Abstract

A nonstationary process of polariton-mode excitation in a two-dimensional (2D) excitonic layer by a light-wave extinction front is analyzed here using the dyadic Green’s function formalism. The electromagnetic response of the 2D excitonic layer includes inhomogeneous radiative exciton-polariton modes corresponding to the complex poles of the matrix Green’s function. On the basis of this theoretical analysis of the nonstationary polaritonic response, an old controversy concerning two different approaches for the description of radiative modes in a 2D excitonic system is resolved.

© 2007 Optical Society of America

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  1. V. M. Agranovich and O. A. Dubovskii, "Effect of retarded interaction on exciton spectra in one- and two-dimensional crystals," JETP Lett. 3, 233-238 (1966).
  2. F. Tassone, F. Bassani, and L. C. Andreani, "Quantum-well reflectivity and exciton-polariton dispersion," Phys. Rev. B 45, 6023-6030 (1992).
    [CrossRef]
  3. E. L. Ivchenko, "Exciton polaritons in periodic quantum well structures," Fiz. Tverd. Tela (Leningrad) 33, 2388-2393 (1991) E. L. Ivchenko,[Sov. Phys. Solid State 33, 1344-1349 (1991)].
  4. E. L. Ivchenko and A. V. Kavokin, "Light reflection from quantum dot structures," Fiz. Tverd. Tela (Leningrad) 34, 1815-1822 (1992). E. L. Ivchenko and A. V. Kavokin,[Sov. Phys. Solid State 34, 968-975 (1992)].
  5. D. S. Citrin, "Radiative lifetimes of excitons in quantum wells: Localization and phase-coherence effects," Phys. Rev. B 47, 3832-3841 (1993).
    [CrossRef]
  6. S. Jorda, U. Rössler, and D. Broido, "Fine structure of excitons and polariton dispersion in quantum wells," Phys. Rev. B 48, 1669-1677 (1993).
    [CrossRef]
  7. R. Atanasov, F. Bassani, and V. M. Agranovich, "Mean-field polariton theory for asymmetric quantum wells," Phys. Rev. B 49, 2658-2666 (1994).
    [CrossRef]
  8. V. V. Popov, T. V. Teperik, N. J. M. Horing, and T. Yu. Bagaeva, "Inhomogeneous radiative decay of polariton modes in a two-dimensional exciton system," Solid State Commun. 127, 589-594 (2003).
    [CrossRef]
  9. V. V. Popov, T. Yu. Bagaeva, T. V. Teperik, N. J. M. Horing, and Y. Ayaz, "Ultrafast radiative decay of polaritons in an interface layer with strong excitonic response," J. Lumin. 112, 225-229 (2005).
    [CrossRef]
  10. E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Ya. Meltzer, and M. A. Kaliteevskii, "Reflection in exciton region of spectrum of structure with a single quantum well. Oblique and normal incidence of light," Fiz. Tekh. Poluprovodn. (S.-Peterburg) 22, 784-788 (1988). E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Y. Meltzer, and M. A. Kaliteevskii,[Sov. Phys. Semicond. 22, 497-501 (1988)].
  11. E. L. Ivchenko, V. P. Kochereshko, P. S. Kop'ev, V. A. Kosobukin, I. N. Uraltsev, and D. R. Yakovlev, "Exciton longitudinal-transverse splitting in GaAs/AsGaAs superlattices and multiple quantum wells," Solid State Commun. 70, 529-534 (1989).
    [CrossRef]
  12. M. W. Berz, L. C. Andreani, E. F. Steingmeier, and F.-K. Reinhart, "Exchange splitting of light hole excitons in Al1−χGaχAs-GaAs quantum wells," Solid State Commun. 80, 553-556 (1991).
    [CrossRef]
  13. E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
    [CrossRef]
  14. L. C. Andreani, F. Tassone, and F. Bassani, "Radiative lifetime of free excitons in quantum wells," Solid State Commun. 77, 641-645 (1991).
    [CrossRef]
  15. B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
    [CrossRef] [PubMed]
  16. A. Vinattieri, J. Shah, T. C. Damen, D. S. Kim, L. N. Pfeiffer, M. Z. Maialle, and L. J. Sham, "Exciton dynamics in GaAs quantum wells under resonant excitation," Phys. Rev. B 50, 10868-10879 (1994).
    [CrossRef]
  17. C.-To Tai, Dyadic Green's Functions in Electromagnetic Theory (Intext Educational Publishers, 1971).
  18. N. J. M. Horing, T. Jena, H. L. Cui, and J. D. Mancini, "Dynamic dielectric properties of a bounded solid-state plasma and a two-dimensional electron sheet: inverse dielectric function and coupled collective modes," Phys. Rev. B 54, 2785-2790 (1996).
    [CrossRef]
  19. N. J. M. Horing and Y. Ayaz, "Dynamic dielectric response of an asymmetric double quantum well near the bounding surface of a semi-infinite dynamic plasmalike host medium," Phys. Rev. B 58, 2001-2007 (1998).
    [CrossRef]
  20. T. Ishihara, J. Takahashi, and T. Goto, "Optical properties due to electronic transitions in two-dimensional semiconductors (CnH2n+1NH3)2PbI4," Phys. Rev. B 42, 11099-11107 (1990).
    [CrossRef]
  21. T. Fujita, Y. Sato, T. Kuitani, and T. Ishihara, "Tunable polariton absorption of distributed feedback microcavities at room temperature," Phys. Rev. B 57, 12428-12434 (1998).
    [CrossRef]
  22. A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, "Polariton effect in distributed feedback microcavities," J. Phys. Soc. Jpn. 70, 1137-1144 (2001).
    [CrossRef]
  23. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Eq. 7.57.
  24. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), p. 340.
  25. E. A. Muljarov, S. G. Tikhodeev, N. A. Gippius, and T. Ishihara, "Excitons in self-organized semiconductor/insulator superlattices: PbI-based porovskite compounds," Phys. Rev. B 51, 14370-14370 (1995).
    [CrossRef]
  26. U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]

