Abstract

A derivation of an analytical expression for the propagation of a parabolic pulse in an optical fiber amplifier with a finite Lorentzian gain bandwidth is presented. Through a separation of variables combined with the method of stationary phase, we derive equations in both time and frequency spaces to obtain the analytical solution. This results in a compact analytical form that has a number of physical meanings. It shows that the finite gain bandwidth seriously limits the performance of parabolic amplification by distorting both the chirp and the frequency envelope, thus preventing efficient pulse compression required for high-power femtosecond pulse generation. The validity of the analytical derivation is verified through numerical simulations using the split-step Fourier method, showing an excellent agreement with the derived analytical solution, in pulse shape, chirp, and the optical spectrum.

© 2006 Optical Society of America

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  1. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
    [CrossRef] [PubMed]
  2. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  3. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
    [CrossRef]
  4. D. B. S. Soh, J. Nilsson, and A. B. Grudinin, "Efficient femtosecond pulse generation using a parabolic amplifier and a pulse compressor. I. Stimulated Raman scattering effects," J. Opt. Soc. Am. B 22, 1-9 (2005).
  5. J. Limpert, T. Schreiber, T. Clausnitzer, K. Zollner, H.-J. Fuchs, E.-B. Kley, H. Zellmer, and A. Tünnermann, "High-power femtosecond Yb-doped fiber amplifier," Opt. Express 10, 628-638 (2002).
    [PubMed]
  6. C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
    [CrossRef]
  7. G. P. Agrawal, "Optical pulse propagation in doped fiber amplifiers," Phys. Rev. A 44, 7493-7501 (1991).
    [CrossRef] [PubMed]
  8. L. W. Liou and G. P. Agrawal, "Solitons in fiber amplifiers beyond the parabolic-gain and rate-equation approximations," Opt. Commun. 124, 500-504 (1996).
    [CrossRef]
  9. C. Paré and P.-A. Bélanger, "Optical solitary waves in the presence of a Lorentzian gain line: limitations of the Ginzburg-Landau model," Opt. Commun. 145, 385-392 (1998).
    [CrossRef]
  10. A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
    [CrossRef]
  11. E. T. Copson, Asymptotic Expansions (Cambridge U. Press, 1965).
    [CrossRef]
  12. D. J. Richardson, V. V. Afanasjev, A. B. Grudinin, and D. N. Payne, "Amplification of femtosecond pulses in a passive, all-fiber soliton source," Opt. Lett. 17, 1596-1598 (1992).
    [CrossRef] [PubMed]
  13. G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).
  14. G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, "Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth," Opt. Lett. 29, 2647-2649 (2004).
    [CrossRef] [PubMed]
  15. E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
    [CrossRef]
  16. W. J. Tomlinson and W. H. Knox, "Limits of fiber-grating optical pulse compression," J. Opt. Soc. Am. B 4, 1404-1411 (1987).
    [CrossRef]
  17. O. E. Martinez, "Grating and prism compressors in the case of finite beam size," J. Opt. Soc. Am. B 3, 929-934 (1986).
    [CrossRef]

2005 (1)

D. B. S. Soh, J. Nilsson, and A. B. Grudinin, "Efficient femtosecond pulse generation using a parabolic amplifier and a pulse compressor. I. Stimulated Raman scattering effects," J. Opt. Soc. Am. B 22, 1-9 (2005).

2004 (2)

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

G. Chang, A. Galvanauskas, H. G. Winful, and T. B. Norris, "Dependence of parabolic pulse amplification on stimulated Raman scattering and gain bandwidth," Opt. Lett. 29, 2647-2649 (2004).
[CrossRef] [PubMed]

2003 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2002 (3)

2000 (1)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

1998 (1)

C. Paré and P.-A. Bélanger, "Optical solitary waves in the presence of a Lorentzian gain line: limitations of the Ginzburg-Landau model," Opt. Commun. 145, 385-392 (1998).
[CrossRef]

1996 (1)

L. W. Liou and G. P. Agrawal, "Solitons in fiber amplifiers beyond the parabolic-gain and rate-equation approximations," Opt. Commun. 124, 500-504 (1996).
[CrossRef]

1992 (1)

1991 (1)

G. P. Agrawal, "Optical pulse propagation in doped fiber amplifiers," Phys. Rev. A 44, 7493-7501 (1991).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

1969 (1)

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

Afanasjev, V. V.

