Abstract

An efficient numerical method based on half-space Green’s function and spherical harmonics expansion is used to study the light scattering from coupled multiple nanospheres on a substrate. The ellipsometric spectra for various geometries of coupled Au nanospheres are calculated and analyzed to realize the effects of plasmonic coupling of closely spaced nanospheres. With only a few parameters to describe the distribution of various coupled nanosphere clusters embedded in a random distribution of nanospheres, the calculated ellipsometric spectra can fit the experimental data very well. This illustrates that our realistic model calculations can be used for determination of the distribution of nanospheres on a substrate or embedded in multilayer structures, such as biological samples.

© 2013 Optical Society of America

Full Article  |  PDF Article

Errata

Huai-Yi Xie and Yia-Chung Chang, "Light scattering from coupled plasmonic nanospheres on a substrate: erratum," J. Opt. Soc. Am. B 31, 2083-2083 (2014)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-31-9-2083

References

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  1. P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
    [CrossRef]
  2. T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
    [CrossRef]
  3. J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
    [CrossRef]
  4. J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
    [CrossRef]
  5. M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
    [CrossRef]
  6. H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
    [CrossRef]
  7. 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. Express 18, 1310–1315 (2010).
    [CrossRef]
  8. H. Y. Xie, Y. C. Chang, G. Li, and S. H. Hsu, “Effect of clustering on ellipsometric spectra of randomly distributed gold nanoparticles on a substrate,” Opt. Express 21, 3091–3102 (2013).
    [CrossRef]
  9. A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74, 033402 (2006).
  10. V. Twersky, “Multiple scattering of waves and optical phenomena,” J. Opt. Soc. Am. 52, 145–171 (1962).
    [CrossRef]
  11. K. A. Fuller and G. W. Kattawar, “Consummate solution to the problem of classical electromagnetic scattering by an ensemble of spheres. II: clusters of arbitrary configuration,” Opt. Lett. 13, 1063–1065 (1988).
    [CrossRef]
  12. D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 433, 599–614 (1991).
    [CrossRef]
  13. Y. I. Xu, “Electromagnetic scattering by an aggregate of spheres,” Appl. Opt. 34, 4573–4588 (1995).
    [CrossRef]
  14. Y. I. Xu, “Electromagnetic scattering by an aggregate of spheres: errata,” Appl. Opt. 37, 6494 (1998).
    [CrossRef]
  15. Y. I. Xu and B. A. S. Gustafson, “Experimental and theoretical results of light scattering by aggregates of spheres,” Appl. Opt. 36, 8026–8030 (1997).
    [CrossRef]
  16. A. Pikulin, A. Afanasiev, N. Agareva, A. P. Alexandrov, V. Bredikhin, and N. Bityurin, “Effects of spherical mode coupling on near-field focusing by clusters of dielectric microspheres,” Opt. Express 20, 9052–9057 (2012).
    [CrossRef]
  17. A. Lakhtakia, “Dyadic Green’s functions for an isotropic chiral half-space bounded by a perfectly conducting plane,” Int. J. Electron. 71, 139–144 (1991).
    [CrossRef]
  18. A. Lakhtakia, “Green’s functions and Brewster condition for a halfspace bounded by an anisotropic impedance plane,” Int. J. Infrared Millim. Waves 13, 161–170 (1992).
    [CrossRef]
  19. Y. C. Chang, G. Li, H. Chu, and J. Opsal, “Efficient finite-element Green’s function approach for CD metrology of 3D gratings on multilayer films,” J. Opt. Soc. Am. A 23, 638–645 (2006).
    [CrossRef]
  20. F. Riazuddin, Quantum Mechanics (World Scientific, 1990).
  21. E. D. Palik, ed., Handbook of Optical Constant of Solid (Academic, 1985), Vol. 1.
  22. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particle (Wiley, 1983).
  23. Here we correct an error of our previous work [8]. Equation (11) of [8] should be replaced by Eq. (22) in the present paper. With this correction and by using a larger (convergent) set of basis functions (with ℓc =6), the best-fit value for the coherent length becomes 3500 nm.
  24. H. Y. Xie, Y. C. Chang, G. Li, and S. H. Hsu, http://arxiv.org/abs/1305.6423 .