2005

V. V. Popov, T. Yu. Bagaeva, T. V. Teperik, N. J. M. Horing, and Y. Ayaz, "Ultrafast radiative decay of polaritons in an interface layer with strong excitonic response," J. Lumin. 112, 225-229 (2005).
[CrossRef]

2003

V. V. Popov, T. V. Teperik, N. J. M. Horing, and T. Yu. Bagaeva, "Inhomogeneous radiative decay of polariton modes in a two-dimensional exciton system," Solid State Commun. 127, 589-594 (2003).
[CrossRef]

2001

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, "Polariton effect in distributed feedback microcavities," J. Phys. Soc. Jpn. 70, 1137-1144 (2001).
[CrossRef]

1998

N. J. M. Horing and Y. Ayaz, "Dynamic dielectric response of an asymmetric double quantum well near the bounding surface of a semi-infinite dynamic plasmalike host medium," Phys. Rev. B 58, 2001-2007 (1998).
[CrossRef]

T. Fujita, Y. Sato, T. Kuitani, and T. Ishihara, "Tunable polariton absorption of distributed feedback microcavities at room temperature," Phys. Rev. B 57, 12428-12434 (1998).
[CrossRef]

1996

N. J. M. Horing, T. Jena, H. L. Cui, and J. D. Mancini, "Dynamic dielectric properties of a bounded solid-state plasma and a two-dimensional electron sheet: inverse dielectric function and coupled collective modes," Phys. Rev. B 54, 2785-2790 (1996).
[CrossRef]

1995

E. A. Muljarov, S. G. Tikhodeev, N. A. Gippius, and T. Ishihara, "Excitons in self-organized semiconductor/insulator superlattices: PbI-based porovskite compounds," Phys. Rev. B 51, 14370-14370 (1995).
[CrossRef]

1994

R. Atanasov, F. Bassani, and V. M. Agranovich, "Mean-field polariton theory for asymmetric quantum wells," Phys. Rev. B 49, 2658-2666 (1994).
[CrossRef]

