Abstract

The influence of quadratic recombination is analyzed theoretically and numerically by formal derivation of the material wave equation including this effect. Analytically, it is shown how the quadratic recombination alters the nonlinear properties of space-charge waves. The influence of higher-harmonic components induces both a nonlinear frequency shift and a nonlinear shift in dissipation. Numerically, it is shown that 12 higher-harmonic components are necessary to cover the entire region of intensity contrast and that the quadratic recombination effect gives significant corrections for small frequencies below the fundamental resonance frequency.

© 1999 Optical Society of America

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References

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  1. Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  2. E. Serrano, M. Cassascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
    [CrossRef] [PubMed]
  3. L. B. Au and L. Solymar, “Higher harmonic grating in photorefractive materials at large modulation with moving fringes,” J. Opt. Soc. Am. A 7, 1554–1561 (1990).
    [CrossRef]
  4. G. A. Brost, K. M. Magde, J. J. Larkin, and M. T. Harris, “Modulation dependence of photorefractive response with moving gratings: numerical analysis and experiment,” J. Opt. Soc. Am. B 11, 1764–1772 (1994).
    [CrossRef]
  5. B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Space-charge waves in photorefractive crystals and their parametric excitation,” J. Opt. Soc. Am. B 10, 1919–1932 (1993).
    [CrossRef]
  6. H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996); “Longitudinal, degenerate, and transversal photorefractive parametric oscillation: theory and experiment,” J. Opt. Soc. Am. B 14, 1418–1427 (1997).
    [CrossRef] [PubMed]
  7. H. C. Pedersen and P. M. Johansen, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996); “Degenerate photorefractive parametric amplification in photorefractive media: theoretical analysis,” J. Opt. Soc. Am. B 13, 590–600 (1996).
    [CrossRef] [PubMed]
  8. E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
    [CrossRef]
  9. P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady-state analysis of ac subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998); “Ac square-wave field-induced subharmonics in photorefractive sillenite: threshold for excitation by including higher harmonics,” J. Opt. Soc. Am. B 16, 103–110 (1999).
    [CrossRef]
  10. T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
    [CrossRef]
  11. H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
    [CrossRef]
  12. B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
    [CrossRef]
  13. N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
    [CrossRef]
  14. T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
    [CrossRef]
  15. P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530–539 (1989).
    [CrossRef]
  16. Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
    [CrossRef]
  17. M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
    [CrossRef]

1998

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

1997

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

1995

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

E. Serrano, M. Cassascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[CrossRef] [PubMed]

1994

1993

1990

1989

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

1985

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

1979

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

1977

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Aguilar, M.

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

Agulló-López, F.

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

E. Serrano, M. Cassascosa, and F. Agulló-López, “Nonperturbative analytical solution for steady-state photorefractive recording,” Opt. Lett. 20, 1910–1912 (1995).
[CrossRef] [PubMed]

Au, L. B.

Brost, G. A.

Cassascosa, M.

Connors, L. M.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Foote, P. D.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Hall, T. J.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Harris, M. T.

Huignard, J.-P.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Jaura, R.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Johansen, P. M.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

Kukhtarev, N. V.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Larkin, J. J.

Magde, K. M.

Mann, M.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Space-charge waves in photorefractive crystals and their parametric excitation,” J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[CrossRef]

Markov, V. B.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

McClelland, T. E.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Micheron, F.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Odoulov, S. G.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Otten, J.

Pedersen, H. C.

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

Peltier, M.

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Podivilov, E. V.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

Rajbenbach, H.

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refregier, Ph.

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Refreiger, Ph.

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ringhofer, K. H.

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Space-charge waves in photorefractive crystals and their parametric excitation,” J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[CrossRef]

Serrano, E.

Solymar, L.

L. B. Au and L. Solymar, “Higher harmonic grating in photorefractive materials at large modulation with moving fringes,” J. Opt. Soc. Am. A 7, 1554–1561 (1990).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

Soskin, M. S.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Sturman, B. I.

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

B. I. Sturman, M. Mann, J. Otten, and K. H. Ringhofer, “Space-charge waves in photorefractive crystals and their parametric excitation,” J. Opt. Soc. Am. B 10, 1919–1932 (1993).
[CrossRef]

Vinetskii, V. L.

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

Webb, D. J.

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

Ferroelectrics

N. V. Kukhtarev, V. B. Markov, S. G. Odoulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals. I. Steady state,” Ferroelectrics 22, 949–960 (1979).
[CrossRef]

IEEE J. Quantum Electron.

P. M. Johansen, “Vectorial solution to the photorefractive band transport model in the spatial and temporal Fourier transformed domain,” IEEE J. Quantum Electron. 25, 530–539 (1989).
[CrossRef]

J. Appl. Phys.

Ph. Refreiger, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving gratings: theory and experiments,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

M. Peltier and F. Micheron, “Volume hologram recording and charge transfer process in Bi12SiO20 and Bi12GeO20,” J. Appl. Phys. 48, 3683–3690 (1977).
[CrossRef]

Ph. Refregier, L. Solymar, H. Rajbenbach, and J.-P. Huignard, “Two-beam coupling in photorefractive Bi12SiO20 crystals with moving grating: theory and experiment,” J. Appl. Phys. 58, 45–57 (1985).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

T. E. McClelland, D. J. Webb, B. I. Sturman, M. Mann, and K. H. Ringhofer, “Low frequency peculiarities of the photorefractive response in sillenite,” Opt. Commun. 113, 371–377 (1995).
[CrossRef]

