Abstract

We develop a theory for light scattering from a random array of nanoparticles spaced much less than an optical wavelength from an optical waveguide. We deal with the randomness in the particle positions by convolving the single-particle Green’s dyadic with a correlation function that describes the average properties of the particle distribution. This allows us to treat free-space and substrate-mediated particle–particle interactions. We show that coherent interactions between particles near a waveguide cause dramatic, qualitative changes to the particle susceptibilities. Hence, the scattering spectra show strong, surface-induced peaks that we associate with the onset of leaky guided waves of the layered substrate. Our predictions produce outstanding agreement with the scattering experiments of Stuart and Hall [Phys. Rev. Lett. 80, 5663 (1998)].

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  50. B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).
    [CrossRef]
  51. M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8, 581–583 (1983).
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    [CrossRef]

2001

2000

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

1999

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

1998

G. S. Agarwal and S. D. Gupta, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1998).
[CrossRef]

S. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[CrossRef]

H. R. Stuart and D. G. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5663 (1998).
[CrossRef]

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

1997

E. Fucile, P. Denti, F. Borghese, R. Saija, and O. I. Sindoni, “Optical properties of a sphere in the vicinity of a plane surface,” J. Opt. Soc. Am. A 14, 1505–1514 (1997).
[CrossRef]

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

S. Nie and S. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

E. Goldstein and P. Meystre, “Dipole–dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[CrossRef]

1996

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic band gap material,” Phys. Rev. Lett. 76, 2484–2487 (1996).
[CrossRef] [PubMed]

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

B. R. Johnson, “Calculation of light scattering from a spherical particle on a surface by the multipole expansion method,” J. Opt. Soc. Am. A 13, 326–337 (1996).
[CrossRef]

1995

F. R. Aussenegg, A. Leitner, and H. Gold, “Optical second-harmonic generation of metal-island films,” Appl. Phys. A 60, 97–101 (1995).
[CrossRef]

R. R. Singer, A. Leitner, and F. R. Aussenegg, “Structure analysis and models for optical constants of discontinuous metallic silver films,” J. Opt. Soc. Am. B 12, 220–228 (1995) and references therein.
[CrossRef]

T. Kobayashi, Q. Zheng, and T. Sekiguchi, “Resonant dipole–dipole interaction in a cavity,” Phys. Rev. A 52, 2835–2846 (1995).
[CrossRef] [PubMed]

1994

M. Tomita and K. Ohosumi, “Enhancement of molecular interactions in strongly scattering dielectric composite optical media,” Phys. Rev. B 50, 10369–10372 (1994).
[CrossRef]

1993

1992

R. M. Emmons, B. N. Kurdi, and D. G. Hall, “Buried-oxide silicon-on-insulator structures. I. Optical waveguide characteristics,” IEEE J. Quantum Electron. 28, 157–163 (1992).
[CrossRef]

B. R. Johnson, “Light scattering from a spherical particle on a conducting plane. I. Normal incidence,” J. Opt. Soc. Am. A 9, 1341–1351 (1992).
[CrossRef]

1991

G. Videen, “Light scattering from a sphere on or near a surface,” J. Opt. Soc. Am. A 8, 483–489 (1991).
[CrossRef]

J. Martorell and N. M. Lawandy, “Spontaneous emission in a disordered dielectric medium,” Phys. Rev. Lett. 66, 887–890 (1991).
[CrossRef] [PubMed]

1990

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[CrossRef] [PubMed]

1989

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

1988

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

1987

P. Bobbert and J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica A 147, 115–141 (1987).
[CrossRef]

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

T. Takemori, M. Inoue, and K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Ga, In, Zn, and, Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

1986

P. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

1985

G. S. Agarwal and S. D. Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B 32, 3607–3611 (1985).
[CrossRef]

M. Meier, A. Wokaun, and P. F. Liao, “Enhanced fields on rough surfaces: dipolar interactions among particles of size exceeding the Rayleigh limit,” J. Opt. Soc. Am. B 2, 931–949 (1985).
[CrossRef]

