Abstract

We present a theoretical model for describing light scattering from randomly distributed Au nanoparticles on a substrate, including the clustering effect. By using the finite-element Green’s function method and spherical harmonic basis functions, we are able to calculate the polarization-dependent reflectivity spectra of the system (modeled by randomly distributed nanoparticles coupled with clusters) efficiently and accurately. The calculated ellipsometric spectra of the system with clusters can adequately describe the experimental data for the whole frequency range. We find that the clustering effect leads to some prominent features in the low frequency range of the ellipsometric spectra, which are attributed to plasmonic resonances associated with the coupling of Au nanoparticles and clusters.

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    [CrossRef]
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2011

2010

2009

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

2008

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C5(5), 1194–1197 (2008).
[CrossRef]

2007

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett.90(18), 181923 (2007).
[CrossRef]

2006

2004

B. Kaplan, T. Novikova, A. De Martino, and B. Drévillon, “Characterization of bidimensional gratings by spectroscopic ellipsometry and angle-resolved Mueller polarimetry,” Appl. Opt.43(6), 1233–1240 (2004).
[CrossRef] [PubMed]

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films455–456, 323–334 (2004).
[CrossRef]

2002

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films419(1-2), 124–136 (2002).
[CrossRef]

2001

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B24(2), 267–284 (2001).
[CrossRef]

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

2000

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(33 Pt B), 4318–4324 (2000).
[CrossRef] [PubMed]

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B104(50), 11965–11971 (2000).
[CrossRef]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000).
[CrossRef]

1999

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc.121(39), 9208–9214 (1999).
[CrossRef]

1997

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

1995

1991

1982

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

1975

1908

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.330(3), 377–445 (1908).
[CrossRef]

Aizpurua, J.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(33 Pt B), 4318–4324 (2000).
[CrossRef] [PubMed]

Apell, P.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(33 Pt B), 4318–4324 (2000).
[CrossRef] [PubMed]

Asano, S.

Bedeaux, D.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B24(2), 267–284 (2001).
[CrossRef]

Booso, B.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Brus, L.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B104(50), 11965–11971 (2000).
[CrossRef]

Chang, Y. C.

Chen, Y. C.

Chu, H.

De Martino, A.

Drévillon, B.

Emory, S. R.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc.121(39), 9208–9214 (1999).
[CrossRef]

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

Gaiduk, A. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Gaponenko, S. V.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Gaylord, T. K.

Gordon, J. P.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

Grann, E. B.

Hilfiker, J. N.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Hofmann, T.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Hsu, S. H.

S. H. Hsu, Y. C. Chang, Y. C. Chen, P. K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express18(2), 1310–1315 (2010).
[CrossRef] [PubMed]

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C5(5), 1194–1197 (2008).
[CrossRef]

Hsu, S.-H.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Jiang, J.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B104(50), 11965–11971 (2000).
[CrossRef]

Jupille, J.

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B24(2), 267–284 (2001).
[CrossRef]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000).
[CrossRef]

Käll, M.

H. Xu, J. Aizpurua, M. Käll, and P. Apell, “Electromagnetic contributions to single-molecule sensitivity in surface-enhanced Raman scattering,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics62(33 Pt B), 4318–4324 (2000).
[CrossRef] [PubMed]

Kaplan, B.

Kim, T. J.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Kim, Y. D.

S. H. Hsu, Y. C. Chang, Y. C. Chen, P. K. Wei, and Y. D. Kim, “Optical metrology of randomly-distributed Au colloids on a multilayer film,” Opt. Express18(2), 1310–1315 (2010).
[CrossRef] [PubMed]

Y. C. Chang, S. H. Hsu, P. K. Wei, and Y. D. Kim, “Optical nanometrology of Au nanoparticles on a multilayer film,” Phys. Status Solidi C5(5), 1194–1197 (2008).
[CrossRef]

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Krug, J. T.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc.121(39), 9208–9214 (1999).
[CrossRef]

Kulakovich, O. S.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Kuo, H. C.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett.90(18), 181923 (2007).
[CrossRef]

Lazzari, R.

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films419(1-2), 124–136 (2002).
[CrossRef]

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B24(2), 267–284 (2001).
[CrossRef]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000).
[CrossRef]

Li, G.

Liao, P. F.

A. Wokaun, J. P. Gordon, and P. F. Liao, “Radiation Damping in Surface-Enhanced Raman Scattering,” Phys. Rev. Lett.48(14), 957–960 (1982).
[CrossRef]

Lin, C. J.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Lin, G. R.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett.90(18), 181923 (2007).
[CrossRef]

Lin, H. S.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett.90(18), 181923 (2007).
[CrossRef]

Liu, E. S.

