Abstract

The effects of the third-order dispersion on the dynamics of solid-state solitary lasers are studied by means of the soliton perturbation theory, with the radiation emitted by the solitary pulse also taken into account. A set of equations for the soliton amplitude and frequency is found and studied. The equations include the contribution of radiation to the frequency shift of the solitary pulse, which is calculated for the first time to our knowledge. The results are in very good agreement with the numerical solutions of the master equation and with the reported experiments.

© 1997 Optical Society of America

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

1995 (5)

1994 (3)

I. M. Uzunov, M. Golles, and F. Lederer, “Soliton interaction in the presence of bandwidth limited amplification near the zero-dispersion wavelength,” Electron. Lett. 30, 882–883 (1994).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. QE-30, 1100–1114 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
[CrossRef] [PubMed]

1993 (7)

1992 (6)

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

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

T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

J. N. Elgin, “Soliton propagation in an optical fiber with third-order dispersion,” Opt. Lett. 17, 1409–1410 (1992).
[CrossRef] [PubMed]

J. P. Gordon, “Dispersive perturbations of solitons of the nonlinear Schrödinger equation,” J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

1991 (7)

1990 (3)

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
[CrossRef]

M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

1987 (1)

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. QE-23, 510–524 (1987).
[CrossRef]

1986 (1)

1984 (2)

1978 (1)

A. C. Newell, “The inverse scattering transform, nonlinear waves, singular perturbations and synchronized solitons,” Rocky Mount. J. Math. 8, 25–52 (1978).
[CrossRef]

1977 (1)

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Afanasjev, V. V.

Andrejco, M. J.

Asaki, M. T.

Blow, K. J.

Brabec, T.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. QE-30, 1100–1114 (1994).
[CrossRef]

T. Brabec and S. M. J. Kelly, “Third-order dispersion as a limiting factor to mode locking in femtosecond solitary lasers,” Opt. Lett. 18, 2002–2004 (1993).
[CrossRef] [PubMed]

P. F. Curley, C. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

T. Brabec, C. Spielmann, and F. Krausz, “Mode locking in solitary lasers,” Opt. Lett. 16, 1961–1963 (1991).
[CrossRef] [PubMed]

Chen, H. H.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[CrossRef] [PubMed]

Curley, P. F.

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. QE-30, 1100–1114 (1994).
[CrossRef]

P. F. Curley, C. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

T. Brabec, C. Spielmann, P. F. Curley, and F. Krausz, “Kerr lens mode locking,” Opt. Lett. 17, 1292–1294 (1992).
[CrossRef] [PubMed]

Doran, N. J.

Duling III, I. N.

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

Elgin, J. N.

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

J. N. Elgin, “Perturbations of optical solitons,” Phys. Rev. A 47, 4331–4341 (1993).
[CrossRef] [PubMed]

J. N. Elgin, “Soliton propagation in an optical fiber with third-order dispersion,” Opt. Lett. 17, 1409–1410 (1992).
[CrossRef] [PubMed]

Fermann, M. E.

Fork, R. L.

Fujimoto, J. G.

H. A. Haus, J. G. Fujimoto, and E. P. Ippen, “Analytic theory of additive pulse and Kerr lens mode locking,” IEEE J. Quantum Electron. QE-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]

Garvey, D.

Golles, M.

I. M. Uzunov, M. Golles, and F. Lederer, “Stabilization of soliton trains in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 12, 1164–1166 (1995).
[CrossRef]

I. M. Uzunov, M. Golles, and F. Lederer, “Soliton interaction in the presence of bandwidth limited amplification near the zero-dispersion wavelength,” Electron. Lett. 30, 882–883 (1994).
[CrossRef]

Golovchenko, E. A.

Gordon, J. P.

Haberl, F.

Hadley, G. R.

Hasegawa, A.

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. QE-23, 510–524 (1987).
[CrossRef]

Haus, H. A.

Hochreiter, H.

Hofer, M.

Huang, C. P.

Ippen, E. P.

Kapteyn, H. C.

Karpman, V. I.

V. I. Karpman, “Radiation by solitons due to higher-order dispersion,” Phys. Rev. E 47, 2073–2082 (1993).
[CrossRef]

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Kean, P. N.

Kelly, S. M. J.

Kodama, Y.