T. Yamaguchi, M. Sakai, and N. Saito, “Optical properties of well-defined granular metal systems,” Phys. Rev. B 32, 2126–2131 (1985).
[CrossRef]

1984

W. R. Holland and D. G. Hall, “Frequency shifts of an electric-dipole resonance near a conducting surface,” Phys. Rev. Lett. 52, 1041–1044 (1984).
[CrossRef]

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[CrossRef]

1983

D. Bedeaux and J. Vlieger, “A statistical theory for the dielectric properties of thin island films: application and comparison with experimental results,” Thin Solid Films 102, 265–281 (1983).
[CrossRef]

B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).
[CrossRef]

M. Meier and A. Wokaun, “Enhanced fields on large metal particles: dynamic depolarization,” Opt. Lett. 8, 581–583 (1983).
[CrossRef] [PubMed]

W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B 27, 7765–7768 (1983).
[CrossRef]

1980

J. Vlieger and D. Bedeaux, “A statistical theory for the dielectric properties of thin island films,” Thin Solid Films 69, 107–130 (1980).
[CrossRef]

1979

1978

1974

1972

T. Yamaguchi, S. Yoshida, and A. Kinbara, “Effect of the dipole interaction between island particles on the optical properties of an aggregated silver film,” Thin Solid Films 13, 261–264 (1972).
[CrossRef]

1969

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792–1796 (1969).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and S. D. Gupta, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1998).
[CrossRef]

G. S. Agarwal and S. D. Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B 32, 3607–3611 (1985).
[CrossRef]

Aussenegg, F. R.

F. R. Aussenegg, A. Leitner, and H. Gold, “Optical second-harmonic generation of metal-island films,” Appl. Phys. A 60, 97–101 (1995).
[CrossRef]

R. R. Singer, A. Leitner, and F. R. Aussenegg, “Structure analysis and models for optical constants of discontinuous metallic silver films,” J. Opt. Soc. Am. B 12, 220–228 (1995) and references therein.
[CrossRef]

A. Leitner, Z. Zhensheng, H. Brunner, F. R. Aussenegg, and A. Wokaun, “Optical properties of a metal island film close to a smooth metal surface,” Appl. Opt. 32, 102–110 (1993).
[CrossRef] [PubMed]

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

Bedeaux, D.

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

D. Bedeaux and J. Vlieger, “A statistical theory for the dielectric properties of thin island films: application and comparison with experimental results,” Thin Solid Films 102, 265–281 (1983).
[CrossRef]

J. Vlieger and D. Bedeaux, “A statistical theory for the dielectric properties of thin island films,” Thin Solid Films 69, 107–130 (1980).
[CrossRef]

Bickel, W.

Bingler, H. G.

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

Bloemer, M. J.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

Bobbert, P.

P. Bobbert and J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica A 147, 115–141 (1987).
[CrossRef]

P. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Bobbert, P. A.

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

Borghese, F.

Brunner, H.

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

A. Leitner, Z. Zhensheng, H. Brunner, F. R. Aussenegg, and A. Wokaun, “Optical properties of a metal island film close to a smooth metal surface,” Appl. Opt. 32, 102–110 (1993).
[CrossRef] [PubMed]

Dasari, R.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Denti, P.

Deussen, M.

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Doicu, A.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Eagen, C. F.

Emmons, R. M.

R. M. Emmons, B. N. Kurdi, and D. G. Hall, “Buried-oxide silicon-on-insulator structures. I. Optical waveguide characteristics,” IEEE J. Quantum Electron. 28, 157–163 (1992).
[CrossRef]

Emory, S.

S. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[CrossRef]

S. Nie and S. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

Feld, M.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Ferrell, T. L.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

Fucile, E.

Genack, A. Z.

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

Göbel, E. O.

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Gold, H.

F. R. Aussenegg, A. Leitner, and H. Gold, “Optical second-harmonic generation of metal-island films,” Appl. Phys. A 60, 97–101 (1995).
[CrossRef]

Goldstein, E.

E. Goldstein and P. Meystre, “Dipole–dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[CrossRef]

Goudonnet, J. P.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

Gupta, S. D.

