly absorbs all of the system energy. Note that the solution parameters of the remaining pulse are determined by the single channel case of Section 1. The system parameters are g 0 = 0.25 , τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 5
Fig. 5

For values of weighting parameter 0.15 < α < 0.245 , the system admits stable solutions with pulses of differing heights and widths. The system parameters are g 0 = 0.25 , τ = 0.10 , γ = 0.2167 , σ = 0.01 , and α = 0.2 .

Fig. 6
Fig. 6

When initial conditions are such that the two channels have no overlap in the time domain, the solution is effectively a pair of one-channel mode-locked pulses as in Section 1. Note that the two channels do interact via the gain. System parameters are τ = 0.10 , γ = 0.2167 , and σ = 0.01

Fig. 7
Fig. 7

When each channel independently saturates ( α = 1 ) , the system can support mode-locked pulses for all channels. System parameters are τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 8
Fig. 8

When each channel saturates independently ( α = 1 ) , the system can support mode-locked pulses for all channels. System parameters are τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 9
Fig. 9

When α = 0.10 , the system effectively supports four uncoupled pulses in channels 1, 3, 6, and 8. System parameters are τ = 0.10 , γ = 0.2167 , and σ = 0.01 .

Fig. 10
Fig. 10

As in the dual-frequency case, when N > 2 , the amplitudes and L 2 norms for the solutions are a function of saturation parameter α. Although asymmetric boundary conditions preclude identical solutions in each channel, it is clear that for sufficient self-saturation, solution amplitudes are nearly identical. The second panel shows that channels 1 and 8 have greater L 2 norms corresponding to a wider pulse shape.

Equations (26)

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

i ( u n z + δ n u n t ) + 1 2 2 u n t 2 + ( 1 i β n ) u n 2 u n i g n ( z ) ( 1 + τ 2 t 2 ) u n + 2 ( u n 1 2 + u n + 1 2 ) u n + i σ n u n 4 u n + i γ u n = 0 ,
g n ( z ) = α g self , n + ( 1 α ) g cav ,
g cav ( z ) = 2 g 0 1 + ( n = 1 N u n 2 ) ( N e 0 ) ,
g self , n ( z ) = 2 g 0 1 + u n 2 e 0 ,
u ( z , t ) = η sech ( ω t ) 1 + i A e i Θ z = u 0 ( t ) e i Θ z ,
Θ + ω 2 ( 1 2 A 2 2 A τ g 1 2 ) = 0 ,
ω 2 ( 1 2 A 2 3 A τ g 1 ) + P η 2 = 0 ,
ω 2 ( A 2 τ g + A τ g ) + ( γ g ) = 0 ,
ω 2 ( A 2 τ g + 3 2 A 2 τ g ) + β η 2 = 0 ,
u ( z , t ) = [ u 0 ( t ) + u 1 ( z , t ) ] e i Θ z ,
d R 1 d z = [ τ g 1 2 1 2 τ g ] t 2 R 1 + [ F + ( R 0 , I 0 ) G ( R 0 , I 0 ) G + ( I 0 , R 0 ) F ( I 0 , R 0 ) ] R 1 ,
F ± ( v , w ) = β ( 3 v 2 + w 2 ) 2 v w ( γ g ) ,
G ± ( v , w ) = ± ( 3 v 2 + w 2 ) + 2 β v w Θ .
g = g sat + g pert ,
g sat = 2 g 0 1 + u 0 2 = constant ,
g pert = g sat 2 ( R 0 R 1 + I 0 I 1 ) d t 1 + u 0 2 .
i g pert ( 1 + τ 2 t 2 ) u 0
2 h g sat 1 + u 0 2 ( I + τ [ D k 0 0 D k ] ) R 0 T R 1 ,
u 1 = u 0 + u 1 , 1 = ( R 0 + i I 0 ) + ( R 1 , 1 + i I 1 , 1 ) ,
u 2 = u 0 + u 2 , 1 = ( R 0 + i I 0 ) + ( R 2 , 1 + i I 2 , 1 ) ,
d R d z = ( [ τ g 1 2 0 0 1 2 τ g 0 0 0 0 τ g 1 2 0 0 1 2 τ g ] t 2 + [ W 1 W 2 W 2 W 1 ] + G ) R ,
W 1 = [ F + ( R 0 , I 0 ) G ( R 0 , I 0 ) 2 ( R 0 2 + I 0 2 ) G + ( I 0 , R 0 ) + 2 ( R 0 2 + I 0 2 ) F ( I 0 , R 0 ) ] ,
W 2 = 4 [ R 0 I 0 I 0 2 R 0 2 R 0 I 0 ] .
G self = g sat 1 + u 0 2 2 h [ M 0 0 M ] ,
G cav = g sat 1 + u 0 2 h [ M M M M ] ,
M = ( I + τ [ D k 0 0 D k ] ) R 0 T R 1 .

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