Abstract

For decades, crystalline silicon (Si) has been the semiconductor of choice for the majority of applications in microelectronics. Recent advances in material science have focused attention on the silicon-on-insulator (SOI) platform, a submicrometer-thick layer of single crystal Si resting on an insulating silicon dioxide (SiO2) layer. Here we calculate the lifetime of an electric dipole moment oscillating in the cover region of several canonical Si waveguiding structures. We show that the vicinity just above SOI produces the most dramatic changes to the radiative lifetime and thus the power spectrum of the emitting dipole. We demonstrate that SOI stands apart from other Si-based optoelectronic platforms in its ability to transport energy, in the form of light, away from an oscillating electric dipole via highly localized, optical- and IR-frequency guided waves.

© 2001 Optical Society of America

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References

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  1. L. Geppert, “Solid state semiconductors. 1999 technology analysis and forecast,” IEEE Spectrum 1, 52–56 (1999).
    [CrossRef]
  2. See, for example, the spatial issue on Silicon-on-Insulator Technology (seven papers) of MRS Bull. 23, 13–40 (1998).
  3. K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
    [CrossRef]
  4. L. Peters, “SOI takes over where silicon leaves off,” Semicond. Int. 16, 48–51 (1993).
  5. T. G. Brown, A. E. Bieber, “Coupling, switching, and modulation in silicon-based optoelectronic structures,” in Silicon-Based Monolithic and Hybrid Optoelectronic Devices, D. C. Houghton, B. Jalali, eds., Proc. SPIE3007, 12–21 (1997).
    [CrossRef]
  6. J. Y. L. Ho, K. S. Wong, “High-speed and high-sensitivity silicon-on-insulator metal-semiconductor-metal photodetector with trench structure,” Appl. Phys. Lett. 69, 16–19 (1996).
    [CrossRef]
  7. C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
    [CrossRef]
  8. M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
    [CrossRef]
  9. H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
    [CrossRef]
  10. B. J. Soller, D. G. Hall, manuscript available from the authors.
  11. See for example, M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
    [CrossRef]
  12. K. H. Drexhage, “Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1974), Vol. 12, pp. 163–232.
  13. W. H. Weber, C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236–238 (1979).
    [CrossRef] [PubMed]
  14. R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.
  15. See, for example, W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison–Wesley, Reading, Mass., 1956), Chap. 20 and Eqs. (21)–(23).
  16. See, for example, O. Svelto, Principles of Lasers, 2nd ed. (Plenum, New York, 1982), Chap. 2.
  17. K. G. Sullivan, D. G. Hall, “Enhancement and inhibi-tion of electromagnetic radiation in plane-layered media. I. Plane-wave spectrum approach to modeling classical effects,” J. Opt. Soc. Am. B 14, 1149–1159 (1997).
    [CrossRef]
  18. G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
    [CrossRef]
  19. We take the quantum yield to be unity because we are interested in radiation effects. This implies that our forthcoming definition of efficiency is with respect to the total amount of light radiated (or reradiated in the case of scatterers) and notthe total amount of energy available in the excited state.
  20. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  21. Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, Calif., 1985, 1991), Vols. I and II.
  22. See, for example, R. Soref, “Applications of silicon-based optoelectronics,” MRS Bull. 23, 20–24 (1998), and references therein.
  23. See, for example, D. G. Hall, “Survey of silicon-based integrated optics,” IEEE Comput. Mag. 20, 25–32 (1987), and references therein.
    [CrossRef]
  24. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).
  25. W. R. Holland, D. G. Hall, “Waveguide mode enhancement of molecular fluorescence,” Opt. Lett. 10, 414–416 (1985).
    [CrossRef] [PubMed]
  26. K. G. Sullivan, D. G. Hall, “Enhancement and inhibition of electromagnetic radiation in plane-layered media. II. Enhanced fluorescence in optical waveguide sensors,” J. Opt. Soc. Am. B 14, 1160–1166 (1997).
    [CrossRef]
  27. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

1999 (2)

L. Geppert, “Solid state semiconductors. 1999 technology analysis and forecast,” IEEE Spectrum 1, 52–56 (1999).
[CrossRef]

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

1998 (3)

See, for example, the spatial issue on Silicon-on-Insulator Technology (seven papers) of MRS Bull. 23, 13–40 (1998).

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

See, for example, R. Soref, “Applications of silicon-based optoelectronics,” MRS Bull. 23, 20–24 (1998), and references therein.