A. Vinattieri, J. Shah, T. C. Damen, D. S. Kim, L. N. Pfeiffer, M. Z. Maialle, and L. J. Sham, "Exciton dynamics in GaAs quantum wells under resonant excitation," Phys. Rev. B 50, 10868-10879 (1994).
[CrossRef]

1993

D. S. Citrin, "Radiative lifetimes of excitons in quantum wells: Localization and phase-coherence effects," Phys. Rev. B 47, 3832-3841 (1993).
[CrossRef]

S. Jorda, U. Rössler, and D. Broido, "Fine structure of excitons and polariton dispersion in quantum wells," Phys. Rev. B 48, 1669-1677 (1993).
[CrossRef]

1992

F. Tassone, F. Bassani, and L. C. Andreani, "Quantum-well reflectivity and exciton-polariton dispersion," Phys. Rev. B 45, 6023-6030 (1992).
[CrossRef]

E. L. Ivchenko and A. V. Kavokin, "Light reflection from quantum dot structures," Fiz. Tverd. Tela (Leningrad) 34, 1815-1822 (1992). E. L. Ivchenko and A. V. Kavokin,[Sov. Phys. Solid State 34, 968-975 (1992)].

1991

E. L. Ivchenko, "Exciton polaritons in periodic quantum well structures," Fiz. Tverd. Tela (Leningrad) 33, 2388-2393 (1991) E. L. Ivchenko,[Sov. Phys. Solid State 33, 1344-1349 (1991)].

L. C. Andreani, F. Tassone, and F. Bassani, "Radiative lifetime of free excitons in quantum wells," Solid State Commun. 77, 641-645 (1991).
[CrossRef]

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
[CrossRef] [PubMed]

M. W. Berz, L. C. Andreani, E. F. Steingmeier, and F.-K. Reinhart, "Exchange splitting of light hole excitons in Al1−χGaχAs-GaAs quantum wells," Solid State Commun. 80, 553-556 (1991).
[CrossRef]

1990

T. Ishihara, J. Takahashi, and T. Goto, "Optical properties due to electronic transitions in two-dimensional semiconductors (CnH2n+1NH3)2PbI4," Phys. Rev. B 42, 11099-11107 (1990).
[CrossRef]

1989

E. L. Ivchenko, V. P. Kochereshko, P. S. Kop'ev, V. A. Kosobukin, I. N. Uraltsev, and D. R. Yakovlev, "Exciton longitudinal-transverse splitting in GaAs/AsGaAs superlattices and multiple quantum wells," Solid State Commun. 70, 529-534 (1989).
[CrossRef]

1988

E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
[CrossRef]

E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Ya. Meltzer, and M. A. Kaliteevskii, "Reflection in exciton region of spectrum of structure with a single quantum well. Oblique and normal incidence of light," Fiz. Tekh. Poluprovodn. (S.-Peterburg) 22, 784-788 (1988). E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Y. Meltzer, and M. A. Kaliteevskii,[Sov. Phys. Semicond. 22, 497-501 (1988)].

1966

V. M. Agranovich and O. A. Dubovskii, "Effect of retarded interaction on exciton spectra in one- and two-dimensional crystals," JETP Lett. 3, 233-238 (1966).

1961

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Fiz. Tekh. Poluprovodn. (S.-Peterburg)

E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Ya. Meltzer, and M. A. Kaliteevskii, "Reflection in exciton region of spectrum of structure with a single quantum well. Oblique and normal incidence of light," Fiz. Tekh. Poluprovodn. (S.-Peterburg) 22, 784-788 (1988). E. L. Ivchenko, P. S. Kop'ev, V. P. Kochereshko, I. N. Uraltsev, D. R. Yakovlev, S. V. Ivanov, B. Y. Meltzer, and M. A. Kaliteevskii,[Sov. Phys. Semicond. 22, 497-501 (1988)].