2013 (1)

2012 (2)

M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
[CrossRef]

A. Pikulin, A. Afanasiev, N. Agareva, A. P. Alexandrov, V. Bredikhin, and N. Bityurin, “Effects of spherical mode coupling on near-field focusing by clusters of dielectric microspheres,” Opt. Express 20, 9052–9057 (2012).
[CrossRef]

2011 (2)

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
[CrossRef]

2010 (3)

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

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. Express 18, 1310–1315 (2010).
[CrossRef]

2008 (1)

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

2006 (1)

1998 (1)

1997 (1)

1995 (1)

1992 (1)

A. Lakhtakia, “Green’s functions and Brewster condition for a halfspace bounded by an anisotropic impedance plane,” Int. J. Infrared Millim. Waves 13, 161–170 (1992).
[CrossRef]

1991 (2)

A. Lakhtakia, “Dyadic Green’s functions for an isotropic chiral half-space bounded by a perfectly conducting plane,” Int. J. Electron. 71, 139–144 (1991).
[CrossRef]

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 433, 599–614 (1991).
[CrossRef]

1988 (1)

1962 (1)

Afanasiev, A.

Agareva, N.

Alexandrov, A. P.

Bao, J.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Bao, K.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Bardhan, R.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Bityurin, N.

Bohren, C. F.

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

Bredikhin, V.

Capasso, F.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

Chan, C. T.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Chang, Y. C.

Chen, Y. C.

Chu, H.

Coenen, T.

M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
[CrossRef]

Dadosh, T.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Fan, J. A.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

Frimmer, M.

M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
[CrossRef]

Fuller, K. A.

Gustafson, B. A. S.

Halas, N. J.

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Hang, Z. H.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Haran, G.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Hsu, S. H.

Huffman, D. R.

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

Kattawar, G. W.

Kim, Y. D.

Koenderink, A. F.

M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
[CrossRef]

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74, 033402 (2006).

Kundu, J.

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia, “Green’s functions and Brewster condition for a halfspace bounded by an anisotropic impedance plane,” Int. J. Infrared Millim. Waves 13, 161–170 (1992).
[CrossRef]

A. Lakhtakia, “Dyadic Green’s functions for an isotropic chiral half-space bounded by a perfectly conducting plane,” Int. J. Electron. 71, 139–144 (1991).
[CrossRef]

Lassiter, J. B.

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

Letnes, P. A.

P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
[CrossRef]

Li, G.

Li, Z.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Liu, H.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 433, 599–614 (1991).
[CrossRef]

Manoharan, V. N.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Mills, D. L.

P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
[CrossRef]

Ng, J.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Nordlander, P.

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Opsal, J.

Pikulin, A.

Polman, A.

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74, 033402 (2006).

Riazuddin, F.

F. Riazuddin, Quantum Mechanics (World Scientific, 1990).

Shegai, T.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Shvets, G.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Simonsen, I.

P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
[CrossRef]

Sobhani, H.

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

Twersky, V.

Wang, S. B.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Wei, P. K.

Wu, C.

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Xie, H. Y.

Xu, H.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Xu, Y. I.

Zhang, Z.

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Zhu, S. N.

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Appl. Opt. (3)

Int. J. Electron. (1)

A. Lakhtakia, “Dyadic Green’s functions for an isotropic chiral half-space bounded by a perfectly conducting plane,” Int. J. Electron. 71, 139–144 (1991).
[CrossRef]

Int. J. Infrared Millim. Waves (1)

A. Lakhtakia, “Green’s functions and Brewster condition for a halfspace bounded by an anisotropic impedance plane,” Int. J. Infrared Millim. Waves 13, 161–170 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Nano Lett. (1)

J. B. Lassiter, H. Sobhani, J. A. Fan, J. Kundu, F. Capasso, P. Nordlander, and N. J. Halas, “Fano resonances in plasmonic nanoclusters: geometrical and chemical tenability,” Nano Lett. 10, 3184–3189 (2010).
[CrossRef]

New J. Phys. (1)

H. Liu, J. Ng, S. B. Wang, Z. H. Hang, C. T. Chan, and S. N. Zhu, “Strong plasmon coupling between two gold nanospheres on a gold slab,” New J. Phys. 13, 073040 (2011).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (1)

P. A. Letnes, I. Simonsen, and D. L. Mills, “Substrate influence on the plasmonic response of clusters of spherical nanoparticles,” Phys. Rev. B 83, 075426 (2011).
[CrossRef]

Phys. Rev. Lett. (1)

M. Frimmer, T. Coenen, and A. F. Koenderink, “Signature of a Fano resonance in a plasmonic metamolecule’s local density of optical states,” Phys. Rev. Lett. 108, 077404 (2012).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