1997 (2)

1996 (2)

J. Y. L. Ho, K. S. Wong, “High-speed and high-sensitivity silicon-on-insulator metal-semiconductor-metal photodetector with trench structure,” Appl. Phys. Lett. 69, 16–19 (1996).
[CrossRef]

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

1994 (1)

M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
[CrossRef]

1993 (1)

L. Peters, “SOI takes over where silicon leaves off,” Semicond. Int. 16, 48–51 (1993).

1987 (1)

See, for example, D. G. Hall, “Survey of silicon-based integrated optics,” IEEE Comput. Mag. 20, 25–32 (1987), and references therein.
[CrossRef]

1985 (2)

W. R. Holland, D. G. Hall, “Waveguide mode enhancement of molecular fluorescence,” Opt. Lett. 10, 414–416 (1985).
[CrossRef] [PubMed]

See for example, M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

1984 (1)

G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

1979 (1)

Bieber, A. E.

T. G. Brown, A. E. Bieber, “Coupling, switching, and modulation in silicon-based optoelectronic structures,” in Silicon-Based Monolithic and Hybrid Optoelectronic Devices, D. C. Houghton, B. Jalali, eds., Proc. SPIE3007, 12–21 (1997).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Brown, T. G.

T. G. Brown, A. E. Bieber, “Coupling, switching, and modulation in silicon-based optoelectronic structures,” in Silicon-Based Monolithic and Hybrid Optoelectronic Devices, D. C. Houghton, B. Jalali, eds., Proc. SPIE3007, 12–21 (1997).
[CrossRef]

Campbell, J. C.

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.

Chen, E.

M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
[CrossRef]

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Chou, S. Y.

M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
[CrossRef]

Drexhage, K. H.

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1974), Vol. 12, pp. 163–232.

Eagen, C. F.

Ford, G. W.

G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Geppert, L.

L. Geppert, “Solid state semiconductors. 1999 technology analysis and forecast,” IEEE Spectrum 1, 52–56 (1999).
[CrossRef]

Hall, D. G.

Ho, J. Y. L.

J. Y. L. Ho, K. S. Wong, “High-speed and high-sensitivity silicon-on-insulator metal-semiconductor-metal photodetector with trench structure,” Appl. Phys. Lett. 69, 16–19 (1996).
[CrossRef]

Hobart, K. D.

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

Holland, W. R.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Jernigan, G. G.

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

Kub, F. J.

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

Li, R.

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

Liu, M. Y.

M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
[CrossRef]

Moskovits, M.

See for example, M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Peters, L.

L. Peters, “SOI takes over where silicon leaves off,” Semicond. Int. 16, 48–51 (1993).

Prock, A.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.

Schaub, J. D.

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

Schow, C. L.

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.

Sommerfeld, A.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

Soref, R.

See, for example, R. Soref, “Applications of silicon-based optoelectronics,” MRS Bull. 23, 20–24 (1998), and references therein.

Stuart, H. R.

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

Sullivan, K. G.

Svelto, O.

See, for example, O. Svelto, Principles of Lasers, 2nd ed. (Plenum, New York, 1982), Chap. 2.

Thompson, P. E.

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

Twigg, M. E.

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

Weber, W. H.

G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

W. H. Weber, C. F. Eagen, “Energy transfer from an excited dye molecule to the surface plasmons of an adjacent metal,” Opt. Lett. 4, 236–238 (1979).
[CrossRef] [PubMed]

Wong, K. S.

J. Y. L. Ho, K. S. Wong, “High-speed and high-sensitivity silicon-on-insulator metal-semiconductor-metal photodetector with trench structure,” Appl. Phys. Lett. 69, 16–19 (1996).
[CrossRef]

Appl. Phys. Lett. (3)

J. Y. L. Ho, K. S. Wong, “High-speed and high-sensitivity silicon-on-insulator metal-semiconductor-metal photodetector with trench structure,” Appl. Phys. Lett. 69, 16–19 (1996).
[CrossRef]

M. Y. Liu, E. Chen, S. Y. Chou, “140-GHz metal-semiconductor-metal photodetectors on silicon-on-insulator substrate with a scaled active layer,” Appl. Phys. Lett. 65, 887–888 (1994).
[CrossRef]

H. R. Stuart, D. G. Hall, “Absorption enhancement in silicon-on-insulator waveguides using metal island films,” Appl. Phys. Lett. 69, 2327–2329 (1996).
[CrossRef]

Electron. Lett. (1)