Fiz. Tverd. Tela (Leningrad)

E. L. Ivchenko, "Exciton polaritons in periodic quantum well structures," Fiz. Tverd. Tela (Leningrad) 33, 2388-2393 (1991) E. L. Ivchenko,[Sov. Phys. Solid State 33, 1344-1349 (1991)].

E. L. Ivchenko and A. V. Kavokin, "Light reflection from quantum dot structures," Fiz. Tverd. Tela (Leningrad) 34, 1815-1822 (1992). E. L. Ivchenko and A. V. Kavokin,[Sov. Phys. Solid State 34, 968-975 (1992)].

J. Lumin.

V. V. Popov, T. Yu. Bagaeva, T. V. Teperik, N. J. M. Horing, and Y. Ayaz, "Ultrafast radiative decay of polaritons in an interface layer with strong excitonic response," J. Lumin. 112, 225-229 (2005).
[CrossRef]

J. Phys. Soc. Jpn.

A. L. Yablonskii, E. A. Muljarov, N. A. Gippius, S. G. Tikhodeev, T. Fujita, and T. Ishihara, "Polariton effect in distributed feedback microcavities," J. Phys. Soc. Jpn. 70, 1137-1144 (2001).
[CrossRef]

JETP Lett.

V. M. Agranovich and O. A. Dubovskii, "Effect of retarded interaction on exciton spectra in one- and two-dimensional crystals," JETP Lett. 3, 233-238 (1966).

Phys. Rev.

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Phys. Rev. B

E. A. Muljarov, S. G. Tikhodeev, N. A. Gippius, and T. Ishihara, "Excitons in self-organized semiconductor/insulator superlattices: PbI-based porovskite compounds," Phys. Rev. B 51, 14370-14370 (1995).
[CrossRef]

A. Vinattieri, J. Shah, T. C. Damen, D. S. Kim, L. N. Pfeiffer, M. Z. Maialle, and L. J. Sham, "Exciton dynamics in GaAs quantum wells under resonant excitation," Phys. Rev. B 50, 10868-10879 (1994).
[CrossRef]

F. Tassone, F. Bassani, and L. C. Andreani, "Quantum-well reflectivity and exciton-polariton dispersion," Phys. Rev. B 45, 6023-6030 (1992).
[CrossRef]

D. S. Citrin, "Radiative lifetimes of excitons in quantum wells: Localization and phase-coherence effects," Phys. Rev. B 47, 3832-3841 (1993).
[CrossRef]

S. Jorda, U. Rössler, and D. Broido, "Fine structure of excitons and polariton dispersion in quantum wells," Phys. Rev. B 48, 1669-1677 (1993).
[CrossRef]

R. Atanasov, F. Bassani, and V. M. Agranovich, "Mean-field polariton theory for asymmetric quantum wells," Phys. Rev. B 49, 2658-2666 (1994).
[CrossRef]

E. Hanamura, "Rapid radiative decay and enhanced optical nonlinearity of excitons in a quantum well," Phys. Rev. B 38, 1228-1234 (1988).
[CrossRef]

N. J. M. Horing, T. Jena, H. L. Cui, and J. D. Mancini, "Dynamic dielectric properties of a bounded solid-state plasma and a two-dimensional electron sheet: inverse dielectric function and coupled collective modes," Phys. Rev. B 54, 2785-2790 (1996).
[CrossRef]

N. J. M. Horing and Y. Ayaz, "Dynamic dielectric response of an asymmetric double quantum well near the bounding surface of a semi-infinite dynamic plasmalike host medium," Phys. Rev. B 58, 2001-2007 (1998).
[CrossRef]

T. Ishihara, J. Takahashi, and T. Goto, "Optical properties due to electronic transitions in two-dimensional semiconductors (CnH2n+1NH3)2PbI4," Phys. Rev. B 42, 11099-11107 (1990).
[CrossRef]

T. Fujita, Y. Sato, T. Kuitani, and T. Ishihara, "Tunable polariton absorption of distributed feedback microcavities at room temperature," Phys. Rev. B 57, 12428-12434 (1998).
[CrossRef]

Phys. Rev. Lett.