T. Shegai, Z. Li, T. Dadosh, Z. Zhang, H. Xu, and G. Haran, “Managing light polarization via plasmon–molecule interactions within an asymmetric metal nanoparticle trimer,” Proc. Natl. Acad. Sci. USA 105, 16448–16453 (2008).
[CrossRef]

Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. (1)

D. W. Mackowski, “Analysis of radiative scattering for multiple sphere configurations,” Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 433, 599–614 (1991).
[CrossRef]

Science (1)

J. A. Fan, C. Wu, K. Bao, J. Bao, R. Bardhan, N. J. Halas, V. N. Manoharan, P. Nordlander, G. Shvets, and F. Capasso, “Self-assembled plasmonic nanoparticle clusters,” Science 328, 1135–1138 (2010).
[CrossRef]

Other (6)

A. F. Koenderink and A. Polman, “Complex response and polariton-like dispersion splitting in periodic metal nanoparticle chains,” Phys. Rev. B74, 033402 (2006).

F. Riazuddin, Quantum Mechanics (World Scientific, 1990).

E. D. Palik, ed., Handbook of Optical Constant of Solid (Academic, 1985), Vol. 1.

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

Here we correct an error of our previous work [8]. Equation (11) of [8] should be replaced by Eq. (22) in the present paper. With this correction and by using a larger (convergent) set of basis functions (with ℓc =6), the best-fit value for the coherent length becomes 3500 nm.

H. Y. Xie, Y. C. Chang, G. Li, and S. H. Hsu, http://arxiv.org/abs/1305.6423 .

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Figures (7)

Fig. 1.
Fig. 1.

Schematic drawing of (a) a chain of nanoparticles (top view) and close-packed clustering nanoparticles such as (b) a trimer and (c) a heptamer (top view).

Fig. 2.
Fig. 2.

Electric field (at the origin as indicated in Fig. 1 with z=0), |E| as a function of photon energy for light scattering from a trimer of Au nanospheres of the same diameter d=80nm [Fig. 1(b) and set φd=0] with two gaps: (a) large gap g=10nm and (b) small gap g=2nm calculated by the present Green’s function method (dashed curves) with (a) (max,Nk,Nz)=(6,101,100) and (b) (max,Nk,Nz)=(9,121,100) and previous coupled-sphere approach [1115] (solid curves).

Fig. 3.
Fig. 3.

Calculated reflectance for s- and p-polarized light for isolated clusters of Au-NPs (with d=80nm and set φd=0 in Fig. 1) on glass substrate with three different angles of incidence: 55° (green), 60° (red), and 65° (blue) for specular reflection for five different arrangements: (a) a chain of two nanoparticles, (b) a chain of three nanoparticles, (c) a chain of four nanoparticles, (d) a trimer, and (e) a heptamer. The gap used is 2 nm.

Fig. 4.
Fig. 4.

Calculated reflectance for s- and p-polarized light for isolated clusters of Au-NPs (with d=80nm and φd=0) on glass substrate with three different angles of incidence: 55° (green), 60° (red), and 65° (blue) for off-specular reflection for five different arrangements: (a) a chain of two nanoparticles, (b) a chain of three nanoparticles, (c) a chain of four nanoparticles, (d) a trimer, and (e) a heptamer. The gap used is 2 nm.

Fig. 5.
Fig. 5.

Orientation-averaged ellipsometric parameters, Ψ and Δ as functions of photon energy obtained by the Green’s function method for a random distribution of clustering nanoparticles for five different arrangements: (a) chains of two nanoparticles, (b) chains of three nanoparticles, (c) chains of four nanoparticles, (d) trimmers, and (e) heptamers for three different angles of incidence: 55° (solid line), 60° (dashed line), and 65° (dashed–dotted line) on the substrate. The particle size is 80 nm and the gap is 2 nm.

Fig. 6.
Fig. 6.

Spectroscopic ellipsometry measurement (solid curves) and model calculations (dashed–dotted curves) of randomly distributed Au nanoparticles with various arrangements. The nominal sizes of nanoparticles are (a) 20, (b) 40, (c) 60, and (d) 80 nm for angle of incidence of 55°, 60°, and 65°.

Fig. 7.
Fig. 7.