G. S. Agarwal and S. D. Gupta, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1998).
[CrossRef]

G. S. Agarwal and S. D. Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B 32, 3607–3611 (1985).
[CrossRef]

Guss, W.

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Hall, D. G.

B. J. Soller, H. R. Stuart, and D. G. Hall, “Energy transfer at optical frequencies to silicon-on-insulator structures,” Opt. Lett. 26, 1421–1423 (2001).
[CrossRef]

H. R. Stuart and D. G. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5663 (1998).
[CrossRef]

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

R. M. Emmons, B. N. Kurdi, and D. G. Hall, “Buried-oxide silicon-on-insulator structures. I. Optical waveguide characteristics,” IEEE J. Quantum Electron. 28, 157–163 (1992).
[CrossRef]

W. R. Holland and D. G. Hall, “Frequency shifts of an electric-dipole resonance near a conducting surface,” Phys. Rev. Lett. 52, 1041–1044 (1984).
[CrossRef]

W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B 27, 7765–7768 (1983).
[CrossRef]

Haskins, W. E.

S. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[CrossRef]

Holland, W. R.

W. R. Holland and D. G. Hall, “Frequency shifts of an electric-dipole resonance near a conducting surface,” Phys. Rev. Lett. 52, 1041–1044 (1984).
[CrossRef]

W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B 27, 7765–7768 (1983).
[CrossRef]

Hopmeier, M.

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Iafelice, V.

Inoue, M.

T. Takemori, M. Inoue, and K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Itzkan, I.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

James, D. R.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

John, S.

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic band gap material,” Phys. Rev. Lett. 76, 2484–2487 (1996).
[CrossRef] [PubMed]

Johnson, B. R.

Jupille, J.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

Kinbara, A.

T. Yamaguchi, S. Yoshuda, and A. Kinbara, “Effect of retarded dipole–dipole interactions between island particles on the optical plasma-resonance absorption of a silver-island film,” J. Opt. Soc. Am. 64, 1563–1568 (1974).
[CrossRef]

T. Yamaguchi, S. Yoshida, and A. Kinbara, “Effect of the dipole interaction between island particles on the optical properties of an aggregated silver film,” Thin Solid Films 13, 261–264 (1972).
[CrossRef]

Klenke, M.

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

Kneipp, H.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Kneipp, K.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Kobayashi, T.

T. Kobayashi, Q. Zheng, and T. Sekiguchi, “Resonant dipole–dipole interaction in a cavity,” Phys. Rev. A 52, 2835–2846 (1995).
[CrossRef] [PubMed]

Ku, J. C.

Kurdi, B. N.

R. M. Emmons, B. N. Kurdi, and D. G. Hall, “Buried-oxide silicon-on-insulator structures. I. Optical waveguide characteristics,” IEEE J. Quantum Electron. 28, 157–163 (1992).
[CrossRef]

Kurizki, G.

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[CrossRef] [PubMed]

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

Lawandy, N. M.

J. Martorell and N. M. Lawandy, “Spontaneous emission in a disordered dielectric medium,” Phys. Rev. Lett. 66, 887–890 (1991).
[CrossRef] [PubMed]

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Lazzari, R.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

Leitner, A.

R. R. Singer, A. Leitner, and F. R. Aussenegg, “Structure analysis and models for optical constants of discontinuous metallic silver films,” J. Opt. Soc. Am. B 12, 220–228 (1995) and references therein.
[CrossRef]

F. R. Aussenegg, A. Leitner, and H. Gold, “Optical second-harmonic generation of metal-island films,” Appl. Phys. A 60, 97–101 (1995).
[CrossRef]

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

A. Leitner, Z. Zhensheng, H. Brunner, F. R. Aussenegg, and A. Wokaun, “Optical properties of a metal island film close to a smooth metal surface,” Appl. Opt. 32, 102–110 (1993).
[CrossRef] [PubMed]

Liao, P. F.

Liebsch, A.

B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).
[CrossRef]

Mahrt, R. F.

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Mantovani, J. G.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

Martorell, J.