K. D. Hobart, F. J. Kub, G. G. Jernigan, M. E. Twigg, P. E. Thompson, “Fabrication of SOI substrates with ultra-thin Si layers,” Electron. Lett. 32, 1265–1267 (1998).
[CrossRef]

IEEE Comput. Mag. (1)

See, for example, D. G. Hall, “Survey of silicon-based integrated optics,” IEEE Comput. Mag. 20, 25–32 (1987), and references therein.
[CrossRef]

IEEE J. Quantum Electron. (1)

C. L. Schow, R. Li, J. D. Schaub, J. C. Campbell, “Design and implementation of high-speed planar Si photodiodes fabricated on SOI substrates,” IEEE J. Quantum Electron. 35, 1478–1482 (1999).
[CrossRef]

IEEE Spectrum (1)

L. Geppert, “Solid state semiconductors. 1999 technology analysis and forecast,” IEEE Spectrum 1, 52–56 (1999).
[CrossRef]

J. Opt. Soc. Am. B (2)

MRS Bull. (2)

See, for example, R. Soref, “Applications of silicon-based optoelectronics,” MRS Bull. 23, 20–24 (1998), and references therein.

See, for example, the spatial issue on Silicon-on-Insulator Technology (seven papers) of MRS Bull. 23, 13–40 (1998).

Opt. Lett. (2)

Phys. Rep. (1)

G. W. Ford, W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113, 195–287 (1984).
[CrossRef]

Rev. Mod. Phys. (1)

See for example, M. Moskovits, “Surface-enhanced spectroscopy,” Rev. Mod. Phys. 57, 783–826 (1985).
[CrossRef]

Semicond. Int. (1)

L. Peters, “SOI takes over where silicon leaves off,” Semicond. Int. 16, 48–51 (1993).

Other (11)

T. G. Brown, A. E. Bieber, “Coupling, switching, and modulation in silicon-based optoelectronic structures,” in Silicon-Based Monolithic and Hybrid Optoelectronic Devices, D. C. Houghton, B. Jalali, eds., Proc. SPIE3007, 12–21 (1997).
[CrossRef]

B. J. Soller, D. G. Hall, manuscript available from the authors.

We take the quantum yield to be unity because we are interested in radiation effects. This implies that our forthcoming definition of efficiency is with respect to the total amount of light radiated (or reradiated in the case of scatterers) and notthe total amount of energy available in the excited state.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, San Diego, Calif., 1985, 1991), Vols. I and II.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

R. R. Chance, A. Prock, R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” in Advances in Chemical Physics, I. Prigogine, S. A. Rice, eds. (Wiley, New York, 1978), Vol. 37, pp. 1–65.

See, for example, W. K. H. Panofsky, M. Phillips, Classical Electricity and Magnetism (Addison–Wesley, Reading, Mass., 1956), Chap. 20 and Eqs. (21)–(23).

See, for example, O. Svelto, Principles of Lasers, 2nd ed. (Plenum, New York, 1982), Chap. 2.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949).

K. H. Drexhage, “Interaction of light with monomolecular dye layers,” in Progress in Optics, E. Wolf, ed. (North–Holland, Amsterdam, 1974), Vol. 12, pp. 163–232.

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Figures (6)

Fig. 1
Fig. 1

(a) Example of a device fabricated with the SOI platform. The thin silicon layer acts as an optical-frequency waveguide, supporting two TE modes and one TM mode at 165-nm thickness. (b) Photocurrent enhancement due to the presence of metal nanoparticles measured by Stuart and Hall,9 in a detector similar to that shown in (a). The enhancement is defined as the ratio of the measured photocurrent in the device with the nanoparticles to that without and is due to resonant coupling of the incident light to the bound modes of the SOI via electric dipole currents induced in the metal nanoparticles.

Fig. 2
Fig. 2

Electric dipole positioned a distance xs from a stratified medium shown supporting a TE-polarized guided wave.

Fig. 3
Fig. 3

Semilog plot of the calculated power spectrum S(u) at wavelength λ=850 nm for vertical (dashed curves) and horizontal (solid curves) electric dipoles placed 30 nm above an SOI structure, with a 160-nm guiding-layer thickness. TE0, TE1, and TM0 designate the excitation of transverse-electric and transverse-magnetic optical-waveguide modes. Note that the electric dipole placed vertically couples only to TM modes, while the horizontal dipole couples to both TE and TM modes.

Fig. 4
Fig. 4

Linear plot of the calculated power spectrum S(u) at wavelength λ=850 nm for vertical (dashed curves) and horizontal (solid curves) electric dipoles placed above an SOI structure, with a 1-µm-thick guiding layer.

