Abstract

Time evolution of a surface plasmon guided by a metal surface owing to rapid ionization in a thin semiconductor film placed on the metal surface is theoretically investigated. When the plasma frequency of the created plasma in the film is close to the frequency of the initial surface plasmon, plasma oscillations in the film are resonantly excited by the surface plasmon, and the energy of the initial surface plasmon starts to move back and forth between these oscillations and the surface plasmon. Initially fast surface plasmons also produce significant transient radiation that propagates from the metal surface into vacuum. The general picture of transient processes that occur after ionization can be applied for ultrafast transient spectroscopy of an electron–hole plasma in metal and semiconductor films.

© 2001 Optical Society of America

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  1. V. M. Agranovich and A. G. Mal’shukov, “Surface polariton spectra if the resonance with the transition layer exists,” Opt. Commun. 11, 169–171 (1974).
    [Crossref]
  2. Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
    [Crossref]
  3. V. M. Agranovich and D. L. Mills, eds., Surface Polaritons (North-Holland, New York, 1982), Chaps. 5 and 6.
  4. T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
    [Crossref]
  5. I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
    [Crossref]
  6. S. I. Bozhevolnyi and F. A. Pudonin, “Two-dimensional micro-optics of surface plasmons,” Phys. Rev. Lett. 78, 2823–2826 (1997).
    [Crossref]
  7. J. A. Sánchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 60, 8359–8367 (1999).
    [Crossref]
  8. M. Kahl and E. Vodes, “Analysis of plasmon resonance and silver-enhanced Raman scattering on periodic silver structures,” Phys. Rev. B 61, 14078–14088 (2000).
    [Crossref]
  9. V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
    [Crossref]
  10. S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
    [Crossref] [PubMed]
  11. K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
    [Crossref]
  12. M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).
  13. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000).
    [Crossref]
  14. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999).
    [Crossref]
  15. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, Berlin, 1984), Chap. 5.
  16. For a review, see for example, W. Steinmann, “Optical plasma resonances in solids,” Phys. Status Solidi 28, 437–462 (1968).
    [Crossref]
  17. M. I. Bakunov and A. V. Maslov, “Transient input of an electromagnetic wave into an open waveguide coated with nonstationary plasma film,” J. Appl. Phys. 83, 3885–3891 (1998).
    [Crossref]
  18. A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
    [Crossref]
  19. M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
    [Crossref]

2000 (5)

M. Kahl and E. Vodes, “Analysis of plasmon resonance and silver-enhanced Raman scattering on periodic silver structures,” Phys. Rev. B 61, 14078–14088 (2000).
[Crossref]

K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
[Crossref]

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
[Crossref]

M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000).
[Crossref]

1999 (2)

M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999).
[Crossref]

J. A. Sánchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 60, 8359–8367 (1999).
[Crossref]

1998 (1)

M. I. Bakunov and A. V. Maslov, “Transient input of an electromagnetic wave into an open waveguide coated with nonstationary plasma film,” J. Appl. Phys. 83, 3885–3891 (1998).
[Crossref]

1997 (1)

S. I. Bozhevolnyi and F. A. Pudonin, “Two-dimensional micro-optics of surface plasmons,” Phys. Rev. Lett. 78, 2823–2826 (1997).
[Crossref]

1996 (2)

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[Crossref] [PubMed]

M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).

1982 (1)

I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
[Crossref]

1981 (1)

V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
[Crossref]

1978 (1)

T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
[Crossref]

1975 (1)

Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
[Crossref]

1974 (1)

V. M. Agranovich and A. G. Mal’shukov, “Surface polariton spectra if the resonance with the transition layer exists,” Opt. Commun. 11, 169–171 (1974).
[Crossref]

1968 (1)

For a review, see for example, W. Steinmann, “Optical plasma resonances in solids,” Phys. Status Solidi 28, 437–462 (1968).
[Crossref]

Abelès, F.

T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
[Crossref]

Agranovich, V. M.

V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
[Crossref]

V. M. Agranovich and A. G. Mal’shukov, “Surface polariton spectra if the resonance with the transition layer exists,” Opt. Commun. 11, 169–171 (1974).
[Crossref]

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, Berlin, 1984), Chap. 5.

Bakunov, M. I.

M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
[Crossref]

M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000).
[Crossref]

M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999).
[Crossref]

M. I. Bakunov and A. V. Maslov, “Transient input of an electromagnetic wave into an open waveguide coated with nonstationary plasma film,” J. Appl. Phys. 83, 3885–3891 (1998).
[Crossref]

M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).

Barnes, W. L.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi and F. A. Pudonin, “Two-dimensional micro-optics of surface plasmons,” Phys. Rev. Lett. 78, 2823–2826 (1997).
[Crossref]

Brillante, A.

I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
[Crossref]

Capasso, F.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Cho, A. Y.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Gildenburg, V. B.

M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
[Crossref]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, Berlin, 1984), Chap. 5.

Gmachl, C.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Hutchinson, A. L.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Inagaki, Y.

K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
[Crossref]

Kahl, M.