J. Martorell and N. M. Lawandy, “Spontaneous emission in a disordered dielectric medium,” Phys. Rev. Lett. 66, 887–890 (1991).
[CrossRef] [PubMed]

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

Meier, M.

Meystre, P.

E. Goldstein and P. Meystre, “Dipole–dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[CrossRef]

Morawitz, H.

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792–1796 (1969).
[CrossRef]

Nie, S.

S. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[CrossRef]

S. Nie and S. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

Ohosumi, K.

M. Tomita and K. Ohosumi, “Enhancement of molecular interactions in strongly scattering dielectric composite optical media,” Phys. Rev. B 50, 10369–10372 (1994).
[CrossRef]

Ohtaka, K.

T. Takemori, M. Inoue, and K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Perelman, L.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Persson, B. N. J.

B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).
[CrossRef]

Quang, T.

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic band gap material,” Phys. Rev. Lett. 76, 2484–2487 (1996).
[CrossRef] [PubMed]

Roux, S.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

Saija, R.

Saito, N.

T. Yamaguchi, M. Sakai, and N. Saito, “Optical properties of well-defined granular metal systems,” Phys. Rev. B 32, 2126–2131 (1985).
[CrossRef]

Sakai, M.

T. Yamaguchi, M. Sakai, and N. Saito, “Optical properties of well-defined granular metal systems,” Phys. Rev. B 32, 2126–2131 (1985).
[CrossRef]

Schatz, G. C.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Ga, In, Zn, and, Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Scott, G. D.

Sekiguchi, T.

T. Kobayashi, Q. Zheng, and T. Sekiguchi, “Resonant dipole–dipole interaction in a cavity,” Phys. Rev. A 52, 2835–2846 (1995).
[CrossRef] [PubMed]

Simonsen, I.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

Sindoni, O. I.

Singer, R. R.

Sipe, J. E.

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[CrossRef]

Soller, B. J.

Stuart, H. R.

B. J. Soller, H. R. Stuart, and D. G. Hall, “Energy transfer at optical frequencies to silicon-on-insulator structures,” Opt. Lett. 26, 1421–1423 (2001).
[CrossRef]

H. R. Stuart and D. G. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5663 (1998).
[CrossRef]

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

Sudoh, A.

Takahashi, H.

Takemori, T.

T. Takemori, M. Inoue, and K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Taubenblatt, M.

Tomita, M.

M. Tomita and K. Ohosumi, “Enhancement of molecular interactions in strongly scattering dielectric composite optical media,” Phys. Rev. B 50, 10369–10372 (1994).
[CrossRef]

Tran, T.

Truong, V. V.

Turner, M.

Videen, G.

Vlieger, J.

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

P. Bobbert and J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica A 147, 115–141 (1987).
[CrossRef]

P. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

D. Bedeaux and J. Vlieger, “A statistical theory for the dielectric properties of thin island films: application and comparison with experimental results,” Thin Solid Films 102, 265–281 (1983).
[CrossRef]

J. Vlieger and D. Bedeaux, “A statistical theory for the dielectric properties of thin island films,” Thin Solid Films 69, 107–130 (1980).
[CrossRef]

Wang, Y.

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

Warmack, R. J.

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

Weber, W. H.

Wind, M. M.

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

Wokaun, A.

Wolfe, W.

Wriedt, T.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Wylie, J. M.

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[CrossRef]

Yamaguchi, T.

T. Yamaguchi, M. Sakai, and N. Saito, “Optical properties of well-defined granular metal systems,” Phys. Rev. B 32, 2126–2131 (1985).
[CrossRef]

T. Yamaguchi, H. Takahashi, and A. Sudoh, “Optical behavior of a metal island film,” J. Opt. Soc. Am. 68, 1039–1044 (1978).
[CrossRef]

T. Yamaguchi, S. Yoshuda, and A. Kinbara, “Effect of retarded dipole–dipole interactions between island particles on the optical plasma-resonance absorption of a silver-island film,” J. Opt. Soc. Am. 64, 1563–1568 (1974).
[CrossRef]

T. Yamaguchi, S. Yoshida, and A. Kinbara, “Effect of the dipole interaction between island particles on the optical properties of an aggregated silver film,” Thin Solid Films 13, 261–264 (1972).
[CrossRef]

Yoshida, S.