Y. Kodama, M. Romagnoli, S. Wabnitz, and M. Midrio, “Role of third-order dispersion on soliton instabilities and interactions in optical fibers,” Opt. Lett. 19, 165–167 (1994).
[CrossRef] [PubMed]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. QE-23, 510–524 (1987).
[CrossRef]

Krausz, F.

Kuehl, H. H.

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
[CrossRef]

Lederer, F.

I. M. Uzunov, M. Golles, and F. Lederer, “Stabilization of soliton trains in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 12, 1164–1166 (1995).
[CrossRef]

I. M. Uzunov, M. Golles, and F. Lederer, “Soliton interaction in the presence of bandwidth limited amplification near the zero-dispersion wavelength,” Electron. Lett. 30, 882–883 (1994).
[CrossRef]

Lee, Y. C.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[CrossRef] [PubMed]

Lenzner, M.

Liu, L. Y.

Martinez, O. E.

Maslov, E. M.

V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons,” Sov. Phys. JETP 46, 281–291 (1977).

Menyuk, C. R.

Midrio, M.

Mollenauer, L. F.

Moores, J. D.

Murnane, M. M.

Nelson, L. E.

Newell, A. C.

A. C. Newell, “The inverse scattering transform, nonlinear waves, singular perturbations and synchronized solitons,” Rocky Mount. J. Math. 8, 25–52 (1978).
[CrossRef]

Ober, M. H.

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

Pilipetskii, A. N.

Romagnoli, M.

Schmidt, A. J.

Sibbett, W.

Silberberg, Y.

Smith, K.

Spence, D. E.

Spielmann, C.

Stingl, A.

Stolen, R. H.

Szipocs, R.

Turi, L.

Uzunov, I. M.

I. M. Uzunov, M. Golles, and F. Lederer, “Stabilization of soliton trains in optical fibers in the presence of third-order dispersion,” J. Opt. Soc. Am. B 12, 1164–1166 (1995).
[CrossRef]

I. M. Uzunov, M. Golles, and F. Lederer, “Soliton interaction in the presence of bandwidth limited amplification near the zero-dispersion wavelength,” Electron. Lett. 30, 882–883 (1994).
[CrossRef]

Wabnitz, S.

Wai, P. K. A.

P. K. A. Wai, H. H. Chen, and Y. C. Lee, “Radiations by “solitons” at the zero group-dispersion wavelength of single-mode optical fibers,” Phys. Rev. A 41, 426–439 (1990).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[CrossRef] [PubMed]

Weiner, A. M.

Wintner, E.

P. F. Curley, C. Spielmann, T. Brabec, F. Krausz, E. Wintner, and A. J. Schmidt, “Operation of a femtosecond Ti:sapphire solitary laser in the vicinity of zero group-delay dispersion,” Opt. Lett. 18, 54–56 (1993).
[CrossRef] [PubMed]

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

Wood, D.

Zhang, C. Y.

H. H. Kuehl and C. Y. Zhang, “Effects of higher-order dispersion on envelope solitons,” Phys. Fluids B 2, 889–900 (1990).
[CrossRef]

Zhou, J.

Electron. Lett. (3)

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

I. M. Uzunov, M. Golles, and F. Lederer, “Soliton interaction in the presence of bandwidth limited amplification near the zero-dispersion wavelength,” Electron. Lett. 30, 882–883 (1994).
[CrossRef]

Y. Kodama, M. Romagnoli, and S. Wabnitz, “Soliton stability and interactions in fibre lasers,” Electron. Lett. 28, 1981–1983 (1992).
[CrossRef]

IEEE J. Quantum Electron. (4)

F. Krausz, M. E. Fermann, T. Brabec, P. F. Curley, M. Hofer, M. H. Ober, C. Spielmann, E. Wintner, and A. J. Schmidt, “Femtosecond solid-state laser,” IEEE J. Quantum Electron. QE-28, 2097–2121 (1992).
[CrossRef]

C. Spielmann, P. F. Curley, T. Brabec, and F. Krausz, “Ultrabroadband femtosecond lasers,” IEEE J. Quantum Electron. QE-30, 1100–1114 (1994).
[CrossRef]

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

Y. Kodama and A. Hasegawa, “Nonlinear pulse propagation in a monomode dielectric guide,” IEEE J. Quantum Electron. QE-23, 510–524 (1987).
[CrossRef]

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

Opt. Commun. (1)

