Abstract

An intense coherent laser field couples two electron subbands in quantum wells. Using generalized self-consistent density functional theory, we study the nature of the Coulomb and light-induced electronic renormalization. We show electric-field-induced transparency and light amplification without population inversion of a weak probe beam in such a system. The effects of the pump field strength, electron density, and δ-doping position are investigated.

© 1994 Optical Society of America

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References

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  1. B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
    [CrossRef]
  2. F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
    [CrossRef]
  3. G. Grynberg, C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993).
    [CrossRef]
  4. S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); O. Kocharovskaya, Yu. I. Khanin, Sov. Phys. JETP 48, 581 (1988).
    [CrossRef] [PubMed]
  5. A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
    [CrossRef] [PubMed]
  6. J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
    [CrossRef]
  7. E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
    [CrossRef] [PubMed]
  8. W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
    [CrossRef] [PubMed]
  9. S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
    [CrossRef]
  10. K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
    [CrossRef] [PubMed]
  11. P. J. Harshman, O. Blum, T. K. Gustafson, P. L. Kelley, Opt. Lett. 18, 1706 (1993).
    [CrossRef] [PubMed]
  12. K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
    [CrossRef] [PubMed]
  13. J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
    [CrossRef] [PubMed]
  14. S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
    [CrossRef]
  15. G. D. Mahan, Phys. Rev. 153, 882 (1967).
    [CrossRef]
  16. L. Hedin, B. I. Lundvist, J. Phys. C 4, 2064 (1971).
    [CrossRef]
  17. H. Haug, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990), p. 12.
  18. D. H. Huang, C. Wu, Y. Zhao, “Coulomb and light-induced electronic renormalizations in quantum wells for electromagnetically induced transparency and light amplification without inversion, ” submitted toJ. Opt. Soc. Am. B.

1993 (4)

G. Grynberg, C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993).
[CrossRef]

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

P. J. Harshman, O. Blum, T. K. Gustafson, P. L. Kelley, Opt. Lett. 18, 1706 (1993).
[CrossRef] [PubMed]

1992 (1)

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

1991 (3)

K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
[CrossRef] [PubMed]

K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
[CrossRef] [PubMed]

1990 (1)

S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
[CrossRef]

1989 (2)

S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); O. Kocharovskaya, Yu. I. Khanin, Sov. Phys. JETP 48, 581 (1988).
[CrossRef] [PubMed]

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

1988 (1)

S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
[CrossRef]

1977 (1)

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

1972 (1)

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

1971 (1)

L. Hedin, B. I. Lundvist, J. Phys. C 4, 2064 (1971).
[CrossRef]

1967 (1)

G. D. Mahan, Phys. Rev. 153, 882 (1967).
[CrossRef]

Blum, O.

Boller, K.-J.

K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

Chemla, D. S.

S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
[CrossRef]

Cohen-Tannoudji, C.

G. Grynberg, C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993).
[CrossRef]

Dönszelmann, A.

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

Ducliy, M.

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Ezekiel, S.

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Field, J. E.

J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
[CrossRef] [PubMed]

S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
[CrossRef]

Fry, E. S.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Gao, J.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Grynberg, G.

G. Grynberg, C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993).
[CrossRef]

Guo, C.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Guo, X.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Gustafson, T. K.

Hahn, K. H.

J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
[CrossRef] [PubMed]

Hakuta, K.

K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
[CrossRef] [PubMed]

Harris, S. E.

J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
[CrossRef] [PubMed]

K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
[CrossRef]

S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); O. Kocharovskaya, Yu. I. Khanin, Sov. Phys. JETP 48, 581 (1988).
[CrossRef] [PubMed]

Harshman, P. J.

Haug, H.

S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
[CrossRef]

H. Haug, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990), p. 12.

Hedin, L.

L. Hedin, B. I. Lundvist, J. Phys. C 4, 2064 (1971).
[CrossRef]

Huang, D. H.

D. H. Huang, C. Wu, Y. Zhao, “Coulomb and light-induced electronic renormalizations in quantum wells for electromagnetically induced transparency and light amplification without inversion, ” submitted toJ. Opt. Soc. Am. B.