T. Yamaguchi, S. Yoshida, and A. Kinbara, “Effect of the dipole interaction between island particles on the optical properties of an aggregated silver film,” Thin Solid Films 13, 261–264 (1972).
[CrossRef]

Yoshuda, S.

Zeman, E. J.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Ga, In, Zn, and, Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

Zheng, Q.

T. Kobayashi, Q. Zheng, and T. Sekiguchi, “Resonant dipole–dipole interaction in a cavity,” Phys. Rev. A 52, 2835–2846 (1995).
[CrossRef] [PubMed]

Zhensheng, Z.

Appl. Opt.

Appl. Phys. A

F. R. Aussenegg, A. Leitner, and H. Gold, “Optical second-harmonic generation of metal-island films,” Appl. Phys. A 60, 97–101 (1995).
[CrossRef]

Appl. Phys. Lett.

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

IEEE J. Quantum Electron.

R. M. Emmons, B. N. Kurdi, and D. G. Hall, “Buried-oxide silicon-on-insulator structures. I. Optical waveguide characteristics,” IEEE J. Quantum Electron. 28, 157–163 (1992).
[CrossRef]

J. Am. Chem. Soc.

S. Emory, W. E. Haskins, and S. Nie, “Direct observation of size-dependent optical enhancement in single metal nanoparticles,” J. Am. Chem. Soc. 120, 8009–8010 (1998).
[CrossRef]

J. Chem. Phys.

H. G. Bingler, H. Brunner, M. Klenke, A. Leitner, F. R. Aussenegg, and A. Wokaun, “Enhanced second harmonic generation in a silver-spacer-islands multilayer system,” J. Chem. Phys. 99, 7499–7505 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for Ag, Au, Cu, Li, Na, Ga, In, Zn, and, Cd,” J. Phys. Chem. 91, 634–643 (1987).
[CrossRef]

J. Phys. Soc. Jpn.

T. Takemori, M. Inoue, and K. Ohtaka, “Optical response of a sphere coupled to a metal substrate,” J. Phys. Soc. Jpn. 56, 1587–1602 (1987).
[CrossRef]

Opt. Commun.

T. Wriedt and A. Doicu, “Light scattering from a particle on or near a surface,” Opt. Commun. 152, 376–384 (1998).
[CrossRef]

Opt. Lett.

Phys. Rev.

H. Morawitz, “Self-coupling of a two-level system by a mirror,” Phys. Rev. 187, 1792–1796 (1969).
[CrossRef]

Phys. Rev. A

J. M. Wylie and J. E. Sipe, “Quantum electrodynamics near an interface,” Phys. Rev. A 30, 1185–1193 (1984).
[CrossRef]

G. Kurizki, “Two-atom resonant radiative coupling in photonic band structures,” Phys. Rev. A 42, 2915–2924 (1990).
[CrossRef] [PubMed]

E. Goldstein and P. Meystre, “Dipole–dipole interaction in optical cavities,” Phys. Rev. A 56, 5135–5146 (1997).
[CrossRef]

G. S. Agarwal and S. D. Gupta, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1998).
[CrossRef]

T. Kobayashi, Q. Zheng, and T. Sekiguchi, “Resonant dipole–dipole interaction in a cavity,” Phys. Rev. A 52, 2835–2846 (1995).
[CrossRef] [PubMed]

M. Hopmeier, W. Guss, M. Deussen, E. O. Göbel, and R. F. Mahrt, “Microcavity enhanced modification of the dipole–dipole interaction,” Phys. Rev. A 57, 667–670 (1999).

