Abstract

The phenomenon of modulation instability of continuous-wave (cw) solutions of the cubic–quintic complex Ginzburg–Landau equation is studied. It is shown that low-amplitude cw solutions are always unstable. For higher-amplitude cw solutions, there are regions of stability and regions where the cw solutions are modulationally unstable. It is found that there is an indirect relation between the stability of the soliton solutions and the modulation instability of the higher-amplitude cw solutions. However, there is no one-to-one correspondence between the two. We show that the evolution of modulationally unstable cw’s depends on the system parameters.

© 2002 Optical Society of America

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    [CrossRef] [PubMed]
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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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]

2000 (1)

P. Coullet, C. Riera, and C. Tresser, “Stable static localized structures in one dimension,” Phys. Rev. Lett. 84, 3069–3072 (2000).
[CrossRef] [PubMed]

1999 (3)

1998 (2)

1997 (2)

B. C. Collings, K. Bergman, and W. H. Knox, “True fundamental solitons in a passively mode-locked short cavity Cr4+:YAG laser,” Opt. Lett. 22, 1098–1100 (1997).
[CrossRef] [PubMed]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A 56, 1582–1591 (1997).
[CrossRef]

1996 (3)

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

J. M. Soto-Crespo, N. N. Akhmediev, and V. V. Afanasjev, “Stability of the pulselike solutions of the quintic complex Ginzburg–Landau equation,” J. Opt. Soc. Am. B 13, 1439–1449 (1996).
[CrossRef]

1995 (6)

1994 (1)

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
[CrossRef]

1993 (2)

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65–70 (1993).
[CrossRef]

J. Hermann, “Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 98, 111–116 (1993).
[CrossRef]

1992 (2)

C. O. Weiss, “Spatio-temporal structures. Part II. Vortices and defects in lasers,” Phys. Rep. 219, 311–338 (1992).
[CrossRef]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

1991 (3)

1989 (1)

S. M. J. Kelly, “Mode-locking dynamics of a laser coupled to an empty external cavity,” Opt. Commun. 70, 495 (1989).
[CrossRef]

1985 (1)

H. A. Haus and M. N. Islam, “Theory of soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1177 (1985).
[CrossRef]

1984 (1)

1975 (1)

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Afanasjev, V. V.

Akhmediev, N. N.

Ankiewicz, A.

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

Belanger, P. A.

Bergman, K.

Bhattacharjee, A.

C. S. Ng and A. Bhattacharjee, “Ginzburg–Landau model and single-mode operation of a free-electron laser oscillator,” Phys. Rev. Lett. 82, 2665–2668 (1999).
[CrossRef]

Brabec, T.

Chen, C.-J.

Chernykh, A. I.

Collings, B. C.

Coullet, P.

P. Coullet, C. Riera, and C. Tresser, “Stable static localized structures in one dimension,” Phys. Rev. Lett. 84, 3069–3072 (2000).
[CrossRef] [PubMed]

Dunlop, A. M.

A. M. Dunlop, E. M. Wright, and W. J. Firth, “Spatial soliton laser,” Opt. Commun. 147, 393–401 (1998).
[CrossRef]

Firth, W. J.

A. M. Dunlop, E. M. Wright, and W. J. Firth, “Spatial soliton laser,” Opt. Commun. 147, 393–401 (1998).
[CrossRef]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Fujimoto, J. G.

Haelterman, M.

Haus, H.

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

Haus, H. A.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447–452 (1995).
[CrossRef]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1995).
[CrossRef]

H. A. Haus and E. P. Ippen, “Self-starting of passively mode-locked lasers,” Opt. Lett. 16, 235–237 (1991).
[CrossRef]

H. A. Haus and M. N. Islam, “Theory of soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1177 (1985).
[CrossRef]

Hermann, J.

J. Hermann, “Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 98, 111–116 (1993).
[CrossRef]

Indik, R.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

Ippen, E. P.

Islam, M. N.

