Abstract

The master mode-locking equation is a canonical model for mode locking in solid-state lasers. We consider the dynamics and stability of the localized pulse solutions that this equation admits of. We provide analytic proof of stable mode-locking behavior along with analysis that shows that mode locking can become destabilized as a result of either a radiation-mode or a saddle-node instability. This is to our knowledge the first analytic proof of the stability of the pulse solutions that takes the time-dependent gain saturation mechanism of mode-locked lasers into account.

© 2002 Optical Society of America

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References

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  1. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
    [CrossRef]
  2. H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
    [CrossRef]
  3. C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17, 417–419 (1992).
    [CrossRef] [PubMed]
  4. 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]
  5. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
    [CrossRef]
  6. M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).
  7. I. N. Duling III, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544–545 (1991).
    [CrossRef]
  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 extra-cavity feedback,” Electron. Lett. 28, 1894–1896 (1992).
    [CrossRef]
  10. J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sánchez-Mondragón, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
    [CrossRef]
  11. F. X. Kartner and U. Keller, “Stabilization of solitonlike pulses with a slow saturable absorber,” Opt. Lett. 20, 16–18 (1995).
    [CrossRef] [PubMed]
  12. 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]
  13. S. Tsuda, W. H. Knox, E. A. DeSouza, W. J. Jan, and J. E. Cunningham, “Low-loss intracavity AlAs/AlGaAs saturable Bragg reflector for femtosecond mode locking in solid-state lasers,” Opt. Lett. 20, 1406–1408 (1995).
    [CrossRef] [PubMed]
  14. I. P. Christov, H. C. Kapteyn, M. M. Murnane, C. P. Huang, and J. P. Zhou, “Space–time focusing of femtosecond pulses in Ti:sapphire,” Opt. Lett. 20, 309–311 (1995).
    [CrossRef] [PubMed]
  15. I. P. Christov, V. Stoev, M. Murnane, and H. Kapteyn, “Sub-10-fs operation of Kerr-lens mode-locked lasers,” Opt. Lett. 21, 1493–1495 (1996).
    [CrossRef] [PubMed]
  16. I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. P. Zhou, and C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1466 (1994).
    [CrossRef] [PubMed]
  17. N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
    [CrossRef]
  18. A. Mielke, P. Holmes, and J. N. Kutz, “Global existence and uniqueness for an optical fibre laser model,” Nonlinearity 11, 1489–1504 (1998).
    [CrossRef]
  19. D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
    [CrossRef] [PubMed]
  20. 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]
  21. T. Kapitula and B. Sandstede, “Stability of bright solitary-wave solutions to perturbed nonlinear Schrödinger equations,” Physica D 124, 58–103 (1998).
    [CrossRef]
  22. D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
    [CrossRef]
  23. B. Sandstede and A. Scheel, “Essential instability of pulses, and bifurcations to modulated travelling waves,” Proc. R. Soc. Edinburgh, Sect. A 129, 1263–1290 (1999).
    [CrossRef]

1999 (2)

B. Sandstede and A. Scheel, “Essential instability of pulses, and bifurcations to modulated travelling waves,” Proc. R. Soc. Edinburgh, Sect. A 129, 1263–1290 (1999).
[CrossRef]

J. W. Haus, G. Shaulov, E. A. Kuzin, and J. Sánchez-Mondragón, “Vector soliton fiber lasers,” Opt. Lett. 24, 376–378 (1999).
[CrossRef]

1998 (4)

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]

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

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

A. Mielke, P. Holmes, and J. N. Kutz, “Global existence and uniqueness for an optical fibre laser model,” Nonlinearity 11, 1489–1504 (1998).
[CrossRef]

1997 (1)

1996 (1)

1995 (3)

1994 (2)

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

I. P. Christov, M. M. Murnane, H. C. Kapteyn, J. P. Zhou, and C. P. Huang, “Fourth-order dispersion-limited solitary pulses,” Opt. Lett. 19, 1465–1466 (1994).
[CrossRef] [PubMed]

1993 (1)

M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).