Fig. 5
Fig. 5

Plot of the ratio γwg as a function of Si guiding-layer thickness for VED and HED 30 nm above an SOI substrate at wavelength λ=850 nm. The strong peaks in the HED curve (solid curve) correspond to the excitation of TE-polarized guided waves in the thin Si layer, while the less-prominent peaks in the HED and all of the peaks in the VED (dashed curve) correspond to the excitation of TM-polarized guided modes. The cut-off values of dSi for the TE modes are marked by thin vertical lines. At dSi=160 nm, ∼84% of the light emitted by the horizontal dipole is directed into the waveguide modes of the upper Si layer.

Fig. 6
Fig. 6

Four types of Si-compatible waveguides: (a) buried-oxide, SOI structure of two crystalline Si regions separated by a layer of Si dioxide; (b) glass layer deposited on oxidized Si; (c) ARROW; (d) Si–germanium alloy grown epitaxially onto the surface of bulk Si.

Tables (2)

Tables Icon

Table 1 Power Emitted by an Electric Dipole into the Guided Waves of Several Si Waveguidesa

Tables Icon

Table 2 Material Indicies of Refraction Used to Calculate γmax

Equations (35)

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

p¨i+bop˙i+ωo2pi=q2m(eˆi·ER),
ER=Eo exp(-iΩt),
p=po exp(-iΩt),
bbo=1+q2mωpobo Im(eˆi·ER),
ω2-ωo2=b24-bbo2-q2mpo Re(eˆi·ER).
bo=noq2ωo26πomc3 1γ,
Es=-μonoωo2Jo6πc=iμonoωo3po6πc.
bbo=1+γ Reeˆi·EREs=1+γ Re(Γ),
[xˆ·DRTE(x=xs)]VED=0,
[xˆ·DRTM(x=xs)]VED=3sEs2 0du u3(1-u2)1/2×[rp exp(iksxxs)],
[zˆ·DRTE(x=xs)]HED=3sEs4 0du u(1-u2)1/2 ×[rs exp(iksxxs)],
[zˆ·DRTM(x=xs)]HED=3sEs4 0du u(1-u2)1/2×(1-u2) [rp exp(iksxxs)],
ΓVED=32 Re0du u3(1-u2)1/2[rp exp(2iksxxs)],
ΓHED=34 Re0du u(1-u2)1/2[rs exp(2iksxxs)-(1-u2)rp exp(2iksxxs)].
bboVED=32 Re0du u3(1-u2)1/2×[1+rp exp(2iksxxs)],
bboHED=34 Re0du u(1-u2)1/2{(1-u2)×[1-rp exp(2iksxxs)]+[1+rs exp(2iksxxs)]}.
PPobbo Pu du.
S(u)VEDu3(1-u2)1/2[1+rp exp(2iksxxs)],
S(u)HEDu(1-u2)1/2{(1-u2)[1-rp exp(2iksxxs)]+[1+rs exp(2iksxxs)]},
γwg=nSiO2S(u)du0S(u)du.
××Ds(r)-ωoμoDs(r)=iωoμoJ(r),
J(r)=αˆδ(x-xs)δ(y-ys)δ(z-zs)Jo
=αˆδ(r-rs)Jo,
Ds(r)=iωoμoVG¯¯(r, r)·J(r)d3r,
××G¯¯(r, r)-ko2G¯¯(r, r)=I¯¯δ(r-r),
G¯(r, r)=I¯¯+1k2g(r, r),
g(r, r)=exp(ik|r-r|)4π|r-r|.
Ds(r)=iωoμoεJoI¯¯+1k2·αˆg(r, rs).
Bs(r)=μoJo×αˆg(r, rs).
g(r, r)=exp(ik|r-r|)4π|r-r|
=i4π 0dkρ kρkxJo(kρ|ρ-ρs|)×exp(ikx|x-xs|),
[Dx(r)]VED=-ωoμoJo4πk2 0dkρ kρ3kxJo(kρ|ρ-ρs|)×exp(ikx|x-xs|),
[Bx(r)]VED=0,
[Dx(r)]HED=sgn(x-xs) iJo4πω cos(ϕ)0dkρkρ2×J1(kρ|ρs-ρs|)exp(ikx|x-xs|),
[Bx(r)]HED=iμoJo4π sin(ϕ)0dkρ kρ2kxJ1(kρ|ρ-ρs|)×exp(ikx|x-xs|),

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