Agrawal, G. P.

L. W. Liou and G. P. Agrawal, "Solitons in fiber amplifiers beyond the parabolic-gain and rate-equation approximations," Opt. Commun. 124, 500-504 (1996).
[CrossRef]

G. P. Agrawal, "Optical pulse propagation in doped fiber amplifiers," Phys. Rev. A 44, 7493-7501 (1991).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

Bélanger, P.-A.

C. Paré and P.-A. Bélanger, "Optical solitary waves in the presence of a Lorentzian gain line: limitations of the Ginzburg-Landau model," Opt. Commun. 145, 385-392 (1998).
[CrossRef]

Billet, C.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Chang, G.

Clausnitzer, T.

Copson, E. T.

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, 1965).
[CrossRef]

Dudley, J. M.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Finot, C.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Fuchs, H.-J.

Galvanauskas, A.

Grudinin, A. B.

D. B. S. Soh, J. Nilsson, and A. B. Grudinin, "Efficient femtosecond pulse generation using a parabolic amplifier and a pulse compressor. I. Stimulated Raman scattering effects," J. Opt. Soc. Am. B 22, 1-9 (2005).

D. J. Richardson, V. V. Afanasjev, A. B. Grudinin, and D. N. Payne, "Amplification of femtosecond pulses in a passive, all-fiber soliton source," Opt. Lett. 17, 1596-1598 (1992).
[CrossRef] [PubMed]

Harvey, J. D.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Kley, E.-B.

Knox, W. H.

Kruglov, V. I.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Kruhlak, R. J.

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

Limpert, J.

Liou, L. W.

L. W. Liou and G. P. Agrawal, "Solitons in fiber amplifiers beyond the parabolic-gain and rate-equation approximations," Opt. Commun. 124, 500-504 (1996).
[CrossRef]

Martinez, O. E.

Millot, G.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Nilsson, J.

D. B. S. Soh, J. Nilsson, and A. B. Grudinin, "Efficient femtosecond pulse generation using a parabolic amplifier and a pulse compressor. I. Stimulated Raman scattering effects," J. Opt. Soc. Am. B 22, 1-9 (2005).

Norris, T. B.

Paré, C.

C. Paré and P.-A. Bélanger, "Optical solitary waves in the presence of a Lorentzian gain line: limitations of the Ginzburg-Landau model," Opt. Commun. 145, 385-392 (1998).
[CrossRef]

Payne, D. N.

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, "Self-similar propagation of parabolic pulses in normal-dispersion fiber amplifiers," J. Opt. Soc. Am. B 19, 461-469 (2002).
[CrossRef]

Pitois, S.

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

Richardson, D. J.

Schreiber, T.

Soh, D. B.

D. B. S. Soh, J. Nilsson, and A. B. Grudinin, "Efficient femtosecond pulse generation using a parabolic amplifier and a pulse compressor. I. Stimulated Raman scattering effects," J. Opt. Soc. Am. B 22, 1-9 (2005).

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

Tomlinson, W. J.

Treacy, E. B.

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

Tünnermann, A.

Winful, H. G.

Zellmer, H.

Zollner, K.

IEEE J. Quantum Electron. (1)

E. B. Treacy, "Optical pulse compression with diffraction gratings," IEEE J. Quantum Electron. QE-5, 454-458 (1969).
[CrossRef]

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

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, "Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers," IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Commun. (3)

L. W. Liou and G. P. Agrawal, "Solitons in fiber amplifiers beyond the parabolic-gain and rate-equation approximations," Opt. Commun. 124, 500-504 (1996).
[CrossRef]

C. Paré and P.-A. Bélanger, "Optical solitary waves in the presence of a Lorentzian gain line: limitations of the Ginzburg-Landau model," Opt. Commun. 145, 385-392 (1998).
[CrossRef]

A. C. Peacock, R. J. Kruhlak, J. D. Harvey, and J. M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

G. P. Agrawal, "Optical pulse propagation in doped fiber amplifiers," Phys. Rev. A 44, 7493-7501 (1991).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, "Self-similar propagation and amplification of parabolic pulses in optical fibers," Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

Other (2)

E. T. Copson, Asymptotic Expansions (Cambridge U. Press, 1965).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic, 2001).