H. C. Pedersen, P. M. Johansen, E. V. Podivilov, and D. J. Webb, “Excitation of higher harmonic gratings in ac-field biased photorefractive crystals,” Opt. Commun. 154, 93–99 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev. E

B. I. Sturman, M. Aguilar, F. Agulló-López, and K. H. Ringhofer, “Fundamentals of the nonlinear theory of photorefractive subharmonics,” Phys. Rev. E 55, 6072–6083 (1997).
[CrossRef]

E. V. Podivilov, H. C. Pedersen, P. M. Johansen, and B. I. Sturman, “Transversal parametric oscillation and its external stability in photorefractive sillenite crystals,” Phys. Rev. E 57, 6112–6126 (1998).
[CrossRef]

Prog. Quantum Electron.

T. J. Hall, R. Jaura, L. M. Connors, and P. D. Foote, “The photorefractive effect—a review,” Prog. Quantum Electron. 10, 77–146 (1985).
[CrossRef]

Other

P. M. Johansen, H. C. Pedersen, E. V. Podivilov, and B. I. Sturman, “Steady-state analysis of ac subharmonic generation in photorefractive sillenite crystals,” Phys. Rev. A 58, 1601–1604 (1998); “Ac square-wave field-induced subharmonics in photorefractive sillenite: threshold for excitation by including higher harmonics,” J. Opt. Soc. Am. B 16, 103–110 (1999).
[CrossRef]

H. C. Pedersen and P. M. Johansen, “Longitudinal, degenerate, and transversal parametric oscillation in photorefractive media,” Phys. Rev. Lett. 77, 3106–3109 (1996); “Longitudinal, degenerate, and transversal photorefractive parametric oscillation: theory and experiment,” J. Opt. Soc. Am. B 14, 1418–1427 (1997).
[CrossRef] [PubMed]

H. C. Pedersen and P. M. Johansen, “Observation of nondegenerate photorefractive parametric amplification,” Phys. Rev. Lett. 76, 4159–4162 (1996); “Degenerate photorefractive parametric amplification in photorefractive media: theoretical analysis,” J. Opt. Soc. Am. B 13, 590–600 (1996).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

Modulus of the space-charge (s-c) field divided by m versus δ for three different values of intensity contrast. The nonlinear frequency shift is indicated in the curves by δωK/Ω.

Fig. 2
Fig. 2

Modulus of the space-charge field versus δ for an intensity contrast of m=0.01. The dashed curve is depicted without the effect of linear saturation, whereas the solid curve includes this effect.

Fig. 3
Fig. 3

Amplitude of the fundamental space-charge field versus normalized frequency Ω/ωK for four different values of the intensity contrast both with and without the quadratic recombination effect included.

Fig. 4
Fig. 4

Numerical plot of the term (0s/qNA)·E1 versus normalized distance x/Λ for four different values of the normalized frequency Ω/ωK.

Equations (27)

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

ND+t=NDsI-γRND+n,
ND+t=-μ·nE+kBTqn,
·E=q0s(ND+-NA),
I=I0+I1,
n1n0=I1I0-0sqNA·E1+1ω0·E˙11+0sqNA·E1,
-1μτ·E˙1=ω0qNA0s·E1+E0·+kBTq2+(·E1)+E1·n1n0,
E0·+kBTq2-1μτ·E˙1-ζI0-ω0kBTq2-ω0E0··E1=ζE0·I1+ζkBTq2I1+ζ·(I1E1)̲̲-·(E1·E˙1)̲̲-ω0·(E1·E1)̲̲-ω0I0E0·+kBTq2(I1·E1)̲+ω0ζI0E0·+kBTq2{[ω0(·E1)+(·E˙1)]·E1}̲,
I=I0+I1=I0+I0m cos(Kx-Ωt),
E1(x, t)=kE0ek(t)exp(ikx-iΩkt)+c.c.,
ekt+i(ωk-Ωk-iγk)ek
=m2Ak(δk,K+δk, -K)+m2(Bk,Kek-K+Bk, -Kek+K)+kCk,k(ω0-iΩk)ek-kek,
ωk=ω0E0[Eq(k)-EM(k)][ED(k)+EM(k)]2+E02,
γk=ω0[Eq(k)+ED(k)][ED(k)+EM(k)]+E02[ED(k)+EM(k)]2+E02,
Ak=-iω0Eq(k)E0ED(k)-iE0ED(k)+EM(k)-iE0,
Bk,k=iω0Eq(k)E0Ck,k,
Ck,k=iE0Eq(k)Eq(k)+[ED(k)-iE0]k-kkED(k)+EM(k)-iE0.
ekt+i(ωk-Ωk-iγk)ek
=-im2ωk(δk,K+δk, -K+ek-K+ek+K)
+ikkkΩkek-kek,
ωk=ω0Eq(k)E0,
γk=ω0E02{Eq(k)[ED(k)+EM(k)]+E02},
Ak=Bk,K=-iωk,Ck,k=-kk.
eK,L=m2ωKΩ-ωK+iγK.
[(ωK+δωK-Ω)-i(γK+δγK)]eK=-m2ωK,
e2K=ΩωK-4Ω-2iγ2KeK2,
δωK=5Ω24Ω-ωK(ωK-4Ω)2+4γ2K2|eK|2,
δγK=10Ω2γ2K(ωK-4Ω)2+4γ2K2|eK|2.

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