J. N. Elgin, T. Brabec, and S. M. J. Kelly, “A perturbative theory of soliton propagation in the presence of third order dispersion,” Opt. Commun. 114, 321–328 (1995).
[CrossRef]

Opt. Lett. (20)

L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13–15 (1984).
[CrossRef] [PubMed]

R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984).
[CrossRef] [PubMed]

P. K. A. Wai, C. R. Menyuk, Y. C. Lee, and H. H. Chen, “Nonlinear pulse propagation in the neighborhood of the zero-dispersion wavelength of monomode optical fibers,” Opt. Lett. 11, 464–466 (1986).
[CrossRef] [PubMed]

M. E. Fermann, F. Haberl, M. Hofer, and H. Hochreiter, “Nonlinear amplifying loop mirror,” Opt. Lett. 15, 752–754 (1990).
[CrossRef] [PubMed]

D. E. Spence, P. N. Kean, and W. Sibbett, “60-fsec pulse generation from a self-mode-locked Ti:sapphire laser,” Opt. Lett. 16, 42–44 (1991).
[CrossRef] [PubMed]

M. E. Fermann, M. Hofer, F. Haberl, A. J. Schmidt, and L. Turi, “Additive-pulse-compression mode locking of a neodymium fiber laser,” Opt. Lett. 16, 244–246 (1991).
[CrossRef] [PubMed]

G. R. Hadley, “Transparent boundary condition for beam propagation,” Opt. Lett. 16, 624–626 (1991).
[CrossRef] [PubMed]

S. M. J. Kelly, K. Smith, K. J. Blow, and N. J. Doran, “Average soliton dynamics of a high-gain erbium fiber laser,” Opt. Lett. 16, 1337–1339 (1991).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Bifurcation diagrams. (a) For amplitude; the solid (dashed) curve is the stable (unstable) critical point of Eqs. (8) and (9). The dashed–dotted (dotted) curve is the stable (unstable) critical point of Eqs. (12) and (13), which include the O(β3) correction. The open circles represent the steady-state amplitude as a result of the numerical solutions of the master equation (1). The coefficients used were g=-0.0025, β =0.051, γ1=0.05, and γ2=0.025. (b) For the frequency with the same coefficients; all symbols are the same as for (a).

Fig. 2
Fig. 2

Comparison of the argument of the sech2 factor of Eqs. (8) and (9) as a result of the SPT (solid curve) and an approximate solution η0=1/2, μ0=0, and μr=1/4β3 (dashed curve). Beyond β3=0.1, the approximation breaks and radiation is overestimated. Coefficients are the same as in Fig. 1. The open circles are the result of the numerical integration.

Fig. 3
Fig. 3

Partitioning of the phase plane (η0, μ0) induced by the sign of the z derivatives of the variables [Eqs. (8) and (9)], in particular the regions (1) (η0)z>0, (μ0)z>0; (2) (η0)z>0, (μ0)z <0; (3) (η0)z<0, (μ0)z>0; (4) (η0)z<0, (μ0)z<0. Intersections A and B are the critical points. (a), (b), (c) are, respectively, for β3 equal to 0.1, 0.2, 0.397. Coefficients are the same as in Fig. 1.

Fig. 4
Fig. 4

Bifurcation diagram for the amplitude as a result of the SPT for g=-0.0025, β=0.049, γ1=0.045, and γ2=0.02 (solid and dashed curves). The dotted curve represents the case of Fig. 1.

Fig. 5
Fig. 5

Pulse width as predicted by the amplitude–width relation of the soliton [Eq. (3)] is represented by the solid curve. Open circles are the results of the numerical solutions of the PDE. Coefficients are the same as Fig. 1.

Fig. 6
Fig. 6

Steady-state spectrum as a result of the numerical integration of Eq. (1) for β3=0 (case A) and β3=0.36 (case B). Coefficients are the same as in Fig. 1.