B. Deveaud, F. Clérot, N. Roy, K. Satzke, B. Sermage, and D. S. Katzer, "Enhanced radiative recombination of free excitons in GaAs quantum wells," Phys. Rev. Lett. 67, 2355-2358 (1991).
[CrossRef] [PubMed]

Solid State Commun.

L. C. Andreani, F. Tassone, and F. Bassani, "Radiative lifetime of free excitons in quantum wells," Solid State Commun. 77, 641-645 (1991).
[CrossRef]

V. V. Popov, T. V. Teperik, N. J. M. Horing, and T. Yu. Bagaeva, "Inhomogeneous radiative decay of polariton modes in a two-dimensional exciton system," Solid State Commun. 127, 589-594 (2003).
[CrossRef]

E. L. Ivchenko, V. P. Kochereshko, P. S. Kop'ev, V. A. Kosobukin, I. N. Uraltsev, and D. R. Yakovlev, "Exciton longitudinal-transverse splitting in GaAs/AsGaAs superlattices and multiple quantum wells," Solid State Commun. 70, 529-534 (1989).
[CrossRef]

M. W. Berz, L. C. Andreani, E. F. Steingmeier, and F.-K. Reinhart, "Exchange splitting of light hole excitons in Al1−χGaχAs-GaAs quantum wells," Solid State Commun. 80, 553-556 (1991).
[CrossRef]

Other

C.-To Tai, Dyadic Green's Functions in Electromagnetic Theory (Intext Educational Publishers, 1971).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975), Eq. 7.57.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, 1941), p. 340.

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Figures (3)

Fig. 1
Fig. 1

Schematic of a spatiotemporal distribution of the amplitude of a radiatively decaying exciton-polariton mode of the 2D excitonic layer. The 2D excitonic layer is located at z = 0 .

Fig. 2
Fig. 2

Amplitudes of the driven (dashed curves) and reradiated undriven free (solid curves) EM fields emitted from the 2D excitonic layer into the host medium ( z > 0 ) as functions of frequency of the incident light for various angles of incidence: (a) 20°, (b) 45°, (c) 60°, (d) 85°. Parameters of the 2D excitonic layer are ω e x = 2.4 eV , ω LT = 0.05 eV , ε b = 4.41 ; γ e x = 0 , d = 30 nm . Amplitudes of the induced fields are normalized to the amplitude of the incident light pulse.

Fig. 3
Fig. 3

Same as in Fig. 2 for γ e x = 0.03 eV .

Equations (47)