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Figures (5)

Fig. 1
Fig. 1

Comparison between a numerical simulation of the nonlinear Schrödinger equation (solid curves), the asymptotic solution with infinite gain bandwidth (dotted curves), and the analytical solution derived in this paper (circles). (a) Signal pulse shape in time, (b) chirp, and (c) the optical spectrum. To see clearly the effect of the finite gain bandwidth on the pulse propagation, we also show (d) the optical spectrum of the propagating signal pulse from the numerical simulation.

Fig. 2
Fig. 2

Comparison of pulse shape, chirp, and spectrum from numerical simulation (solid curves), asymptotic solution with infinite gain bandwidth (dotted curves), and analytical solution (circles). Two amplifiers with different gains are considered: (a) 3.5 dB m and (b) 4.5 dB m .

Fig. 3
Fig. 3

(a) Compressed amplified pulse and (b) the peak power of the compressed pulse for different distances between the two diffraction gratings.

Fig. 4
Fig. 4

Compression efficiency (solid curve, open circles) and compressed pulse duration (dashed curve with squares).

Fig. 5
Fig. 5

Fiber length limits versus gain coefficient α for fibers with two dispersions (a) β 2 = 20 ps 2 km 1 and (b) β 2 = 40 ps 2 km 1 . The figure shows the critical fiber length to avoid signal pulse distortion by the SRS (open circles) and the fiber length limit to avoid the signal pulse distortion by the finite gain bandwidth (squares). The patterned area represents the set of acceptable design parameters. Equilines for the same total gain of (a) 26.91 dB and (b) 28.63 dB are also shown (dashed curves). Dotted curves represent the strong Raman interaction characteristic length.[4]

Equations (47)