Imamoglu, A.

K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
[CrossRef]

Jiang, D.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Jiang, Y.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Jin, G.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Kelley, P. L.

Koch, S. W.

H. Haug, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990), p. 12.

Lambropoulos, P.

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

Li, X.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Lundvist, B. I.

L. Hedin, B. I. Lundvist, J. Phys. C 4, 2064 (1971).
[CrossRef]

Lyras, A.

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

Mahan, G. D.

G. D. Mahan, Phys. Rev. 153, 882 (1967).
[CrossRef]

Marmet, L.

K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
[CrossRef] [PubMed]

Mollow, B. R.

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

Nikonov, D.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Padmabandu, G. G.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Schmitt-Rink, S.

S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
[CrossRef]

Scully, M. O.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Smith, A. V.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Stoicheff, B. P.

K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
[CrossRef] [PubMed]

Tang, X.

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

Tittel, F. K.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

van der Veer, W. E.

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

van Diest, R. J. J.

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

van Linden van den Heuvell, H. B.

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

Wang, C.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Wang, D.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Wang, P.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Wilkinson, S. R.

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

Wu, C.

D. H. Huang, C. Wu, Y. Zhao, “Coulomb and light-induced electronic renormalizations in quantum wells for electromagnetically induced transparency and light amplification without inversion, ” submitted toJ. Opt. Soc. Am. B.

Wu, F. Y.

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

Zhang, H.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Zhang, J.

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

Zhao, J.

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

Zhao, Y.

D. H. Huang, C. Wu, Y. Zhao, “Coulomb and light-induced electronic renormalizations in quantum wells for electromagnetically induced transparency and light amplification without inversion, ” submitted toJ. Opt. Soc. Am. B.

J. Phys. C (1)

L. Hedin, B. I. Lundvist, J. Phys. C 4, 2064 (1971).
[CrossRef]

Opt. Commun. (2)

J. Gao, C. Guo, X. Guo, G. Jin, P. Wang, J. Zhao, H. Zhang, Y. Jiang, D. Wang, D. Jiang, Opt. Commun. 93, 323 (1992).
[CrossRef]

G. Grynberg, C. Cohen-Tannoudji, Opt. Commun. 96, 150 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

S. E. Harris, J. E. Field, A. Imamoǧlu, Phys. Lett. 64, 1107 (1990).
[CrossRef]

Phys. Rev. (1)

G. D. Mahan, Phys. Rev. 153, 882 (1967).
[CrossRef]

Phys. Rev. A (2)

A. Lyras, X. Tang, P. Lambropoulos, J. Zhang, Phys. Rev. A 40, 4131 (1989); Y. Zhu, M. Xiao, Y. Zhao, “Intensity characteristics of inversionless lasers from induced atomic coherence,” Phys. Rev. A (to be published).
[CrossRef] [PubMed]

B. R. Mollow, Phys. Rev. A 5, 2217 (1972).
[CrossRef]

Phys. Rev. B (1)

S. Schmitt-Rink, D. S. Chemla, H. Haug, Phys. Rev. B 37, 941 (1988), and references thereinPhys. Status Solidi B 173, 1 (1992).
[CrossRef]

Phys. Rev. Lett. (7)

E. S. Fry, X. Li, D. Nikonov, G. G. Padmabandu, M. O. Scully, A. V. Smith, F. K. Tittel, C. Wang, S. R. Wilkinson, Phys. Rev. Lett. 70, 3235 (1993).
[CrossRef] [PubMed]

W. E. van der Veer, R. J. J. van Diest, A. Dönszelmann, H. B. van Linden van den Heuvell, Phys. Rev. Lett. 70, 3243 (1993).
[CrossRef] [PubMed]

F. Y. Wu, S. Ezekiel, M. Ducliy, B. R. Mollow, Phys. Rev. Lett. 38, 1077 (1977).
[CrossRef]

S. E. Harris, Phys. Rev. Lett. 62, 1033 (1989); O. Kocharovskaya, Yu. I. Khanin, Sov. Phys. JETP 48, 581 (1988).
[CrossRef] [PubMed]