H. A. Haus and M. N. Islam, “Theory of soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1177 (1985).
[CrossRef]

Jagadish, C.

Jakobsen, P. K.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

Jian, P.-S.

Jung, I. D.

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

Kapitula, T.

Kärtner, F. X.

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

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

Keller, U.

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

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

Kelly, S. M. J.

S. M. J. Kelly, “Mode-locking dynamics of a laser coupled to an empty external cavity,” Opt. Commun. 70, 495 (1989).
[CrossRef]

Khatri, F. I.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447–452 (1995).
[CrossRef]

Knox, W. H.

Krausz, F.

Lederer, F.

Lederer, M. J.

Lenz, G.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447–452 (1995).
[CrossRef]

Luther-Davies, B.

Menyuk, C. R.

Mollenauer, L. F.

Moloney, J. V.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

Moores, J. D.

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447–452 (1995).
[CrossRef]

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65–70 (1993).
[CrossRef]

Newell, A. C.

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

Ng, C. S.

C. S. Ng and A. Bhattacharjee, “Ginzburg–Landau model and single-mode operation of a free-electron laser oscillator,” Phys. Rev. Lett. 82, 2665–2668 (1999).
[CrossRef]

Peschel, U.

Riera, C.

P. Coullet, C. Riera, and C. Tresser, “Stable static localized structures in one dimension,” Phys. Rev. Lett. 84, 3069–3072 (2000).
[CrossRef] [PubMed]

Rozanov, N. N.

N. N. Rozanov, Optical Bistability and Hysteresis in Distributed Nonlinear Systems (Physical and Mathematical Literature, Moscow, 1997).

Sandstede, B.

Scroggie, A. J.

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

Soto-Crespo, J. M.

Spielmann, Ch.

Staliunas, K.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A 56, 1582–1591 (1997).
[CrossRef]

Stolen, R. H.

Tan, H. H.

Taranenko, V. B.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A 56, 1582–1591 (1997).
[CrossRef]

Torruellas, W. E.

Tresser, C.

P. Coullet, C. Riera, and C. Tresser, “Stable static localized structures in one dimension,” Phys. Rev. Lett. 84, 3069–3072 (2000).
[CrossRef] [PubMed]

Trillo, S.

Turitsyn, S. K.

Wai, P. K. A.

Weiss, C. O.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A 56, 1582–1591 (1997).
[CrossRef]

C. O. Weiss, “Spatio-temporal structures. Part II. Vortices and defects in lasers,” Phys. Rep. 219, 311–338 (1992).
[CrossRef]

Wright, E. M.

A. M. Dunlop, E. M. Wright, and W. J. Firth, “Spatial soliton laser,” Opt. Commun. 147, 393–401 (1998).
[CrossRef]

Appl. Phys. B (1)

E. P. Ippen, “Principles of passive mode locking,” Appl. Phys. B 58, 159–170 (1994).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and M. N. Islam, “Theory of soliton laser,” IEEE J. Quantum Electron. QE-21, 1172–1177 (1985).
[CrossRef]

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

F. X. Kärtner, I. D. Jung, and U. Keller, “Soliton mode-locking with saturable absorbers,” IEEE J. Sel. Top. Quantum Electron. 2, 540–556 (1996).
[CrossRef]

J. Appl. Phys. (1)

H. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys. 46, 3049–3058 (1975).
[CrossRef]

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

Opt. Commun. (5)

J. Hermann, “Starting dynamic, self-starting condition and mode-locking threshold in passive, coupled-cavity or Kerr-lens mode-locked solid-state lasers,” Opt. Commun. 98, 111–116 (1993).
[CrossRef]

J. D. Moores, “On the Ginzburg–Landau laser mode-locking model with fifth-order saturable absorber term,” Opt. Commun. 96, 65–70 (1993).
[CrossRef]

F. I. Khatri, J. D. Moores, G. Lenz, and H. A. Haus, “Models for self-limited additive pulse mode-locking,” Opt. Commun. 114, 447–452 (1995).
[CrossRef]