M. Kahl and E. Vodes, “Analysis of plasmon resonance and silver-enhanced Raman scattering on periodic silver structures,” Phys. Rev. B 61, 14078–14088 (2000).
[Crossref]

Katayama, K.

K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
[Crossref]

Kitson, S. C.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[Crossref] [PubMed]

Kravtsov, V. E.

V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
[Crossref]

Leskova, T. A.

V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
[Crossref]

Lopez-Rios, T.

T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
[Crossref]

Mal’shukov, A. G.

V. M. Agranovich and A. G. Mal’shukov, “Surface polariton spectra if the resonance with the transition layer exists,” Opt. Commun. 11, 169–171 (1974).
[Crossref]

Maradudin, A. A.

J. A. Sánchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 60, 8359–8367 (1999).
[Crossref]

Maslov, A. V.

Möbius, D.

I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
[Crossref]

Nazin, V. G.

Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
[Crossref]

Pockrand, I.

I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
[Crossref]

Pudonin, F. A.

S. I. Bozhevolnyi and F. A. Pudonin, “Two-dimensional micro-optics of surface plasmons,” Phys. Rev. Lett. 78, 2823–2826 (1997).
[Crossref]

Sambles, J. R.

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[Crossref] [PubMed]

Sánchez-Gil, J. A.

J. A. Sánchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 60, 8359–8367 (1999).
[Crossref]

Sawada, T.

K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
[Crossref]

Sivco, D. L.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Steinmann, W.

For a review, see for example, W. Steinmann, “Optical plasma resonances in solids,” Phys. Status Solidi 28, 437–462 (1968).
[Crossref]

Tredicucci, A.

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

Vodes, E.

M. Kahl and E. Vodes, “Analysis of plasmon resonance and silver-enhanced Raman scattering on periodic silver structures,” Phys. Rev. B 61, 14078–14088 (2000).
[Crossref]

Vuye, G.

T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
[Crossref]

Yakovlev, Y. A.

Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
[Crossref]

Zharova, N. A.

M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
[Crossref]

Zhizhin, G. N.

Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
[Crossref]

Zhukov, S. N.

Appl. Phys. Lett. (1)

A. Tredicucci, C. Gmachl, F. Capasso, A. L. Hutchinson, D. L. Sivco, and A. Y. Cho, “Single-mode surface-plasmon laser,” Appl. Phys. Lett. 76, 2164–2166 (2000).
[Crossref]

J. Appl. Phys. (1)

M. I. Bakunov and A. V. Maslov, “Transient input of an electromagnetic wave into an open waveguide coated with nonstationary plasma film,” J. Appl. Phys. 83, 3885–3891 (1998).
[Crossref]

J. Chem. Phys. (1)

I. Pockrand, A. Brillante, and D. Möbius, “Exciton-surface plasmon coupling: an experimental investigation,” J. Chem. Phys. 77, 6289–6295 (1982).
[Crossref]

J. Opt. Soc. Am. B (1)

J. Phys. (Paris) (1)

T. Lopez-Rios, F. Abelès, and G. Vuye, “Splitting of the Al surface plasmon dispersion curves by Ag surface layers,” J. Phys. (Paris) 39, 645–650 (1978).
[Crossref]

Opt. Commun. (2)

V. M. Agranovich and A. G. Mal’shukov, “Surface polariton spectra if the resonance with the transition layer exists,” Opt. Commun. 11, 169–171 (1974).
[Crossref]

Y. A. Yakovlev, V. G. Nazin, and G. N. Zhizhin, “Surface polariton splitting due to thin surface film LO vibrations,” Opt. Commun. 15, 293–295 (1975).
[Crossref]

Opt. Lett. (1)

Phys. Plasmas (1)

M. I. Bakunov, V. B. Gildenburg, S. N. Zhukov, and N. A. Zharova, Phys. Plasmas 7, 1035 (2000).
[Crossref]

Phys. Rev. B (3)

K. Katayama, Y. Inagaki, and T. Sawada, “Ultrafast two-step thermalization process of photoexcited electrons at a gold surface: application of a wavelength-selective transient reflecting grating method,” Phys. Rev. B 61, 7332–7335 (2000).
[Crossref]

J. A. Sánchez-Gil and A. A. Maradudin, “Near-field and far-field scattering of surface plasmon polaritons by one-dimensional surface defects,” Phys. Rev. B 60, 8359–8367 (1999).
[Crossref]

M. Kahl and E. Vodes, “Analysis of plasmon resonance and silver-enhanced Raman scattering on periodic silver structures,” Phys. Rev. B 61, 14078–14088 (2000).
[Crossref]

Phys. Rev. Lett. (2)

S. C. Kitson, W. L. Barnes, and J. R. Sambles, “Full photonic band gap for surface modes in the visible,” Phys. Rev. Lett. 77, 2670–2673 (1996).
[Crossref] [PubMed]

S. I. Bozhevolnyi and F. A. Pudonin, “Two-dimensional micro-optics of surface plasmons,” Phys. Rev. Lett. 78, 2823–2826 (1997).
[Crossref]

Phys. Status Solidi (1)

For a review, see for example, W. Steinmann, “Optical plasma resonances in solids,” Phys. Status Solidi 28, 437–462 (1968).
[Crossref]

Plasma Phys. Rep. (1)

M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).

