Abstract

A complete characterization is given of the effects of homogeneous and inhomogeneous gain broadening on the mode-locking dynamics and stability of a laser operating simultaneously at N frequency channels. Using a low-dimensional model for the wavelength-division multiplexing interactions of the governing cubic-quintic master mode-locking equation, the interplay of the gain dynamics can be completely classified. This gives a simple way to characterize the laser performance and the parameter regimes under which stable multifrequency operation can be achieved. The analysis shows that a small amount of inhomogeneous gain broadening is critical for the multifrequency operation. The model further provides a simple framework for understanding the stability of mode-locked pulses at multiple frequencies, thus contributing to the characterization of the increasingly important and timely technology of dual-frequency and multifrequency mode-locked laser cavities.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  7. 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).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  28. B. G. Bale and J. N. Kutz, “Variational method for mode-locked lasers,” J. Opt. Soc. Am. B 25, 1193-1202 (2008).
    [CrossRef]
  29. C. Jirauschek, U. Morgner, and F. X. Kärtner, “Variational analysis of spatio-temporal pulse dynamics in dispersive Kerr media,” J. Opt. Soc. Am. B 19, 1716-1721 (2002).
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2008 (3)

2007 (2)

2006 (3)

2005 (1)

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

2003 (1)

2002 (3)

2001 (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]

2000 (3)

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

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]

1996 (2)

Z. Ahned and N. Onodera, “High repetition rate optical pulse generation by frequency multiplication in actively mode-locked fiber ring lasers,” Electron. Lett. 32, 455455 (1996).

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

1991 (1)

L. F. Mollenauer, S. G. Evangelides, and J. P. Gordon, “Wavelength division multiplexing with solitons in ultra-longdistance transmission using lumped amplifiers,” J. Lightwave Technol. 9, 362-367 (1991); see Appendix.
[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, 455455 (1996).

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]

Antonelli, C.

Bale, B. G.

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

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

B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion lasers,” J. Opt. Soc. Am. B (to be published).

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multi-frequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. (to be published).

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 laser,” Phys. Rev. A 65, 063811 (2002).
[CrossRef]

Buckley, J.

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]

Chartier, T.

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

Chen, J.

Chong, A.

Dennis, M. L.

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

Desurvire, E.

E. Desurvire, Erbium-Doped Fiber Amplifiers Principles and Applications (Wiley-Interscience, 1994).

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).
[CrossRef]

Drazin, P. G.

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

Duling, I. N.

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

Evangelides, S. G.

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

Farnum, E.

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-longdistance 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,” in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Hasegawa, A.

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

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, 1989).
[CrossRef]

Haus, H. A.

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

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

Hideur, A.

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

Jirauschek, C.

Kapitula, T.

Kärtner, F. X.

Kodama, Y.

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

Komarov, A.

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

Kutz, J. N.

Leblond, H.

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

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

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]

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]

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]

Mollenauer, L. F.

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

Morgner, U.

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, 455455 (1996).

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).
[CrossRef]

Renninger, W.

Renninger, W. H.

Salhi, M.

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

Sanchez, F.

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

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

Sandstede, B.

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).
[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,” in Advances in Fiber Lasers, L.N.Duprasula, ed., Proc. SPIE 4974, 43-49 (2003).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

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).
[CrossRef]

Whitham, G.

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

Wise, F.

Wise, F. W.

Wong, W. S.

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

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).
[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,” in Advances in Fiber Lasers, L.N.Duprasula, ed., 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,” in Physics and Simulation of Optoelectronic Devices XII, M.Osinski, H.Amano, and F.Henneberger, eds., Proc. SPIE 5349, 117-121 (2004).
[CrossRef]

Electron. Lett. (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, 455455 (1996).

IEEE J. Quantum Electron. (1)

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]

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]

J. Lightwave Technol. (1)

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

J. Opt. Soc. Am. B (6)

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. Express (2)

Opt. Lett. (2)

Phys. Rev. A (1)

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

Phys. Rev. E (1)

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

Rev. Mod. Phys. (1)

H. A. Haus and W. S. Wong, “Solitons in optical communications,” Rev. Mod. Phys. 68, 423-444 (1996).
[CrossRef]

SIAM Rev. (1)

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

Other (12)

B. G. Bale, J. N. Kutz, A. Chong, W. H. Renninger, and F. W. Wise, “Spectral filtering for high-energy mode-locking in normal dispersion lasers,” J. Opt. Soc. Am. B (to be published).

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

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

A. E. Siegman, Lasers (University Science Books, 1986).

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

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

A. Hasegawa, Optical Solitons in Fibers (Springer-Verlag, 1989).
[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).
[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).
[CrossRef]

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

B. G. Bale, E. Farnum, and J. N. Kutz, “Theory and simulation of passive multi-frequency mode-locking with waveguide arrays,” IEEE J. Quantum Electron. (to be published).

E. Desurvire, Erbium-Doped Fiber Amplifiers Principles and Applications (Wiley-Interscience, 1994).

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Figures (10)

Fig. 1
Fig. 1

When subject to homogeneous gain broadening alone, competition is established between the multifrequency channels so that a single channel eventually dominates the laser cavity. This is evident by the unstable saddle node at ( η 1 , η 2 ) = ( η * , η * ) = ( 1.20 , 1.20 ) and the two stable nodes at ( η * , 0 ) and ( 0 , η * ) . Thus all flow lines terminate (mode-lock) in either one of the fixed points along the η 1 or η 2 axis.