S. M. J. Kelly, “Mode-locking dynamics of a laser coupled to an empty external cavity,” Opt. Commun. 70, 495 (1989).
[CrossRef]

A. M. Dunlop, E. M. Wright, and W. J. Firth, “Spatial soliton laser,” Opt. Commun. 147, 393–401 (1998).
[CrossRef]

Opt. Lett. (9)

Phys. Rep. (1)

C. O. Weiss, “Spatio-temporal structures. Part II. Vortices and defects in lasers,” Phys. Rep. 219, 311–338 (1992).
[CrossRef]

Phys. Rev. A (2)

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A 56, 1582–1591 (1997).
[CrossRef]

P. K. Jakobsen, J. V. Moloney, A. C. Newell, and R. Indik, “Space–time dynamics of wide-gain-section lasers,” Phys. Rev. A 45, 8129–8147 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

P. Coullet, C. Riera, and C. Tresser, “Stable static localized structures in one dimension,” Phys. Rev. Lett. 84, 3069–3072 (2000).
[CrossRef] [PubMed]

W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. 76, 1623–1626 (1996).
[CrossRef] [PubMed]

C. S. Ng and A. Bhattacharjee, “Ginzburg–Landau model and single-mode operation of a free-electron laser oscillator,” Phys. Rev. Lett. 82, 2665–2668 (1999).
[CrossRef]

Other (2)

N. N. Rozanov, Optical Bistability and Hysteresis in Distributed Nonlinear Systems (Physical and Mathematical Literature, Moscow, 1997).

N. N. Akhmediev and A. Ankiewicz, Solitons: Nonlinear Pulses and Beams (Chapman & Hall, London, 1997).

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Figures (13)

Fig. 1
Fig. 1

(a) Locus of singular points for the set of ODEs (7). Only the upper-half for positive M is shown. The solid and the dotted curves correspond to the two solutions of the biquadratic Eq. 7(a). The equation parameters are written inside the figure. D, as in the rest of the figures, is taken to be D=1. The values β, μ, ν, and δ are common for all three curves, and is different. (b) Parametric dependence of a versus ω.

Fig. 2
Fig. 2

(a) Cw amplitude (dotted curve) and the peak amplitude of the plain-soliton solutions (solid curve) versus . (b) ω versus for cw’s and the plain-soliton solutions.

Fig. 3
Fig. 3

Illustration of the stability for the cw solutions. The gain changes sign at the stationary values of the amplitude a. Arrows at the two intervals between the singular points show whether the amplitude at this interval increases or decreases.

Fig. 4
Fig. 4

Real part of the eigenvalue g as a function of the perturbation frequency Ω for the lower-amplitude cw solution. The values of the parameters are β=0.18,δ=-0.1,μ=-0.1,ν=-0.6,=1 (continuous curve), and =1.5 (dotted curve).

Fig. 5
Fig. 5

Maximum growth rate as a function of a. Parameter values are β=0.18,δ=-0.1,μ=-0.2,ν=-0.1, and =1.5 (upper curve), 1 (middle curve), and 0.5 (lowest curve). These values of are the same as in Fig. 1.

Fig. 6
Fig. 6

Maximum growth rate, Re(g)max, as a function of for the cw solutions with M=0. The continuous curve is for the low-amplitude cw solutions, and the dotted curve is for the high-amplitude cw solutions.

Fig. 7
Fig. 7

Region (vertically hatched area) in the plane (δ, ) where all the cw solutions are unstable and the region (horizontally hatched area) where SP solitons are stable.

Fig. 8
Fig. 8

Region (shaded) in the plane (μ, ) where cw solutions are modulationally unstable and the region (dashed) where SP solitons are stable.

Fig. 9
Fig. 9

Region (shaded) in the plane (ν, ) where all cw solutions are modulationally unstable and the region (dashed) where SP solitons are stable.