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

[ I ̂ ( 2 + ω 2 ε h c 2 ) ] E ( r , ω ) + 4 π i ω c 2 d r σ ̂ 2 D ( r , r ; ω ) E ( r , ω ) = 4 π i ω c 2 J e x t ( r , ω ) ,
[ I ̂ ( 2 + ω 2 ε h c 2 ) ] G ̂ ( r , r ; ω ) + 4 π i ω c 2 d r σ ̂ 2 D ( r , r ; ω ) G ̂ ( r , r ; ω ) = I ̂ δ ( r r ; ω ) .
( G ̂ 3 D ) 1 = I ̂ ( 2 + ω 2 ε h c 2 ) ,
G ̂ ( r , r , ω ) = G ̂ 3 D ( r , r ; ω ) 4 π i ω c 2 d r d r G ̂ 3 D ( r , r , ω ) σ ̂ 2 D ( r , r ; ω ) G ̂ ( r , r ; ω ) .
G ̂ = G ̂ 3 D 4 π i ω c 2 G ̂ 3 D σ ̂ 2 D G ̂ ,
G ̂ 3 D 1 = G ̂ 1 4 π i ω c 2 σ ̂ 2 D .
G ̂ 1 E = G ̂ 3 D 1 E + 4 π i ω c 2 σ ̂ 2 D E = 4 π i ω c 2 J .
E + 4 π i ω c 2 G ̂ 3 D σ ̂ 2 D E = E 0 ,
E 0 = 4 π i ω c 2 G ̂ 3 D J ,
G ̂ = K ̂ G ̂ 3 D or K ̂ = G ̂ G ̂ 3 D 1 ,
K ̂ = I ̂ 4 π i ω c 2 G ̂ 3 D σ ̂ 2 D K ̂ .
δ E ( 1 ) δ E 0 ( 2 ) δ E δ E 0 , δ E 0 ( 1 ) δ E 0 ( 2 ) = I ̂ δ ( 1 2 ) I ̂ ,
δ E δ E 0 + 4 π i ω c 2 G ̂ 3 D σ ̂ 2 D δ E δ E 0 = I ̂ .
K ̂ δ E δ E 0 , K ̂ ( r , r ; ω ) δ E ( r , ω ) δ E 0 ( r , ω ) .
E ( r , ω ) = d r K ̂ ( r , r ; ω ) E 0 ( r , ω ) ,
E = K ̂ E 0 = G ̂ G ̂ 3 D 1 E 0 = G ̂ ( G ̂ 1 4 π i ω c 2 σ ̂ 2 D ) E 0 ,
E = E 0 4 π i ω c 2 G ̂ σ ̂ 2 D E 0 .
ε ( ω ) = ε b ( 1 + ω LT ω e x ω i γ e x ) ,
G ̂ ( z , z , k ; ω ) = G ̂ 3 D ( z , z , k ; ω ) 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( z , 0 , k ; ω ) G ̂ ( 0 , z , k ; ω ) .
G ̂ ( z , z , k ; ω ) = G ̂ 3 D ( z , z , k ; ω ) 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( z , 0 , k ; ω ) [ I ̂ + 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( 0 , 0 , k ; ω ) ] 1 G ̂ 3 D ( 0 , z , k ; ω ) .
G ̂ 3 D ( k , ω ) = I ̂ kk k 2 ω 2 ε h c 2 k 2 + kk k 2 ω 2 ε h c 2 ,
G ̂ 3 D ( z , z , k , ω ) = 1 2 i k z { I ̂ c 2 ω 2 ε h [ k k + ( k e z + e z k ) 1 i z + e z e z ( 1 i z ) 2 ] } exp ( i k z z z ) ,
G ̂ 3 D ( z , z , k , ω ) = 1 2 i k z { I ̂ c 2 ω 2 ε h [ k k + k z sgn ( z ) ( k e z + e z k ) + e z e z ( k z 2 2 i k z δ ( z z ) ) ] } exp ( i k z z z ) ,
k z = ( ω 2 c 2 ) ε h k 2 .
G ̂ 3 D ( 0 , 0 , k ; ω ) = 1 2 i k z { I ̂ c 2 ω 2 ε h [ k k + e z e z ( k z 2 2 i k z d ) ] } .
E ( z , k , ω ) = E 0 ( z , k , ω ) 4 π i ω c 2 σ 2 D ( ω ) G ̂ ( z , 0 , k ; ω ) E 0 ( 0 , k , ω ) .
G ̂ ( z , 0 , k ; ω ) = G ̂ 3 D ( z , 0 , k ; ω ) { I ̂ 4 π i ω c 2 σ 2 D ( ω ) [ I ̂ + 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( 0 , 0 , k ; ω ) ] 1 G ̂ 3 D ( 0 , 0 , k ; ω ) } .
