Abstract

We develop a statistical model to study all the relevant phenomena that can give rise to an anomalous intensity saturation in the propagation of incoherent pulses in a laser amplifier. The interplay among diffraction, self-focusing, group-velocity dispersion, gain narrowing, and gain saturation is investigated. Changes in the temporal and spatial characteristics of the pulses are shown.

© 1998 Optical Society of America

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References

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  1. O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
    [CrossRef] [PubMed]
  2. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
    [CrossRef] [PubMed]
  3. R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 (1983).
    [CrossRef]
  4. C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
    [CrossRef] [PubMed]
  5. J. Garnier, J. P. Fouque, C. Gouédard, L. Videau, and A. Migus, J. Opt. Soc. Am. B 14, 2563 (1997).
    [CrossRef]
  6. G. A. Pasmanik, Sov. Phys. JETP 39, 234 (1975).
  7. V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
    [CrossRef]
  8. V. P. Nayyar, J. Opt. Soc. Am. B 14, 2248 (1997).
    [CrossRef]
  9. D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
    [CrossRef]
  10. R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
    [CrossRef]
  11. S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
    [CrossRef]
  12. D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
    [CrossRef]
  13. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, 1984), pp. 9–75.
  14. J. Garnier, C. Gouédard, L. Videau, and A. Migus, J. Opt. Soc. Am. A 14, 1928 (1997).
    [CrossRef]
  15. Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
    [CrossRef]
  16. A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).
  17. R. T. Glassey, J. Math. Phys. 18, 1794 (1977).
    [CrossRef]
  18. H. A. Rose and D. F. DuBois, Phys. Fluids B 5, 590 (1993).
    [CrossRef]
  19. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  20. L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
    [CrossRef]

1997

1995

O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
[CrossRef] [PubMed]

1993

D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
[CrossRef]

Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
[CrossRef]

H. A. Rose and D. F. DuBois, Phys. Fluids B 5, 590 (1993).
[CrossRef]

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

1989

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

1988

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

1987

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
[CrossRef]

1983

R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 (1983).
[CrossRef]

1981

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
[CrossRef]

1977

R. T. Glassey, J. Math. Phys. 18, 1794 (1977).
[CrossRef]

1975

G. A. Pasmanik, Sov. Phys. JETP 39, 234 (1975).

1963

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Aleshkevich, V. A.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
[CrossRef]

Averbukh, I. S.

O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
[CrossRef] [PubMed]

Ayral, H.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Bodner, S. E.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Chuang, Y. H.

Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
[CrossRef]

Craxton, R. S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

DuBois, D. F.

H. A. Rose and D. F. DuBois, Phys. Fluids B 5, 590 (1993).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Fouque, J. P.

Frantz, L. M.

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Garnier, J.

Glassey, R. T.

R. T. Glassey, J. Math. Phys. 18, 1794 (1977).
[CrossRef]

Gouédard, C.

J. Garnier, C. Gouédard, L. Videau, and A. Migus, J. Opt. Soc. Am. A 14, 1928 (1997).
[CrossRef]

J. Garnier, J. P. Fouque, C. Gouédard, L. Videau, and A. Migus, J. Opt. Soc. Am. B 14, 2563 (1997).
[CrossRef]

D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
[CrossRef]

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Husson, D.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Kessler, T.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

Kinrot, O.

O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
[CrossRef] [PubMed]

Lauriou, J.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Lebedev, S. S.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
[CrossRef]

Lehmberg, R. H.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
[CrossRef]

R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 (1983).
[CrossRef]

Letzring, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

Li, Z. W.

C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Martin, O.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Matveev, A. N.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
[CrossRef]

Meyer, B.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Meyerhofer, D. D.

Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
[CrossRef]

Migus, A.

Nayyar, V. P.

Nodvik, J. S.

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

Obenschain, S. P.

R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 (1983).
[CrossRef]

Pasmanik, G. A.

G. A. Pasmanik, Sov. Phys. JETP 39, 234 (1975).

Prior, Y.

O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
[CrossRef] [PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Radzewicz, C.

C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Raymer, G.

C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Rose, H. A.

H. A. Rose and D. F. DuBois, Phys. Fluids B 5, 590 (1993).
[CrossRef]

Rostaing, M.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Sauteret, C.

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Schmitt, A. J.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
[CrossRef]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Short, R. W.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

Skupsky, S.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

Soures, J. M.