Equations (49)

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iqz+β22qtt+|q|2q+iβ3qttt
=igq+iβqtt+iγ1|q|2q-iγ2|q|4q,
iqz+½qtt+|q|2q+iβ3qttt=0.
qs(z, t)=2η0 sech[2η0(t-ξ)]exp[i(2μ0(t-ξ)+δ)],
ξz=2μ0,
δz=2(η02+μ02),
-½ ωr2+β3ωr3=2η02.
(η0)z=23η03g-4β(η02+3μ02)+8γ1η02-1285γ2η04,
(μ0)z=-163βμ0η02.
-+|q|2dt=4η0-1π-+ ln(1-|b(z, μ)|2)dμ,
12-+(qqt*-q*qt)dt=-8iμ0η0+1π-+2iμ×ln(1-|b(z, μ)|2)dμ.
(η0)z=23η03g-4β(η02+3μ02)+8γ1η02-1285γ2η04-2β32π2μr2(μr2+η02)2|μr-6β3μr2-μ0|×sech2π2η0(μ0-μr),
(μ0)z=-163βμ0η02+(μ0-μr)2β32π2μr2(μr2+η02)2η0|μr-6β3μr2-μ0|×sech2π2η0(μ0-μr),
μr2-4β3μr3+η02+μ02-2μ0μr=0.
q(z, t)=qs(z, t)exp[iωs(t-ξ)],
(η0)z=23η03g-4β(η02+3μ02)+8γ1η02-1285γ2η04-2β32π2μr2(μr2+η02)2|μr-6β3μr2-μ0+2β3(η02+3μ02)|×sech2π2η0(μ0-μr),
(μ0)z=-163βμ0η02+(μ0-μr)2β32π2μr2(μr2+η02)2η0|μr-6β3μr2-μ0+2β3(η02+3μ02)|×sech2π2η0(μ0-μr)-83β3η021-6π2×[3g+8β(η02+3μ02)].
-+|q|2dt=4η0-1π-+ ln(1-|b(z, μ)|2)dμ,
12-+(qqt*-q*qt)dt=-8iμ0η0+1π-+2iμ×ln(1-|b(z, μ)|2)dμ,
qs(z, t)=2η0 sech[2η0(t-ξ)]exp[i2μ0(t-ξ)+iδ],
ξz=2μ0,
δz=2(η02+μ02).
-+(qzq*+qqz*)dt
=4(η0)z-1π-+[ln(1-|b(z, μ)|2)]zdμ,
-+(qzqt*-qz*qt)dt
=-8i(μ0η0)z+1π-+2iμ×[ln(1-|b(z, μ)|2)]zdμ.
(η0)z=23η03g-4β(η02+μ02)+8γ1η02-1285γ2η04-14π-+(|b|2)zdμ,
(μ0)z=-163βμ0η02+μ0η014π-+(|b|2)zdμ-1η014π-+μ(|b|2)zdμ.
fˆ(z, μ)=b*(z, μ)4(μ2+η02).
ifz=-12ftt-iβ3fttt+iβ32(qs)t,
f(z, t)=12π-+ fˆ(z, ω)exp[iωt]dω,
fˆz=-iω22-β3ω3fˆ+iβ3ω2qˆs,
qˆs(z, ω)=-+qs(z, t)exp[-iωt]dt=π sechπ4η0(2μ0-ω)exp[i(δ-ωξ)].
fˆ=Fˆ exp[i(δ-ωξ)],
Fˆz=-iD(ω)Fˆ+Qˆ(ω),
D(ω)=ω22-β3ω3+δz-ξzω,
Qˆ(ω)=iβ3ωπ2sechπ4η0(2μ0-ω),
Fˆ=Qˆ(ω)iD(ω)[1-exp(-iD(ω)z)].
(|b|2)z=32(μ2+η02)2|Qˆ(ω)|2sin[D(ω)z]D(ω).
δ[D(2µ)]=1dD(2µ)dμδ(μ-μr),
D(2μr)=2(μr2-4β3μr3+η02+μ02-2μ0μr)=0,
-+(|b|2)zdμ-+8π(μ2+η02)2|Qˆ(2µ)|2|μ-6β3μ2-μ0|×δ(μ-μr)dμ,
-+(|b|2)zdμ8β32π3μr2(μr2+η02)2|μr-6β3μr2-μ0|×sech2π2η0(μ0-μr).
Fˆ=Qˆ(ω)A(ω)-iD(ω)(exp{[A(ω)-iD(ω)]z}-1),
q(z, t)=qs(z, t)exp[iωs(t-ξ)],
ξz=2μ0-4β3(η02+3μ02),
δz=2(η02+μ02)-16β3μ0(μ02-η02).
|μ-6β3μ2-μ0+2β3(η02+3μ02)|.
ωs-8β3η021-6π2.

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