G. R. Lin, Y. C. Chang, E. S. Liu, H. C. Kuo, and H. S. Lin, “Low refractive index Si nanopillars on Si substrate,” Appl. Phys. Lett.90(18), 181923 (2007).
[CrossRef]

Liu, E.-S.

S.-H. Hsu, E.-S. Liu, Y. C. Chang, J. N. Hilfiker, Y. D. Kim, T. J. Kim, C. J. Lin, and G. R. Lin, “Characterization of Si nanorods by spectroscopic ellipsometry with efficient theoretical modeling,” Phys. Status Solidi A205(4), 876–879 (2008).
[CrossRef]

Maskevich, S. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Mewe, A.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films455–456, 323–334 (2004).
[CrossRef]

Michaels, A. M.

A. M. Michaels, J. Jiang, and L. Brus, “Ag nanocrystal junctions as the site for surface-enhanced Raman scattering of single rhodamine 6G molecules,” J. Phys. Chem. B104(50), 11965–11971 (2000).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur optik trüber medien, speziell kolloidaler metallösungen,” Ann. Phys.330(3), 377–445 (1908).
[CrossRef]

Moharam, M. G.

Moirangthem, R. S.

Nie, S.

J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc.121(39), 9208–9214 (1999).
[CrossRef]

Nie, S. M.

S. M. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science275(5303), 1102–1106 (1997).
[CrossRef] [PubMed]

Novikova, T.

Opsal, J.

Poelsema, B.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films455–456, 323–334 (2004).
[CrossRef]

Pommet, D. A.

Prokhorov, O. A.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Rekveld, S.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films455–456, 323–334 (2004).
[CrossRef]

Roux, S.

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000).
[CrossRef]

Sarangan, A.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Schmidt, D.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Schubert, E.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Schubert, M.

D. Schmidt, B. Booso, T. Hofmann, E. Schubert, A. Sarangan, and M. Schubert, “Monoclinic optical constants, birefringence, and dichroism of slanted titanium nanocolumns determined by generalized ellipsometry,” Appl. Phys. Lett.94(1), 011914 (2009).
[CrossRef]

Shelekhina, V. M.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Simonsen, I.

R. Lazzari and I. Simonsen, “GRANFILM: a software for calculating thin-layer dielectric properties and Fresnel coefficients,” Thin Solid Films419(1-2), 124–136 (2002).
[CrossRef]

R. Lazzari, I. Simonsen, D. Bedeaux, J. Vlieger, and J. Jupille, “Polarizability of truncated spheroidal particles supported by a substrate: model and applications,” Eur. Phys. J. B24(2), 267–284 (2001).
[CrossRef]

I. Simonsen, R. Lazzari, J. Jupille, and S. Roux, “Numerical modeling ot the optical response of supported metallic particles,” Phys. Rev. B61(11), 7722–7733 (2000).
[CrossRef]

Stefan Kooij, E.

H. Wormeester, E. Stefan Kooij, A. Mewe, S. Rekveld, and B. Poelsema, “Ellipsometric characterisation of heterogeneous 2D layers,” Thin Solid Films455–456, 323–334 (2004).
[CrossRef]

Strekal, N. D.

S. V. Gaponenko, A. A. Gaiduk, O. S. Kulakovich, S. A. Maskevich, N. D. Strekal, O. A. Prokhorov, and V. M. Shelekhina, “Raman scattering enhancement using crystallographic surface of a colloidal crystal,” JETP Lett.74(6), 309–311 (2001).
[CrossRef]

Videen, G.

Vlieger, J.

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[CrossRef]

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[CrossRef]

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J. T. Krug, G. D. Wang, S. R. Emory, and S. Nie, “Efficient Raman enhancement and intermittent light emission observed in single gold nanocrystals,” J. Am. Chem. Soc.121(39), 9208–9214 (1999).
[CrossRef]

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[CrossRef]

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[CrossRef] [PubMed]

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

SEM pictures of samples with random distribution of Au nanoparticles with diameters: (a) 20nm, (b) 40nm, (c) 60nm, and (d) 80nm.

Fig. 2
Fig. 2

(a) a picture which describes randomly distributed identical nanoparticles with variable distances between the centers of particles R i and the origin O (top view); (b) equivalent spheroid model to describe clusters with three different diameters (top view).