Phys. Rev. B

W. R. Holland and D. G. Hall, “Surface-plasmon dispersion relation: shifts induced by the interaction with localized plasma resonances,” Phys. Rev. B 27, 7765–7768 (1983).
[CrossRef]

M. J. Bloemer, J. G. Mantovani, J. P. Goudonnet, D. R. James, R. J. Warmack, and T. L. Ferrell, “Observation of driven surface-plasmon modes in metal particulates above tunnel junctions,” Phys. Rev. B 35, 5947–5954 (1987).
[CrossRef]

G. S. Agarwal and S. D. Gupta, “Interaction between surface plasmons and localized plasmons,” Phys. Rev. B 32, 3607–3611 (1985).
[CrossRef]

M. Tomita and K. Ohosumi, “Enhancement of molecular interactions in strongly scattering dielectric composite optical media,” Phys. Rev. B 50, 10369–10372 (1994).
[CrossRef]

T. Yamaguchi, M. Sakai, and N. Saito, “Optical properties of well-defined granular metal systems,” Phys. Rev. B 32, 2126–2131 (1985).
[CrossRef]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling of the optical response of supported metallic particles,” Phys. Rev. B 61, 7722–7733 (2000).
[CrossRef]

B. N. J. Persson and A. Liebsch, “Optical properties of two-dimensional systems of randomly distributed particles,” Phys. Rev. B 28, 4247–4254 (1983).
[CrossRef]

Phys. Rev. Lett.

W. R. Holland and D. G. Hall, “Frequency shifts of an electric-dipole resonance near a conducting surface,” Phys. Rev. Lett. 52, 1041–1044 (1984).
[CrossRef]

J. Martorell and N. M. Lawandy, “Spontaneous emission in a disordered dielectric medium,” Phys. Rev. Lett. 66, 887–890 (1991).
[CrossRef] [PubMed]

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic band gap material,” Phys. Rev. Lett. 76, 2484–2487 (1996).
[CrossRef] [PubMed]

J. Martorell and N. M. Lawandy, “Observation of inhibited spontaneous emission in a periodic dielectric structure,” Phys. Rev. Lett. 65, 1877–1880 (1990).
[CrossRef] [PubMed]

K. Kneipp, Y. Wang, H. Kneipp, L. Perelman, I. Itzkan, R. Dasari, and M. Feld, “Single molecule detection using surface-enhanced Raman scattering (SERS),” Phys. Rev. Lett. 78, 1667–1670 (1997).
[CrossRef]

G. Kurizki and A. Z. Genack, “Suppression of molecular interactions in periodic dielectric structures,” Phys. Rev. Lett. 61, 2269–2271 (1988).
[CrossRef] [PubMed]

H. R. Stuart and D. G. Hall, “Enhanced dipole–dipole interaction between elementary radiators near a surface,” Phys. Rev. Lett. 80, 5663–5663 (1998).
[CrossRef]

Physica A

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “Optical properties of 2D-systems of small particles on a substrate,” Physica A 157, 269–278 (1989).
[CrossRef]

P. Bobbert and J. Vlieger, “The polarizability of a spheroidal particle on a substrate,” Physica A 147, 115–141 (1987).
[CrossRef]

P. Bobbert and J. Vlieger, “Light scattering by a sphere on a substrate,” Physica A 137, 209–242 (1986).
[CrossRef]

Science

S. Nie and S. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275, 1102–1106 (1997).
[CrossRef] [PubMed]

Thin Solid Films

T. Yamaguchi, S. Yoshida, and A. Kinbara, “Effect of the dipole interaction between island particles on the optical properties of an aggregated silver film,” Thin Solid Films 13, 261–264 (1972).
[CrossRef]

M. M. Wind, P. A. Bobbert, J. Vlieger, and D. Bedeaux, “The polarizability of truncated spheres and oblate spheroids on a substrate: comparison with experimental results,” Thin Solid Films 164, 57–62 (1988).
[CrossRef]

D. Bedeaux and J. Vlieger, “A statistical theory for the dielectric properties of thin island films: application and comparison with experimental results,” Thin Solid Films 102, 265–281 (1983).
[CrossRef]

J. Vlieger and D. Bedeaux, “A statistical theory for the dielectric properties of thin island films,” Thin Solid Films 69, 107–130 (1980).
[CrossRef]

Other

M. Born and E. Wolf, Principles of Optics, 6th ed. (Cambridge University, New York, 1980).

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

E. Palik, ed., Handbook of Optical Constants of Solids (Academic, New York, 1985).

Due to the factor of V in the polarizability of a small particle, the scattering cross section is proportional to V2, whereas the absorption cross section is proportional to V.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chaps. 5 and 12.