K. Hakuta, L. Marmet, B. P. Stoicheff, Phys. Rev. Lett. 66, 596 (1991).
[CrossRef] [PubMed]

K.-J. Boller, A. Imamoǧlu, S. E. Harris, Phys. Rev. Lett. 66, 2593 (1991).
[CrossRef] [PubMed]

J. E. Field, K. H. Hahn, S. E. Harris, Phys. Rev. Lett. 67, 3062 (1991).
[CrossRef] [PubMed]

Other (2)

H. Haug, S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990), p. 12.

D. H. Huang, C. Wu, Y. Zhao, “Coulomb and light-induced electronic renormalizations in quantum wells for electromagnetically induced transparency and light amplification without inversion, ” submitted toJ. Opt. Soc. Am. B.

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Figures (3)

Fig. 1
Fig. 1

βabs(ωt) (solid curve) and nr(ωt) (dashed curve) as functions of ħωt. The parameters for the numerical calculations are L = 10 mm, ΔEc = 280 meV, b = 12.5, m* = 0.067me, ħωp = 96.24076 meV, γ = 3.46 meV, ɛp = 0, and N 1 2 D = 1.0 + 10 12 cm - 2. The δ-doping position is 10 nm to the left of the quantum-well center.

Fig. 2
Fig. 2

Calculated βabs(ωt) (solid curve) and nr(ωt) (dashed curve) as functions of ħωt. The parameters are the same as in Fig. 1, except that ɛp = 5 × 106 Vcm−1.

Fig. 3
Fig. 3

βabs(ωt) (solid curve) and nr(ωt) (dashed curve) at ɛp = 5 × 106 Vcm−1 as functions of ħωt. The parameters are the same as those in Fig. 1, except that the δ doping is at the quantum-well center.

Equations (7)

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

{ - 2 2 m * d 2 d z 2 + U ext ( z ) + V H ( z ) + V xc [ n ( 0 ) ( z ) ] } ψ j ( 1 ) ( z ) = E j ( 1 ) ψ j ( 1 ) ( z ) ,
E j ( 1 ) ( k ) = 2 k 2 2 m j * 1 2 [ E 2 ( 1 ) - E 1 ( 1 ) - ω p ] ,
E 1 ( 2 ) ( k ) = { 1 2 ( E 1 ( 1 ) ( k ) + E 2 ( 1 ) ( k ) + { [ E 1 ( 1 ) ( k ) - E 2 ( 1 ) ( k ) ] 2 + 4 μ E p 2 } 1 / 2 ) E 1 ( 1 ) ( k ) > E 2 ( 1 ) ( k ) 1 2 ( E 1 ( 1 ) ( k ) + E 2 ( 1 ) ( k ) - { [ E 1 ( 1 ) ( k ) - E 2 ( 1 ) ( k ) ] 2 + 4 μ E p 2 } 1 / 2 ) E 1 ( 1 ) ( k ) < E 2 ( 1 ) ( k ) ,
ψ j k ( 2 ) ( z ) = μ E p ψ 1 ( 1 ) ( z ) - [ E 1 ( 1 ) ( k ) - E j ( 2 ) ( k ) ] ψ 2 ( 1 ) ( z ) { μ E p 2 + [ E 1 ( 1 ) ( k ) - E j ( 2 ) ( k ) ] 2 } 1 / 2 .
α L ( Δ ω ) = - e 2 A π j , j 0 + k d k z j , j ( k ) × { f 0 [ E j ( k ) - E F ] - f 0 [ E j ( k ) - E F ] Δ ω + i γ - E j ( k ) + E j ( k ) } × { z j , j ( k ) + [ U j , j ( k ; Δ ω ) e E t ] } ,
β abs ( Δ ω ) = Δ ω 0 c L A n r ( Δ ω ) [ ρ ph ( Δ ω ) + 1 ] Im α L ( Δ ω ) ,
n r ( Δ ω ) = 1 2 ( b + Re α L ( Δ ω ) 0 L A + { [ b + Re α L ( Δ ω ) 0 L A ] 2 + [ Im α L ( Δ ω ) 0 L A ] 2 } 1 / 2 ) .

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