1992 (4)

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

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (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]

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Soliton fiber ring laser,” Opt. Lett. 17, 417–419 (1992).
[CrossRef] [PubMed]

1991 (3)

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

I. N. Duling III, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544–545 (1991).
[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]

1990 (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

1977 (1)

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Afanasjev, V.

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Andrejco, M. J.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).

Bergman, K.

Chen, C.-J.

Christov, I. P.

Collings, B. C.

Cundiff, S.

Cunningham, J. E.

Dennis, M. L.

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

DeSouza, E. A.

Duling III, I. N.

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

I. N. Duling III, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544–545 (1991).
[CrossRef]

Fermann, M. E.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[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 (1991).
[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]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (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]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Haus, J. W.

Holmes, P.

Huang, C. 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]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (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]

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Structures for additive pulse mode locking,” J. Opt. Soc. Am. B 8, 2068–2076 (1991).
[CrossRef]

Jan, W. J.

Kapitula, T.

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]

Kapteyn, H.

Kapteyn, H. C.

Kartner, F. X.

Kaup, D. J.

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Keller, U.

Kivshar, Y.

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Knox, W. H.

Kutz, J. N.

Kuzin, E. A.

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]

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]

Menyuk, C. R.

Mielke, A.

A. Mielke, P. Holmes, and J. N. Kutz, “Global existence and uniqueness for an optical fibre laser model,” Nonlinearity 11, 1489–1504 (1998).
[CrossRef]

Murnane, M.

Murnane, M. M.

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]

Pelinovsky, D.

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Pereira, N. R.

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[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]

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]

Sánchez-Mondragón, J.

Sandstede, B.

B. Sandstede and A. Scheel, “Essential instability of pulses, and bifurcations to modulated travelling waves,” Proc. R. Soc. Edinburgh, Sect. A 129, 1263–1290 (1999).
[CrossRef]

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]

Scheel, A.

B. Sandstede and A. Scheel, “Essential instability of pulses, and bifurcations to modulated travelling waves,” Proc. R. Soc. Edinburgh, Sect. A 129, 1263–1290 (1999).
[CrossRef]

Shaulov, G.

Silverberg, Y.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).

Stenflo, L.

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Stock, M. L.

M. E. Fermann, M. J. Andrejco, Y. Silverberg, 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).

Stoev, V.

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.

Wai, P. K. A.

Weinstein, M.

Zhou, J. P.

Electron. Lett. (4)

I. N. Duling III, “Subpicosecond all-fiber erbium laser,” Electron. Lett. 27, 544–545 (1991).
[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 extra-cavity 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. (2)

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

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse modelocking and Kerr lens modelocking,” IEEE J. Quantum Electron. 28, 2086–2096 (1992).
[CrossRef]

J. Opt. Soc. Am. B (3)

Nonlinearity (1)

A. Mielke, P. Holmes, and J. N. Kutz, “Global existence and uniqueness for an optical fibre laser model,” Nonlinearity 11, 1489–1504 (1998).
[CrossRef]

Opt. Lett. (8)

Phys. Fluids (1)

N. R. Pereira and L. Stenflo, “Nonlinear Schrödinger equation including growth and damping,” Phys. Fluids 20, 1733–1734 (1977).
[CrossRef]

Phys. Rev. A (1)

D. J. Kaup, “Perturbation theory for solitons in optical fibers,” Phys. Rev. A 42, 5689–5694 (1990).
[CrossRef] [PubMed]

Physica D (2)

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

D. Pelinovsky, Y. Kivshar, and V. Afanasjev, “Internal modes of envelope solitons,” Physica D 116, 121–142 (1998).
[CrossRef]

Proc. R. Soc. Edinburgh, Sect. A (1)

B. Sandstede and A. Scheel, “Essential instability of pulses, and bifurcations to modulated travelling waves,” Proc. R. Soc. Edinburgh, Sect. A 129, 1263–1290 (1999).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Stable mode-locking evolution for β=0.034 [see Eqs. (25) for the other parameters]. Note that in (a) the evolution quickly settles to the steady-state pulse solution. The stable mode-locking process is further characterized by the L2 and H1 norms in (b), which are related to pulse energy and momentum, respectively, and which settle to fixed values for Z400.