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Thiell, G.

D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
[CrossRef]

Véron, D.

D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
[CrossRef]

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Videau, L.

Zheng, L.

Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
[CrossRef]

IEEE J. Quantum Electron.

Y. H. Chuang, L. Zheng, and D. D. Meyerhofer, IEEE J. Quantum Electron. 29, 270 (1993).
[CrossRef]

J. Appl. Phys.

R. H. Lehmberg, A. J. Schmitt, and S. E. Bodner, J. Appl. Phys. 62, 2680 (1987).
[CrossRef]

S. Skupsky, R. W. Short, T. Kessler, R. S. Craxton, S. Letzring, and J. M. Soures, J. Appl. Phys. 66, 3456 (1989).
[CrossRef]

L. M. Frantz and J. S. Nodvik, J. Appl. Phys. 34, 2346 (1963).
[CrossRef]

J. Math. Phys.

R. T. Glassey, J. Math. Phys. 18, 1794 (1977).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

D. Véron, G. Thiell, and C. Gouédard, Opt. Commun. 97, 259 (1993).
[CrossRef]

R. H. Lehmberg and S. P. Obenschain, Opt. Commun. 46, 27 (1983).
[CrossRef]

D. Véron, H. Ayral, C. Gouédard, D. Husson, J. Lauriou, O. Martin, B. Meyer, M. Rostaing, and C. Sauteret, Opt. Commun. 65, 42 (1988).
[CrossRef]

Phys. Fluids B

H. A. Rose and D. F. DuBois, Phys. Fluids B 5, 590 (1993).
[CrossRef]

Phys. Rev. A

C. Radzewicz, Z. W. Li, and G. Raymer, Phys. Rev. A 37, 2039 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett.

O. Kinrot, I. S. Averbukh, and Y. Prior, Phys. Rev. Lett. 75, 3822 (1995).
[CrossRef] [PubMed]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Sov. J. Quantum Electron.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, Sov. J. Quantum Electron. 11, 647 (1981).
[CrossRef]

Sov. Phys. JETP

G. A. Pasmanik, Sov. Phys. JETP 39, 234 (1975).

Other

A. C. Newell and J. V. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, 1984), pp. 9–75.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

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Figures (4)

Fig. 1
Fig. 1

Ratio of the output energy E(z) (computed by inclusion of the gain-narrowing effects) over the expected optimal output E0eγz as a function of the value of the B integral. The numerical values could correspond to the final section of a high-power Nd-doped glass laser. We have taken γ=20 cm-1, γz=4, λ0=1.05 μm, and T2=100 fs. The incident beam has a flat macroscopic profile (i.e., p=q=), with coherence time τc=1.6 ps, which corresponds to a spectrum of full bandwidth extent (FWHM) 1.2 nm. We considered spatial modulations with radii of 1 cm, 5 mm, and 2 mm.

Fig. 2
Fig. 2

Output energy as function of input energy for four input intensity profiles. The solid curve corresponds to a flat profile with contrast c as shown, and the dotted curve to a static speckle pattern. The two other curves represent the amplification of time-varying speckle patterns consisting of two independent speckle patterns (dashed curve) and ten independent speckle patterns (dotted–dashed curve). The small-signal gain is taken to be equal to γz=4, and the saturation energy to Es=5000 J.

Fig. 3
Fig. 3

Energy loss owing to the small-scale intensity fluctuations in the regime that correspond to high depletion as a function of the contrast of the time-integrated pattern. When the intensity profile is flat, the contrast is zero and the energy output is taken as a reference.

Fig. 4
Fig. 4

Output energy and contrast as functions of input energy for several input pulses. The incident pulses consist of the succession of N independent speckle patterns. We assume that the spatial macroscopic envelope is uniform (i.e., p=) and that η3(z)=0.2 for E0=25 J. The small-signal gain is taken to be equal to γz=4, and the saturation energy to Es=5000 J.