Fig. 3
Fig. 3

The calculate field strength, | E | at the top of the sphere as a function of photon energy for light scattering from an isolated Au sphere obtained by the present Green’s function method with different cutoffs: ( max ,Nk) = (2,101) (dash-dotted), (4,51) (dotted), and (4,101) (dashed) and Mie scattering theory (solid). (a) d = 20 nm, (b) d = 40 nm, (c) d = 60 nm, and (d) d = 80 nm.

Fig. 4
Fig. 4

SE measurements (solid curves) and model calculations (dash-dotted curves) without clusters for random distribution of nanoparticles with nominal sizes of (a) 20 nm, (b) 40 nm, (c) 60 nm, and (d) 80 nm for incident angles of 55°, 60°, and 65°.

Fig. 5
Fig. 5

The ellipsometric parameters, Ψ and Δ as functions of photon energy obtained by the SHGF method for four different isolated pancakes of diameters (a) 2d, (b) 2.5d, (c) 3d and (d) 3.5d with d = 60nm at three different angles of incidence: 55° (solid line), 60 o (dashed line) and 65 o (dash-dotted line) on the substrate.

Fig. 6
Fig. 6

SE measurements (solid curves) and model calculations (dashed dot curves) including clusters for random distribution of nanoparticles with nominal sizes of (a) 40, (b) 60, and (c) 80 nm for incident angles of 55°, 60°, and 65°.

Tables (1)

Tables Icon

Table 1 Best-fit parameters used in the theoretical modeling for Au nanoparticles without and with clusters.

Equations (15)

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E( r )= 1 N i e i k 0 R i u i ( r i ) ,
e i k 0 R i u i ( r i )= N E 0 ( r )+ j=1 N d r G( r, r ) V j ( r ) e i k 0 R j u j ( r j ) ,
G( r, r )= 1 ( 2π ) 2 d k n e i k n ( ρ ρ ) g n ( z, z ),
u i ( r i )= N e i k 0 R i E 0 ( r )+ j=1 N d ϕ n ( 2π ) 2 k n d k n d r j e i k n ( ρ i ρ j ) g n ( z, z ) e i K n ( R j R i ) V j ( r j ) u j ( r j )
u i ( r i )= N e i k 0 R i E 0 ( r )+ d ϕ n ( 2π ) 2 k n d k n S( K n ) d r i e i k n ( ρ i ρ i ) g n ( z, z ) V 1 ( r i ) u i ( r i ) ,
S( K n )= 1 N i j e i K n ( R j R i )
S( K n )=N G δ K n ,G ,
S( K n )=1+ f S 1 ( K n ),
S 1 ( K n )= 1 N i ji e i K n ( R j R i ) .
S 1 ( K n )= 1 N i ji e i K n ( R j R i ) j1 e i K n ( R j R 1 ) 1 A cell dϕ R u e i K n R RdR = δ k n , k 0 2 J 1 ( K n R u ) K n R u ,
S 1 ( K n )= j1 e i K n ( R j R 1 ) e ( R j R 1 ) 2 / λ c 2 π λ c 2 A cell e K n 2 λ c 2 /2 2π A cell 0 R u J 0 ( K n R u ) e R 2 / λ c 2 RdR .
u 1 ( r )= N E 0 ( r )+ d ϕ n ( 2π ) 2 k n d k n d r e i k n ( ρ ρ ) g n ( z, z ){ [ 1+ S 1 ( K n ) f u f ] V 1 ( r ) u 1 ( r ) + α p α f S 1 ( K n ) V α ( r ) c α ( r ) },
c α ( r )= N E 0 ( r )+ d ϕ n ( 2π ) 2 k n d k n d r e i k n ( ρ ρ ) g n ( z, z ) { [ 1+ S 1 ( K n ) p α f ] V α ( r ) c α ( r ) + α α S 1 ( K n ) p α f V α ( r ) c α ( r ) + f u f S 1 ( K n ) V 1 ( r ) u 1 ( r ) }
p( r )= N e i k 0 R i E 0 ( r )+ d ϕ n ( 2π ) 2 k n d k n [ 1+ S p ( K n )f ] d r i e i k n ( ρ ρ ) g n ( z, z ) V 1 ( r )p( r ) ,
E( z=0 )= E 0 ( z=0 ) + 1 A cell dz g 0 ( 0, z ) d ϕ ρ d ρ e i k 0 ρ [ f u V 1 ( r ) u 1 ( r )+ α p α V α ( r ) c α ( r ) + f p V 1 ( r )p( r )].

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