Abstract

We discover and analytically describe self-similar pulses existing in homogeneously broadened amplifying linear media in a vicinity of an optical resonance. We demonstrate numerically that the discovered pulses serve as universal self-similar asymptotics of any near-resonant short pulses with sharp leading edges, propagating in coherent linear amplifiers. We show that broadening of any low-intensity seed pulse in the amplifier has a diffusive nature: Asymptotically the pulse width growth is governed by the simple diffusion law. We also compare the energy gain factors of short and long self-similar pulses supported by such media.

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  1. J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
    [CrossRef]
  2. S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fibers,” Opt. Lett. 16, 1478–1480 (1991).
    [CrossRef] [PubMed]
  3. C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
    [CrossRef] [PubMed]
  4. T. M. Monroe, P. D. Millar, P. L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. 23, 268–270 (1998).
    [CrossRef]
  5. M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
    [CrossRef]
  6. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear optical fibers,” J. Opt. Soc. Am. B 10, 1185–1190 (1993).
    [CrossRef]
  7. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21, 68–70 (1996).
    [CrossRef] [PubMed]
  8. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]
  9. F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
    [CrossRef] [PubMed]
  10. B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
    [CrossRef]
  11. W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
    [CrossRef]
  12. S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
    [CrossRef] [PubMed]
  13. S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659–1661 (2007).
    [CrossRef] [PubMed]
  14. L. Wu, J.-F. Zhang, L. Li, and Q. Tian, “Similaritons in nonlinear optical systems,” Opt. Express 16, 6352–6360 (2008).
    [CrossRef] [PubMed]
  15. S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010).
    [CrossRef]
  16. K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
    [CrossRef] [PubMed]
  17. S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56, 37–44 (1982).
  18. I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983).
    [CrossRef]
  19. I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).
  20. S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010).
    [CrossRef]
  21. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., 1975).
  22. P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).
  23. V. V. Kozlov and S. Wabnitz, “Quasi-parabolic pulses in a coherent nonlinear optical amplifier,” Opt. Lett. 35, 2058–2060 (2010).
    [CrossRef] [PubMed]
  24. A small deviation from the universal asymptotics in the pulse tails can be explained by limited accuracy of our sharp leading edge approximation.
  25. M. Abramowitz and I. A. SteganHandbook of Mathematical Functions (Dover, 1972).

2010

B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
[CrossRef]

S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010).
[CrossRef]

S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010).
[CrossRef]

V. V. Kozlov and S. Wabnitz, “Quasi-parabolic pulses in a coherent nonlinear optical amplifier,” Opt. Lett. 35, 2058–2060 (2010).
[CrossRef] [PubMed]

2008

2007

S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659–1661 (2007).
[CrossRef] [PubMed]

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

2006

S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

2004

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

2003

K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

2000

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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

1996

1993

1992

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

1991

1984

I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).

1983

I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983).
[CrossRef]

1982

S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56, 37–44 (1982).

Abramowitz, M.

M. Abramowitz and I. A. SteganHandbook of Mathematical Functions (Dover, 1972).

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659–1661 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Allen, L.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., 1975).

An, S.

Anderson, D.

Buckley, J. R.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Chong, A.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
[CrossRef]

Clark, W. G.

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

de Sterke, C. M.

Desaix, M.

Dudley, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

Dudley, J. M.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Eberly, J. H.

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., 1975).

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

Fibich, G.

K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

Finot, C.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Gabitov, I. R.

I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).

I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983).
[CrossRef]

Gaeta, A.

K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

Haghgoo, S.

S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010).
[CrossRef]

S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010).
[CrossRef]

Harvey, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

Ilday, F. O.

B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Karlsson, M.

Kozlov, V. V.

Kruglov, V. I.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

Levi, D.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Li, L.

Lisak, M.

Manakov, S. V.

I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983).
[CrossRef]

S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56, 37–44 (1982).

Menyuk, C. R.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Mikhailov, A. V.

I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).

Millar, P. D.

Millot, G.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Milonni, P. W.

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).

Moll, K. D.

K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

Monroe, T. M.

Nakazawa, M.

Oktem, B.

B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Poladian, P. L.

Ponomarenko, S. A.

S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010).
[CrossRef]

S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010).
[CrossRef]

S. A. Ponomarenko and G. P. Agrawal, “Optical similaritons in nonlinear waveguides,” Opt. Lett. 32, 1659–1661 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Quiroga-Teixeiro, M. L.

Renninger, W. H.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
[CrossRef]

Richardson, D. J.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Segev, M.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

Sipe, J. E.

Soljacic, M.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

Stegan, I. A.

M. Abramowitz and I. A. SteganHandbook of Mathematical Functions (Dover, 1972).

Tamura, K.

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

Tian, Q.

Ulgudur, F. O.

B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Wabnitz, S.

Winternitz, P.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Wise, F. W.

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
[CrossRef]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Wu, L.

Zakharov, V. E.

I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).

Zhang, J.-F.

J. Opt. Soc. Am. B

Nat. Photon.

B. Oktem, F. O. Ulgudur, and F. O. Ilday, “Soliton-similariton fibre laser,” Nat. Photon. 4, 307–311 (2010).
[CrossRef]

Nat. Phys.