I ̂ = [ I ̂ + 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( 0 , 0 , k ; ω ) ] 1 [ I ̂ + 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( 0 , 0 , k ; ω ) ] ,
G ̂ ( z , 0 , k ; ω ) = G ̂ 3 D ( z , 0 , k ; ω ) [ I ̂ + 4 π i ω c 2 σ 2 D ( ω ) G ̂ 3 D ( 0 , 0 , k ; ω ) ] 1 .
k ω 2 = ω 2 ε h c 2 ,
G ̂ ( z , 0 , k ; ω ) = e i k z z 2 i k z { e x e x D 1 ( 1 k z 2 k ω 2 ) + e y e y D 3 + e z e z D 2 ( 1 1 k ω 2 [ k z 2 2 i k z δ ( z ) ] ) k x k z k ω 2 sgn ( z ) [ e x e z D 2 + e z e x D 1 ] } ,
D 1 = 1 + 2 π ω c 2 k z σ 2 D ( ω ) ( 1 k x 2 k ω 2 ) ,
D 2 = 1 + 2 π ω c 2 k z σ 2 D ( ω ) [ 1 1 k ω 2 ( k z 2 2 i k z d ) ] ,
D 3 = 1 + 2 π ω c 2 k z σ 2 D ( ω ) .
E ( r , z , t ) = 1 ( 2 π ) 3 d ( 2 ) k d ω E ( z , k , ω ) exp [ i ( k r ω t ) ] .
E 0 ( r , z , t ) = E 0 θ ( τ ) exp ( i ω 0 τ ) ,
E 0 ( z , k x , ω ) = E 0 ( 2 π ) 2 i ω ( ω 0 + i 0 + ) δ ( k x k x 0 ω ω 0 ) δ ( k y ) exp ( i k z 0 ω ω 0 z ) .
E ( r , z , ω ) E 0 ( r , z , ω ) = 2 π i ω c 2 σ 2 D k ω cos θ 0 exp [ i k ω ( z cos θ 0 + x sin θ 0 ) ] 1 ω ( ω 0 + i 0 + ) × [ e x e x D 1 cos 2 θ 0 + e y e y D 3 + e z e z D 2 sin 2 θ 0 sin θ 0 cos θ 0 sgn ( z ) ( e x e z D 2 + e z e x D 1 ) ] E 0 ,
D 1 = ( ω e x i γ e x ) ω ϖ ( L ) ϖ ( L ) ( ω ω e x + i γ e x ) , ϖ ( L ) = ω e x i γ e x 1 + i ε h 1 2 ( cos θ 0 ) d ω LT 2 c ,
D 2 = ( ω e x + ω LT i γ e x ) ω ϖ ( Z ) ϖ ( Z ) ( ω ω e x + i γ e x ) ,
ϖ ( Z ) = ω e x + ω LT i γ e x 1 + i ε h 1 2 ( sin θ 0 tan θ 0 ) d ω LT 2 c ,
D 3 = ( ω e x i γ e x ) ω ϖ ( T ) ϖ ( T ) ( ω ω e x + i γ e x ) , ϖ ( T ) = ω e x i γ e x 1 + i ε h 1 2 d ω LT ( 2 c cos θ 0 ) .
E ± ( x , z , t ) = E 0 ( e x cos θ 0 + e z sin θ 0 ) θ ( τ ) exp ( i ω 0 τ ) E 0 i ε b ω LT d 2 c { θ ( τ ± ) ( A ± + B ± ) exp ( i ω 0 τ ± ) + θ ( τ ± ) [ A ± ϖ ( L ) ω 0 exp ( i ϖ ( L ) τ ± ) + B ± ϖ ( Z ) ω 0 exp ( i ϖ ( Z ) τ ± ) ] } ,
τ ± = t ( k x 0 ω 0 ) x ( k z 0 ω 0 ) z ,
A ± = [ e x cos θ 0 e z sin θ 0 ] ϖ ( L ) ω 0 cos θ 0 ( ω e x i γ e x ) ( ω 0 ϖ ( L ) ) ,
B ± = [ e z sin θ 0 e x cos θ 0 ] ϖ ( Z ) ω 0 sin θ 0 tan θ 0 ( ω e x + ω LT i γ e x ) ( ω 0 ϖ ( Z ) ) .
E ± ( x , z , t ) = E 0 e y θ ( τ ) exp ( i ω 0 τ ) + E 0 e y i ε b ω LT d 2 c × ϖ ( T ) ω 0 ( ω e x i γ e x ) ( ϖ T ω 0 ) cos θ 0 [ θ ( τ ± ) exp ( i ω 0 τ ± ) + ϖ ( T ) ω 0 θ ( τ ± ) exp ( i ϖ ( T ) τ ± ) ] .

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