Equations (52)

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

i Az-12k0 ΔrA-σ 2At2-k22 |A|2A=0,
C0,R,T(ρ, τ)=A0(r1, t1)A0*(r2, t2),
C0,R,T(ρ, τ)=I0(R, T)exp-|ρ|2ρc2+τ2τc2.
I0(R, T)=I0 exp-|R|pR0p+TqT0q,
Cz,R,T(ρ, τ)=A(z, r1, t1)A*(z, r2, t2),
c2(z)
=|A|4(z, r, t)d2rdt-Cz,R,T2(0, 0)d2RdTCz,R,T2(0, 0)d2RdT.
c2(z)=|A|4(R, T)-|A|22(R, T)d2RdT|A|22(R, T)d2RdT.
E  |A|2(z, r, t)d2rdt,
E0=4πR02T0 I0pq Γ2pΓ1q,
H   12k0 |rA|2+σAt2-k24 |A|4d2rdt,
H0=2k0ρc2+2στc2-k2I022+2p+1qE0.
V(z)  2k0|r|2+t2σ|A|2d2rdt.
2Vz2=8H-k2|A|4d2rdt.
A(z=0, r, t)=I11/2 exp-|r|2ρc2+t2τc2
A(z, r, t)=I(z)1/2 exp-|r|2ρ(z)2+t2τ(z)2×exp i[ϕ0(z)+ϕ1(z)|r|2+ϕ2(z)t2].
2ρ(z)z2=4k02ρ3(z)-k2I1τcρc22k0τ(z)ρ3(z),
2τ(z)z2=16σ2τ3(z)-σk2I1τcρc2τ(z)2ρ2(z),
I(z)=I1 ρc2τcρ2(z)τ(z).
Cz,R,T(ρ, τ)
=C0,R,T(ρ, τ)1+B02(z, R, T)1-exp-2τ2τc2-2|ρ|2ρc22,
|rA|2d2rdt=-ΔρCz,R,T(0, 0)d2RdT,
At2d2rdt=- 2Cz,R,Tτ2 (0, 0)d2RdT.
|rA|2d2rdt=[1+4(3-2/p-1/q)B02(z)] 4ρc2 E0,
At2d2rdt=[1+4(3-2/p-1/q)B02(z)] 2τc2 E0,
c(z)2=1+16(2/3)2/p+1/qB0(z)zk0ρc2+σzτc2.
i Az-12k0 ΔrA-σ 2At2-k22 |A|2A=P2,
T2 Pt+P=iγA,
Cz,R,T(ρ, τ)=eγzC0,R,T(ρ, τ)1+Bγ2(z, R, T)1-exp-2τ2τc2-2|ρ|2ρc22,
Bγ(z)=k2I02 eγz-1γ
E(z)=E0eγz.
|rA|2d2rdt=[1+4(3-2/p-1/q)Bγ2(z)] 4ρrc2 E0eγz,
At2d2rdt=[1+4(3-2/p-1/q)Bγ2(z)] 2τc2 E0eγz.
Hz=-γk24 |A|4d2rdt+γH.
c(z)2=1+2η1(z),
η1(z)=16(2/3)2/p+1/qBγ(z)1γk0ρc2+σγτc2.
E(z)=E0eγz{1-2(T2/τc)2γz-4(1/3)2/p+1/q(T2/τc)2Bγ2(z)[1+η2(z)]},
η2(z)=(4/3)2/p+1/qBγ(z)723k0γρc2+176σ3γτc2.
c(z)2=1+2η1(z)-32(1/2)2/p+1/qBγ2(z)(T2/τc)2.
γt+γTγ=i2 fs (A*P-AP*),
γ(z, r, t)=γ0 exp-1fs -t|A|2(z, r, s)ds.
i Az+12k0 ΔrA+σ 2At2+k22 |A|2A
=iγ02 A exp-1fs -t|A|2(z, r, s)ds.
Eunif(z)=Es ln{1+[exp(E0/Es)-1]exp(γ0z)},
Eunif(z)E0eγ0z1-E0 exp(γ0z)2Es.
E(z)
=Es0 ln1+expuE0Es-1exp(γ0z)pn(u)du,
E(z)E0 exp(γ0z)1-(1+c2) E0 exp(γ0z)2Es.
A0(R-ρ/2)A0*(R+ρ/2)=I0 exp-|R|pR0p-|ρ|2ρc2.
E(z)=E0 exp(γ0z)×1-E0 exp(γ0z)2Es1+τcT0 [1+η3(z)],
η3(z)=32(2/3)2/pBγ0(z)k0γ0ρc2.
c(z)2=1+η3(z)-(2/3)2/p E0 exp(γ0z)2Es×(9/8)2/pη3(z)+τcT0 [4+6(9/8)2/pη3(z)],

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