J. M. Dudley, C. Finot, D. J. Richardson, and G. Millot, “Self-similarity in ultrafast nonlinear optics,” Nat. Phys. 3, 597–603 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

S. A. Ponomarenko and S. Haghgoo, “Self-similarity and optical kinks in resonant nonlinear media,” Phys. Rev. A 82, 051801(R) (2010).
[CrossRef]

S. A. Ponomarenko and S. Haghgoo, “Spatial optical similaritons in conservative nonintegrable systems,” Phys. Rev. A 81, 051801(R) (2010).
[CrossRef]

W. H. Renninger, A. Chong, and F. W. Wise, “Self-similar pulse evolution in an all-normal-dispersion laser,” Phys. Rev. A 82, 021805(R) (2010).
[CrossRef]

Phys. Rev. E

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

Phys. Rev. Lett.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. 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]

F. O. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, “Do soliton-like self-similar waves exist in nonlinear optical media?” Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

I. R. Gabitov and S. V. Manakov, “Propagation of ultrashort pulses in degenerate laser amplifiers,” Phys. Rev. Lett. 50, 495–498 (1983).
[CrossRef]

K. D. Moll, A. Gaeta, and G. Fibich, “Self-similar optical wave collapse: observation of the townes profile,” Phys. Rev. Lett. 90, 203902 (2003).
[CrossRef] [PubMed]

Sov. Phys. JETP

S. V. Manakov, “Propagation of an ultrashort optical pulse in a two-level laser amplifier,” Sov. Phys. JETP 56, 37–44 (1982).

I. R. Gabitov, V. E. Zakharov, and A. V. Mikhailov, “Nonlinear theory of superfluorescence,” Sov. Phys. JETP 59, 703–709 (1984).

Other

L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications Inc., 1975).

P. W. Milonni and J. H. Eberly, Lasers (Wiley, 1985).

A small deviation from the universal asymptotics in the pulse tails can be explained by limited accuracy of our sharp leading edge approximation.

M. Abramowitz and I. A. SteganHandbook of Mathematical Functions (Dover, 1972).

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Figures (5)

Fig. 1
Fig. 1

Schematics of a pump-probe three-level system modeling coherent linear amplifier. The resonant transition takes place between levels e and g.

Fig. 2
Fig. 2

Normalized intensity of a short self-similar pulse as a function of dimensionless time T = τ/T and propagation distance Z = α 0 ζ. The pulse intensity is normalized to its peak value at Z = 0. The initial pulse width is chosen to be t p = T .

Fig. 3
Fig. 3

Normalized intensity of a short Gaussian (solid) and self-similar (dashed) pulses as functions of dimensionless time T = τ/T and propagation distance Z = α 0 ζ. The pulse intensities are normalized to their peak values at Z = 0. The initial pulse width is chosen to be t p = T /2 and t p = T for Gaussian and self-similar pulses, respectively. The inset shows pulse dynamics for short propagation distances.

Fig. 4
Fig. 4

Average widths of Gaussian, secant hyperbolic and exponential pulses as functions of the dimensionless propagation distance Z = α 0 ζ. The self-similar asymptotic pulse width dependence on the propagation distance is shown as the solid curve.

Fig. 5
Fig. 5

Energy gain factor G(Z) for a short (solid) and long (dashed) self-similar pulse as a function of the dimensionless propagation distance Z = α 0 ζ.

Equations (26)

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

ζ Ω = i κ σ ;
ζ Ω k Ω , τ Ω ω Ω .
τ σ = - ( γ + i Δ ) σ - i Ω w ,
τ w = - γ | | ( w - w e q ) - i 2 ( Ω * σ - Ω σ * ) .
w w e q = ± 1 ,
τ σ = - ( γ + i Δ ) σ i Ω .
ζ Ω = ± ( α + i β 2 ) Ω .
α = 2 κ γ γ 2 + Δ 2 , β = 2 κ Δ γ 2 + Δ 2 .
Ω ( τ , ζ ) = Ω 0 ( τ ) e ± ( α + i β ) ζ / 2 ,
ζ Ω = i 2 γ α 0 σ ,
τ σ = - γ σ i Ω .
Ω ( ζ , τ ) = γ θ ( τ ) Ω ¯ [ γ τ F ( ζ ) ] e - γ τ ,
σ ( ζ , τ ) = θ ( τ ) G ( ζ ) σ ¯ [ γ τ F ( ζ ) ] e - i γ τ .
F ( ζ ) = α 0 ζ + T / t p ; G ( ζ ) = 1 / F ( ζ ) .
η Ω ¯ " + Ω ¯ ' - Ω ¯ / 2 = 0 ,
η = γ τ ( α 0 ζ + T / t p ) .
Ω ( η , τ ) γ θ ( τ ) 1 F 1 ( 1 / 2 , 1 , - 2 2 η ) exp ( 2 η - γ τ ) ,
Ω ( η , τ ) γ θ ( τ ) I 0 ( 2 η ) exp ( - γ τ ) ,
Ω ( τ , ζ ) γ θ ( τ ) I 0 ( 2 κ ζ τ ) exp ( - γ τ ) .
T 2 ( Z ) 0 d T T 2 | ( T , Z ) | 2 0 d T | ( T , Z ) | 2 .
Δ T ( Z ) 3 Z / 2 .
ω - 1 t s w t p T .
θ ( τ ) [ 1 + tanh ( τ / t s w ) ] / 2 ,
G ( ζ ) = d τ | ( ζ , τ ) | 2 / d τ | E ( 0 , τ ) | 2 ,
G 0 ( ζ ) = exp ( α 0 ζ ) ,
G ( ζ ) e α 0 ζ α 0 ζ .

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