K. 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 (9)

Fig. 1
Fig. 1

One realization of the distribution function W(ρ; x) obtained from a SEM image of a layer of Ag nanoparticles thermally deposited on a smooth LiF substrate. The surface height was normalized to unity for the sake of our scattering model. The actual particles have an average height of approximately 30 nm, depending on the deposition parameters.

Fig. 2
Fig. 2

Geometry and coordinate system referred to throughout this paper. The particle is shown situated above a two-layer substrate for simplicity.

Fig. 3
Fig. 3

Two-dimensional autocorrelation function A(r; x) of the particle distribution function W(ρ; x). In the absence of particle–particle correlations and in the limit of point scatterers, we would have A(r; x)=δ(r).

Fig. 4
Fig. 4

Schematic of the experimental setup used by Stuart and Hall,13 to measure the diffusely scattered light from a collection of interacting noble-metal nanoparticles on the surface of a SOI optical waveguide. Optical frequency guided waves are supported by the thin layer of Si. It is these guided waves that mediate particle–particle interaction in the lateral direction.

Fig. 5
Fig. 5

Spectral density representation of the power dissipated by a Ag nanoparticle in a random, two-dimensional array spaced 30 nm from a waveguide substrate. The region kρ/k(0, 1) corresponds to power loss to the radiation field, whereas kρ/k(1, ) corresponds to power dissipation into waves that are evanescent in the region occupied by the scatterers. The TE and TM designations correspond to the direct excitation of TE and TM guided surface waves in the substrate by the overlayer of scatterers. The waveguide layers from top to bottom are 165-nm Si (n500=4.2992+0.0504i, n800=3.6925+0.0079i), 205-nm SiO2 (n=1.45) on a semi-infinite Si substrate.

Fig. 6
Fig. 6

Measured versus calculated scattering spectra for a collection of Ag nanoparticles in the cover region of an optical waveguide (solid curves) and on a smooth glass substrate (dashed curves). The waveguide layers from top to bottom are 165-nm Si, 205-nm SiO2 (n=1.45) on a semi-infinite Si substrate. The particles are of 33-nm average radius and 50-nm average height and similar average interparticle spacing. The dashed curve in the top (calculated) graph is shown multiplied by a factor of 2 for the purpose of comparison with the measured curves.

Fig. 7
Fig. 7

Measured versus calculated scattering spectra for a collection of Ag nanoparticles in the cover region of an optical waveguide for three different average particle radii. (In the model, the average interparticle spacing was adjusted to match the average particle radius.) The waveguide layers from top to bottom are 165-nm Si, 205-nm SiO2 (n=1.45) on a semi-infinite Si substrate. The overall strength of the calculated curves was adjusted for the purpose of comparison.

Fig. 8
Fig. 8

Calculated scattering spectra per unit volume for a collection of Ag nanoparticles in the cover region of an optical waveguide for three different average particle radii. These curves were generated by use of the effective polarizability per unit volume in Eq. (41). Note the larger field enhancement for smaller particle size, especially at wavelengths near the plasma resonance. The TE and TM designations refer to TE and TM surface-wave coupling in the substrate, causing the scattering spectra to peak.

Fig. 9
Fig. 9

Calculated and measured scattering spectra per unit volume for a collection of Ag nanoparticles separated from a smooth Ag surface by a thin layer of LiF for four different LiF thicknesses. The average particle radius is 50 nm.