Fig. 10
Fig. 10

Evolution of the low-amplitude cw solution perturbed by a weak periodic wave. The perturbation initially grows, but finally the whole solution vanishes because of the absence of stable pulses at these values of the equation parameters.

Fig. 11
Fig. 11

Evolution of the lower-amplitude cw perturbed by a weak periodic wave. The perturbation grows, and the cw is gradually transformed into the higher-amplitude cw solution, which is the only stable solution for these values of the parameters.

Fig. 12
Fig. 12

Transformation of an unstable cw solution into a stable soliton. Parameter values are shown in the figure.

Fig. 13
Fig. 13

Same as Fig. 12 but with a periodical perturbation of different frequency. The period of the perturbation is the length of the x axis.

Equations (43)

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

iψz+D2ψtt+|ψ|2ψ+ν|ψ|4ψ
=iδψ+i|ψ|2ψ+iβψtt+iμ|ψ|4ψ,
ψ(t, z)=ψ0(τ)exp(-iωz)=a(τ)exp[iϕ(τ)-iωz],
ω-D2ϕ2+βϕ+vϕa+2βϕa+D2a
+a3+νa5=0,
-δ+βϕ2+D2ϕa+(Dϕ-v)a
-βa-a3-μa5=0,
ω-D2M2+βM+vMa+2βMa+D2a
+a3+νa5=0,
-δ+βM2+D2Ma+(DM-v)a
-βa-a3-μa5=0,
M=-y(8β2M+2M-2Dv)a(4β2+1)-4βω+4βMv-2δD1+4β2+a2(2D-4β)1+4β2+a4(2Dμ-4βν)1+4β2,
y=M2a-2(Dω+2βδ)1+4β2a-2(D+2β)1+4β2a3-2(Dν+2βμ)1+4β2a5-4βv1+4β2y-2Dv1+4β2Ma,
a=y.
M=y(8β2M+2M)a(4β2+1)-4βω-2δD1+4β2+a2(2D-4β)1+4β2+a4(2Dμ-4βν)1+4β2,
y=M2a-2(Dω+2βδ)1+4β2a-2(D+2β)1+4β2a3-2(Dν+2βμ)1+4β2a5,
a=y.
(Dμ-2βν)a4+(D-2β)a2+(Dδ-2βω)=0,
M2=2[(Dω+2βδ)+(D+2β)a2+(Dν+2βμ)a4]1+4β2.
ψ(t,z)=a1,2 exp(-iωz),
a1,22=-±2-4δμ2µ,
ω=-νa4-a2.
2>4δμ.
Ψ(t, z)=a exp[i(Mt-ωz)],
Ψ(t, z)=[a exp(iMt)+αf(t, z)]exp(-iωz),
ifz+D2-iβftt+(ω-iδ)f+(1-i)
×[2a2f+a2 exp(2iMt)f*]+(ν-iμ)
×[3a4+2a4 exp(2iMt)f*]=0,
f(t, z)=h(t, g)exp(gz),
[ω+i(g-δ)]f˜(Ω)]-Ω2D2-iβf˜(Ω)+(1-i)
×[2a2f˜(Ω)+a2f˜*(2M-Ω)]+(ν-iμ)
×[3a4f˜(Ω)+2a4f˜*(2M-Ω)]=0.
[ω-i(g-δ)]f˜*(2M-Ω)-(2M-Ω)2
×D2+iβf˜*(2M-Ω)+(1+i)
×[2a2f˜*(2M-Ω)+a2f˜(Ω)]+(ν+iμ)
×[3a4f˜*(2M-Ω)+2a4f˜(Ω)]=0.
ig+CPP*S-igA1A2=00,
C=ω-iδ-Ω2D2-iβ+(1-i)2a2+(ν-iμ)3a4,
S=C*-[(2M)2-4MΩ]D2+iβ,
P=(1-i)a2+(ν-iμ)2a4.
g2+ig(S-C)+CS-|P|2=0
g=-Im(C)±|P|2-[Re(C)]2,
Ψ(t, 0)=a+0.001 cos(Ωt),

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