Fig. 2
Fig. 2

Stable mode-locked solutions of the master mode-locking equation for various values of β [see Eqs. (25) for the other parameters]. The steady-state solutions in (a) are significantly affected by the strength of the nonlinear gain. Higher values of β give more energy to the pulse [measured by the L2 norm in (b)] and lead to a smaller signal-to-noise ratio and FWHM.

Fig. 3
Fig. 3

Unstable evolution for β=-0.05 [see Eqs. (25) for the other parameters]. Note that in (a) the evolution quickly evolves into a nonlocalized weak-turbulence regime, which persists over time. This unstable behavior is characterized by the L2 and H1 norms in (b).

Fig. 4
Fig. 4

Unstable evolution for β=0.005 [see Eqs. (25) for the other parameters]. Note that in (a) the evolution quickly evolves into a nonlocalized quasi-periodic solution. Although the cavity energy is controlled by the time-dependent gain of the master mode-locking equation, the quasi-periodic oscillations persist. This unstable behavior is characterized by the L2 and H1 norms in (b).

Fig. 5
Fig. 5

Unstable evolution for β=0.035 [see Eqs. (25) for the other parameters]. Note that in (a) the evolution quickly begins to blow up. The dynamics becomes numerically difficult to resolve beyond Z56. In this case, the cavity energy is not controlled by the time-dependent gain of the master mode-locking equation. Instead, the nonlinear gain dominates the linear attenuation of the filter and loss, and the L2 and H1 norms quickly go toward infinity as shown in (b).

Equations (34)

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

i Uz+λ02D¯4πc 2Ut2+2πn2λ0Aeff|U|2U+iΓU
-i2G01+U2/E01+1Ω2 2t2U=0,
U2=-U(z, t)2dt.
Uz=B|U|2U,
i QZ+[1-iτg(Z)] 2QT2
+(4-iβ)|Q|2Q+i[γ-g(Z)]Q=0,
g(Z)=2g01+Q2/e0
Q(Z, T)=η sech(ωT)1+iA exp(iθZ)
i QZ+2QT2+4|Q|2Q=iτg(Z) 2QT2+iβ|Q|2Q-i[γ-g(Z)]Q.
i QZ+2QT2+4|Q|2Q=0.
Q0(Z, T)=ω2 sech(ωT)exp(iω2Z),
βω3+(β-2τg0)e0ω2-3γω+3e0(2g0-γ)=0.
Q(Z, T)=ω2 sech(ωT)1+iA exp(iω2Z),
A=2τg0e0ω2-γω+e0(2g0-γ)2ω2(e0+ω).
iσ3 VZ+(σ0-iτgσ3) 2VT2+[i(γ-g)σ3-ω2σ0]V
+[8|Q0|2σ0+4(|Q0|2σ1+2iA|Q0|Q1σ2)
+β|Q0|2(σ2-2iσ3)]V+igτ 2Q0T2+Q0
×1-1- Q0*(V1+V2)dT=0,
g=2g01+Q2/e0=2g01+ω/e0,
 Q1=iA ω2 sech(ωT)ln[sech(ωT)].
iλσ3V+(σ0-iτgσ3) 2VT2+[i(γ-g)σ3-ω2σ0]V
+[8|Q0|2σ0+4(|Q0|2σ1+2iA|Q0|Q1σ2)
+β|Q0|2(σ2-2iσ3)]V+igτ 2Q0T2+Q0
×1-1- Q0*(V1+V2)dT=0.
γ>2g01+ω/e0.
ω*=(2g0-γ)e0γ,β*=2τg0γγ+e0(2g0-γ).
λ1=4e02(e0+ω)2 13τg0ω3+γe0ω2+(2γ-3g0)ω+e0(γ-2g0),
λ2=-8τg0e0ω23(e0+ω).
2g0>γ>2g0e0e0+ω,
2g0>γ.
3π/4<arg E(iω2, 0)<7π/4,
E(iω2, 0)=i 32ω2β3.
ω2[β(1+ω)-2τg0]=0,
e0=1.0,g0=0.1,τ=0.1,γ=0.1.

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