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ψ z + i 2 β 2 2 ψ T 2 = i γ s ψ 2 ψ + α 2 P ( ψ ) ,
1 Ω P T + P = ψ ,
ψ a z + i 2 β 2 2 ψ a T 2 = i γ s ψ a 2 ψ a + α 2 ψ a ,
ψ a ( z , T ) = A 0 B T p ( z ) [ 1 T 2 T p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + α 12 β 2 [ T p 2 ( z ) 2 T 2 ] } ) ,
F [ ψ ( T ) ] = 1 2 π ψ ( T ) exp ( i ω T ) d T .
ψ ̃ a ( z , ω ) = 1 i ( 3 β 2 α ) 2 3 A 0 B ω p ( z ) [ 1 ω 2 ω p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + 3 β 2 4 α [ ω p 2 ( z ) + 2 ω 2 ] } ) ,
P ( ψ ) = F 1 [ 1 1 + i ω Ω F [ ψ ] ] ,
P [ ψ ( z , T ) ] = 1 1 + i h ( T ) Ω ψ ( z , T ) exp [ i ξ ( z , T ) ] ,
P ( ψ ( z , T ) ) = 1 1 + i h ( T ) Ω ψ ( z , T ) ,
ψ ( z , T ) = A 0 B T p p ( z ) [ 1 T 2 T p p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + α 12 β 2 1 1 + i h ( T ) Ω [ 1 + i h ( T ) Ω 2 T p p 2 ( z , T ) 2 T 2 ] } ) ,
T p p ( z , T ) = B exp [ α z 3 1 1 + i h ( T ) Ω ] ,
h ( T ) = ε Ω 2 [ 1 + ( 2 α 3 β 2 Ω T ) 2 1 ] 1 2 ,
ϕ c ( ω ) = a c ( ω ω 0 ) 2 ( 1 β c ω ω 0 ω 0 ) ,
ω p = ( α U in γ 2 β 2 2 ) 1 3 exp ( α L 3 ) .
L = 3 α ln ( 2 Ω 3 β 2 2 α U in γ ) ,
Φ ( T ) + ω T Φ ( T 0 ) + ω T 0 + 1 2 d 2 Φ ( T 0 ) d T 2 ( T T 0 ) 2 .
F [ f ( T ) ] = 1 2 π A ( T ) exp [ i Φ ( T ) + i ω T ] d T 1 2 π A ( T 0 ) exp { i [ Φ ( T 0 ) + ω T 0 ] } exp [ i 2 d 2 Φ ( T 0 ) d T 2 ( T T 0 ) 2 ] d T .
exp ( i a x 2 ) d x = π i a ,
F [ f ( T ) ] 1 i [ d 2 Φ ( T 0 ) d T 2 ] 1 2 A ( T 0 ) exp { i [ Φ ( T 0 ) + ω T 0 ] } .
F [ f ( T ) ] 1 i n = 0 N [ d 2 Φ ( T n ) d T 2 ] 1 2 A ( T n ) exp { i [ Φ ( T n ) + ω T n ] } .
ψ ̃ a ( z , ω ) = 1 i ( 3 β 2 α ) 2 3 A 0 B ω p ( z ) [ 1 ω 2 ω p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + 3 β 2 4 α [ ω p 2 ( z ) + 2 ω 2 ] } ) ,
ψ a ( z , T ) = A 0 B T p ( z ) [ 1 T 2 T p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + α 12 β 2 [ T p 2 ( z ) 2 T 2 ] } ) ,
ψ z + i 2 β 2 2 ψ T 2 = i γ s ψ 2 ψ + α 2 1 + i h ( T ) Ω ψ .
A z = β 2 A T Φ T + β 2 2 A 2 Φ T 2 + α 2 [ 1 1 + i h ( T ) Ω ] A ,
[ β 2 2 ( Φ T ) 2 Φ z ] A = β 2 2 2 A T 2 γ s A 3 .
θ = f ( z ) 2 exp [ α z 1 + i h ( T ) Ω ] Φ T
d f d z 1 f = B 2 2 2 Φ T 2 + α 2 [ 1 1 + i h ( T ) Ω ] ,
β 2 2 ( Φ T ) 2 1 f 2 Φ z 1 f 2 = γ s F 2 .
β 2 2 θ 2 f 2 exp [ 2 α z 1 + i h ( T ) Ω ] Φ z 1 f 2 = γ s F 2 .
β 2 2 1 f 6 exp [ 2 α z 1 + i h ( T ) Ω ] = a γ s ,
1 f 2 Φ z = γ s ,
F ( θ ) = 1 a θ 2 .
f ( z ) = A 0 exp { α z 3 [ 1 1 + i h ( T ) Ω ] } ,
Φ z = γ s A 0 2 exp [ 2 α z 3 1 1 + i h ( T ) Ω ] ,
2 Φ T 2 = α 3 β 2 1 1 + i h ( T ) Ω .
Φ ( z , T ) = ϕ 0 + 3 γ s A 0 2 1 + i h ( T ) Ω 2 α exp [ 2 α z 3 1 1 + i h ( T ) Ω ] α 6 β 2 T 2 1 + i h ( T ) Ω ,
ψ ( z , T ) = A 0 B T p p ( z ) [ 1 T 2 T p p 2 ( z ) ] 1 2 exp ( i { ϕ 0 + α 12 β 2 1 1 + i h ( T ) Ω [ 1 + i h ( T ) Ω 2 T p p 2 ( z , T ) 2 T 2 ] } ) ,
T p p ( z , T ) = B exp [ α z 3 1 1 + i h ( T ) Ω ] .
α 3 β 2 T 1 + i h ( T ) Ω + ω = 0 .
h ( T ) = Ω 2 [ 1 + ( 2 α 3 β 2 Ω T ) 2 1 ] 1 2 .
2 T 1 + i h ( T ) Ω > h ( T ) Ω 2 d h ( T ) d T 1 + i h ( T ) Ω 3 T 2 .
1 + 2 h ( T ) 2 Ω 2 > h ( T ) 2 2 Ω 2 ,
h ( T ) = ε Ω 2 [ 1 + ( 2 α 3 β 2 Ω T ) 2 1 ] 1 2 .
ψ ̃ exp ( i ω T ) = Φ [ z , h 1 ( ω ) ] + ω h 1 ( ω ) ω T ,
ω ψ ̃ exp ( i ω T ) = ω Φ [ z , h 1 ( ω ) ] + h 1 ( ω ) + ω h 1 ( ω ) ω T = 0 .
ω 1 1 + i h ( T n ) Ω = T n ω T n [ 1 1 + i h ( T n ) Ω ] ,
[ α 3 β 2 h 1 ( ω ) 1 + i ω Ω + ω ] h 1 ( ω ) ω + h 1 ( ω ) T = 0 .

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