Solid State Commun. (1)

V. M. Agranovich, V. E. Kravtsov, and T. A. Leskova, “Thin film on metal: resonance effects in light diffraction at the edge,” Solid State Commun. 40, 687–692 (1981).
[Crossref]

Other (2)

V. M. Agranovich and D. L. Mills, eds., Surface Polaritons (North-Holland, New York, 1982), Chaps. 5 and 6.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion and Excitons (Springer-Verlag, Berlin, 1984), Chap. 5.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem: The initial surface wave propagates along the metal plasma–vacuum boundary coated with a thin semiconductor film that will be ionized by a laser pulse.

Fig. 2
Fig. 2

Kinematic diagram: (1) dispersion curve for the initial surface wave before ionization in the coating; (2) branches of the dispersion curve after ionization in the coating; (3) light-line h=ω/c. Frequencies of the new surface waves correspond to the intersection points of the horizontal line h=h0 with curve (2). Symmetric branches of the dispersion curve (2) in the region of negative values of frequency ω are not shown because of negligibly small amplitudes of corresponding surface waves.

Fig. 3
Fig. 3

Frequencies of the excited surface waves ω+/ω0 (curve 1) and ω-/ω0 (curve 2) as a function of the plasma frequency in the coating for ω0d/c=0.01, n0=1.1, and b=10.

Fig. 4
Fig. 4

Amplitudes of the excited surface waves B+/B0 (curve 1) and B-/B0 (curve 2) for the same parameters as in Fig. 3.

Fig. 5
Fig. 5

Energies of the excited surface waves W+/W0 (curve 1) and W-/W0 (curve 2) for the same parameters as in Fig. 3.

Fig. 6
Fig. 6

Energy localized in the film Wf/W0 (curve 2) and outside the film Wout/W0 (curve 1) for the same parameters as in Fig. 3.

Fig. 7
Fig. 7

Group velocity vg±/c of the excited positive surface mode (curve 1) and negative surface mode (curve 2) for the same parameters as in Fig. 3.

Fig. 8
Fig. 8

Radiated energy Wv/W0 (curve 1) and surface-wave energies W±/W0 (curves 2 and 3) as functions of the plasma frequency in the coating for ω0d/c=0.01, n0=1.002, and b=10.

Fig. 9
Fig. 9

Angular density of the outgoing radiation Wv(θ)/W0 for ω0d/c=0.01, n0=1.002, b=10, and ωf/ω0=1 (when Wv/W00.63).

Equations (29)

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

Bz(x, y, t)=B0 exp(iω0t-ih0x)×exp(-κv1y)ify>0exp(κp1y)ify<0,
h0=ω0c 11+1,
Ex(x, y, t)=cBziω0 -κv1ify>0κp1/1ify<0,
Ey(x, y, t)=ch0Bzω0 1ify>01/1ify<0.
 y 1 by-h02+ s2c2b=F(x, y, s),
F(x, y, s)=-s+iω0γc2Bz(x, y, 0)-Ex(x, y, 0) c y γ,
{b}=0,1 by+γcEx(x, y, 0)=0,
b(x, y, s)=Bz(x, y, 0)s-iω0+Av(s)×exp[-ih0x-κv(y-d)],
Av(s)=-B0 isω0ωf2h02d(s-iω0)(s2+ω02+ωf2)D(s),
D(s)=s2[h02d+f(κp/p+κv)],
κpp+κv-bP h0ω02(ω2-ω02),
P=(2n02-1) 2n02(n02-1)+12n05n02-1.
ω±2=ωf2+ω022±Q22,
Q2=(ωf2-ω02)2+2h0dbPω02(ωf2+ω02).
ω±=ωf+ω02±Ω2,
Ω=(ωf-ω0)2+h0dbPω02.
Bα=B0 ω03ωf2ωα(ωα-ω0)(ω02+ωf2-ωα2) h0dbP βα 1(ωα-ωβ),
B±=±B0 ω024Ω(ω±-ω0) h0dbP.
W0=B028π cn0ω0P.
Wout±W0=B±2B02 ω02cn0 1Pn02ω02ω±2×1κv±+2-p±p±2κp±-1-p±p±κp±,
Wf±W0=B±2B02 ω02ω±2(ω±2-ωf2)2 h0dbP.
Wout(t)=Wperm+Wexc cos2Ω2t,
Wf(t)=Wexc sin2Ω2t.
WexcW0=h0dbP ω02Ω2,
WpermW0=1-WexcW0.
Wv=-π/2π/2Wv(θ)dθ,
Wv(θ)=c2h016π2|Av(s=iω)|2 cot2 θ,
Av(s)=B0 isω0ωf2(s-iω0)(s2+ω02+ωf2)D(s)×κpp-s2κv1ω02f[1-cosh(κfd)]-κf-s2f2κv1κpω02pκfsinh(κfd),
D(s)=s2κf+f2κvκppκfsinh(κfd)+fκpp+κvcosh(kfd),

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