Fig. 2
Fig. 2

When gain is strictly inhomogeneously broadened, all frequency channels in the laser cavity are mode-locked simultaneously. In this case, the solutions settle to the stable node (mode-locked solution) demonstrated here.

Fig. 3
Fig. 3

Interplay of homogenous and inhomogeneous gain broadening effects is the most physically relevant case considered and allows for dual-frequency mode-locked pulse streams of different (top) or the same (bottom) amplitudes. The parameter α, which measures the degree of homogeneous to inhomogeneous gain broadening, is the critical bifurcation parameter that determines the mode-locking behavior. For α 0.3 the fixed point at ( η 1 , η 2 ) = ( η * , η * ) = ( 1.20 , 1.20 ) changes from an unstable saddle to a stable node. The two figures represent the dynamics at α = 0.25 (top) and α = 0.35 (bottom). If homogeneous gain broadening dominates ( α < 0.2 ) , single channel operation is supported as shown in Fig. 1.

Fig. 4
Fig. 4

Amplitude η 1 as function of α depicts the pitchfork bifurcation structure. For α < 0.20 (region I), only a single pulse is supported. For a narrow range of α values (region II), dual pulse operation is feasible with different pulse heights. For a sufficiently high degree of inhomogeneous broadening (region III), identical dual pulse operation is stable.

Fig. 5
Fig. 5

Mode-locking behavior for asymmetric laser cavity parameters. By taking different gain bandwidths, τ 1 = 0.1 < τ 2 = 0.2 , the more energetically favorable channel one produces a higher amplitude mode-locked pulse stream. Here, the parameter α = 0.4 , which produced the identical mode-locked pulses for symmetric parameters.

Fig. 6
Fig. 6

Numerical solution of Eq. (1) for dual-channel operation (top panels) along with comparison to the analytic reduction (10) (bottom panel). Here, only homogenous gain broadening is considered ( α = 0 ) . As predicted, the gain competition between channels shows that one channel eventually dominates the laser cavity. Since the dominant channel has δ 1 = 0.5 , the group-velocity produces the observed drift.

Fig. 7
Fig. 7

Numerical solution of Eq. (1) for dual-channel operation (top panels) along with comparison to the analytic reduction (10) (bottom panel). Both homogeneous and inhomogenous gain broadening effects are included with α = 0.25 . Mode-locking of two frequencies occurs with two different amplitudes.

Fig. 8
Fig. 8

Numerical solution of Eq. (1) for dual-channel operation (top panels) along with comparison to the analytic reduction (10) (bottom panel). Both homogeneous and inhomogenous gain broadening effects are included with α = 0.4 . Mode-locking of two frequencies occurs with equal amplitudes.

Fig. 9
Fig. 9

Amplitudes as a function of propagation distance given by the reduced model (10) with six frequency channels. Here, asymmetry is introduced into the initial conditions. For significant inhomogeneous gain broadening (top panel with α = 0.5 ), all six channels have identical amplitudes. As the homogenous gain broadening is more pronounced (bottom two panels with α < 0.5 ), some channels dominate while others are dropped.

Fig. 10
Fig. 10

Amplitudes as a function of propagation distance given by the reduced model (10) with six frequency channels. Here, symmetric initial conditions are considered with asymmetry introduced into the system through the gain bandwidth parameters: τ 1 = 0.07 , τ 2 = 0.08 , τ 3 = 0.09 , τ 4 = 0.1 , τ 5 = 0.11 , and τ 6 = 0.12 . Depending on the degree of inhomogeneous gain broadening, this can result in all channels operating with different amplitudes (top panel). For the case where homogenous gain broadening effects are more pronounced, certain channels can become more dominant while others are dropped (bottom panels).

Equations (11)

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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 γ u n i g n ( z ) ( 1 + τ n 2 t 2 ) u n + i σ n u n 4 u n + 2 j = 1 ( j n ) N u j 2 u n = 0 ,
g n ( z ) = α G n ( z ) + ( 1 α ) G h ( z ) ,
G h ( z ) = 2 g 0 1 + ( n = 1 N u n 2 ) ( N e 0 ) ,
G n ( z ) = 2 g 0 1 + u n 2 e 0 .
L = n = 1 N { i 2 ( u n u n * z u n * u n z ) + i δ n 2 ( u n u n * t u n * u n t ) 1 2 u n t 2 + 1 2 u n 4 + j = 1 ( j n ) N u j 2 u n 2 } ,
R [ u n ] = i { τ g n ( z ) 2 u n t 2 + ( g n ( z ) γ n ) u n + β n u n 2 u n σ n u n 4 u n } ,
δ L δ p j = 2 R { R [ u ¯ n ] u ¯ n * p j d T } ,
u n ( z , t ) = η n ( z ) sech ( ω n ( z ) t ) exp ( i ϕ n ( z ) ) ,
ω n = η n 2 + 2 j = 1 ( j n ) N η j 2 .
d η n d z = η n 6 ( g n ( 3 τ n ω n 2 ) 3 γ + 2 β n η n 2 8 5 σ n η n 4 ) ,
g n = 2 g 0 ( 1 + ( 2 α ) η n 2 ω n + α j = 1 ( j n ) N η j 2 ω j ) ( 1 + 2 η n 2 ω n ) ( 1 + η n 2 ω n + j = 1 ( j n ) N η j 2 ω j ) .

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