Equations (41)

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pn=αn·Eill=αn·(Eo+Edep+Escatt),
pn=αn(Eo+Edep);αn=Vn[(ω)-1],
Edep=1[k2ΠV+(·ΠV)],
ΠV(rn)=1VSSΠpt(rn-rm)d3rm,
Πpt(rn-rm)=pm4πoeik|rn-rm||rn-rm|
Edep(rn)ξdep(rn, rm)·p(rm),
pn=Vn[(ω)-1](Eo+ξ·pn).
pS=npn=VS[(ω)-1](Einc+ξ·pS).
pi=VS[(ω)-1]1-VS[(ω)-1]ξii(0)Ei,
pi=VS[(ω)-1]1+[(ω)-1]LeffEi,
pi=αeff,i(ω)Ei,
Leff=-VSξii(0)
αeff=4πa3(-1)3+(-1)[1-(ka)2-23i(ka)3].
pn(r)=(αn)eff·(Eo+Eself), =(αn)eff·Eo+ω2μom=1NG(r, rm)·pm,
G(r, rm)=I+1ko2G(r, rm),
iGixiˆVED=14π1+1ko2·xˆ0dkρikρkx×{exp(ikx|x-xm|)+rp exp[ikx(x+xm)]}×J0[kρ(ρ-ρm)],
iGiziˆHED=14π1+1ko2·zˆ0dkρikρkx×{exp(ikx|x-xm|)+rs×exp[ikx(x+xm)]}J0[kρ(ρ-ρm)]+xˆ cos ϕ0dkρ(rs+rp)×exp[ikx(x+xm)]J1[kρ(ρ-ρm)],
pn=(αn)eff·Eo+ω2μom=1NpmxiGix(r, rn)iˆ+pmziGiz(r, rn)iˆ.
pn=(αn)eff·Eo+ω2μopmxiGix(r, rn)iˆ+pmziGiz(r, rn)iˆS(r),
G(r, rn)S(r)=1VSallspaceG(rn-rm)S(rm)ΔVm,
pn=(αn)eff·Eo+ω2μopmziGiz(r, rn)iˆ+pmziGiz(r, rn)iˆW(r),
Eint=1Nlimd 14d2-ddd2rEint(r, x)W(r; x),
Eint(r, x)=1Nlimd14d2-dd d2rEint(r+r, x)W(r; x).
Eint(r, x)=1Nlimd14d2×-dd d2bEint(b, x)W(b-r; x).
Eint(b, x)=1Vlimd-dx×-dd d2bEpt(b, x-x)W(b-b; x),
Eint(r, x)=1NVlimd14d2-dx-dd d2b-dd d2bEpt(b; x, x)W(b-b; x)W(b-r; x)=1NVlimd-dx-dd d2bEpt(b; x, x)14d2 -dd d2bW(b-b; x)W(b-r; x).
A(r-b; x)limd14d2 -dd d2cW(c; x)W[c-(r-b); x],
Eint(r, x)=1NV limd-dx×-dd d2bEpt(b; x, x)A(r-b; x).
Eint(r, x)=1NV-dxEpt(r; x, x)A(r; x),
·(E×H)+E·tD+H·tB+J·E=0.
P=-12V Re{J*·E}dV=ω2VV Im{p*·E}dV,
J=tP=-iωpV.
pnz=(αn eff)zz[Eoz+ω2μopmGzz(r, rn)W(r)].
pave=(αave)eff[Eoz+ω2μopaveGzz(r, rn)W(r)].
pave=(αave)eff1-ω2μopaveGzz(r, rn)W(r)(αave)effEoz.
Pave=ω2pave* Im(Eintz),
Eintz=1NV-dxEpt(r; x, x)A(r; x),=paveNV-dx0dkρf(r; x, x; kρ)A(r; x)=paveNV0dkρ-dxf(r; x, x; kρ)A(r; x),
f(r; x, x; kρ)=ko2{exp[±ikx(x-x)]+rs exp[ikx(x+x)]}×ikρkxJ0[kρ(ρ-ρ)]+{exp[±ikx(x-x)]+rs exp[ikx(x+x)]}ikρkx+ikx(rs+rp)exp[ikx(x+x)]C[kρ(ρ-ρ)],
C(kρρ)=kρ22cos2ϕ[J0(kρρ)-J2(kρρ)]-sin2ϕJ1(kρρ)ρ,
Pave=0dPdkρdkρ=0S(kρ)dkρ,
Pff=ω2NV|pave|2×Im0kodkρ-dxf(r; x, x;kρ)A(r; x).

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