Spectroscopic ellipsometry measurement (solid curves) and model calculations (dashed–dotted curves) of random distribution of Au nanoparticles including the effect of clusters which are modeled by aggregations of nanoparticles with various arrangements. The nominal sizes of nanoparticles are (a) 40, (b) 60, and (c) 80 nm for angle of incidence 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 (32)

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

E(r)=E0(r)+q=1N(εqε0)VqG(r,r)E(r)d3r,
Eγ(r)=mαqmγj(kqr)Ym(Ω),rVq,q=1,2,,N,
Q(kp)αpmj=eik0·RpFjpm+q=1N(εqεa)γmGjγpq,mmαqmγ,p=1,2,,N,
Gjγpq,mm=nαβeikndqpSn,αjG¯n,αβpq,mmSn,βγ,p,q=1,2,,N.
Sn=(cosφnsinφn0sinφncosφn0001)=(sinφ˜ncosφ˜n0cosφ˜nsinφ˜n0001),
G¯n,xxpq,mm=qn2u˜n{R˜nEnmp+[Bnmq++r˜nBnmq]+r˜nEnmp[R˜nBnmq++Bnmq]}qn2Mnmmpq+,
G¯n,xzpq,mm=ikn2u˜n{R˜nEnmp+[Bnmp+r˜nBnmp]+r˜nEnmp[R˜nBnmp+Bnmp]}ikn2Mnmmpq,
G¯n,yypq,mm=k022qnun{RnEnmp+[Bnmp++r¯nBnmp]+r¯nEnmp[RnBnmp++Bnmp]}+k022qnMnmmpq+
G¯n,zxpq,mm=ikn2u˜n{R˜nEnmp+[Bnmp++r˜nBnmp]+r˜nEnmp[R˜nBnmp++Bnmp]}ikn2Mnmmpq,
G¯n,zzpq,mm=kn22qnu˜n{R˜nEnmp+[Bnmp+r˜nBnmp]+r˜nEnmp[R˜nBnmp+Bnmp]}+kn22qnMnmmpq+Mnmmpq0
Fjpm=4πi[T0jeik0zaYm*(Ωk0)+R0jeik0zaYm*(πθk0,φk0)]0ar2[j(kpr)]*j(k0r)dr,
Bnmq±=eimφ˜naadze±qnzeqnaInmq(z),Enmp±=eimφ˜naadze±qnzeqna[Inmp(z)]*,
Inmq(z)=2π0a2z2dρρj(kqρ2+z2)×P˜m(zρ2+z2)Jm(knρ),q=1,2,,N,
Mnmmpq±=eimφ˜neimφ˜naadz[Inmp(z)]*[azdzeqn(zz)Inmq(z)±zadzeqn(zz)Inmq(z)],
Mnmmpq0=eimφ˜neimφ˜naadz[Inmp(z)]*Inmq(z),
j(kpρ2+z2)P˜m(zρ2+z2)={ν=0,2,4,2+4Cmνρν,misevenν=1,3,5,2+5Cmνρν,misodd,ρ[0,a],
Inmp(z)=2π{ν=0,2,4,2+4ν=1,3,5,2+5}Cmν0a2z2dρρν+1Jm(knρ)=2π{ν=0,2,4,2+4ν=1,3,5,2+5}Cmν(a2z2)ν+10a2z2dρ(ρa2z2)ν+1Jm(knρ){ν=0,2,4,2+4ν=1,3,5,2+5}Cmν(z)Wz(ν+1,m),
12π02πdφ˜nei(mm)φ˜neikndqp=ei(mm)φqpJmm(kndqp),
12π02πdφ˜nei(mm)φ˜neikndqpsinφ˜n=ei(mm)φqp2{i[Jmm+1(kndqp)Jmm1(kndqp)]cosφqp[Jmm+1(kndqp)+Jmm1(kndqp)]sinφqp},
12π02πdφ˜nei(mm)φ˜neikndqpcosφ˜n=ei(mm)φqp2{[Jmm+1(kndqp)+Jmm1(kndqp)]cosφqpi[Jmm+1(kndqp)Jmm1(kndqp)]sinφqp},
u1(r)=NE0(r)+dϕn(2π)2kndknS(Kn)dreikn(ρρ)Gn(z,z)V1(r)u1(r),
S(Kn)=1NijeiKn(RjRi)=1+fS1(Kn),
S1(Kn)=j1eiKn(RjR1)e(RjR1)2/2λc22πλc2AcelleKn2λc2/22πAcell0RuJ0(KnR)eR2/2λc2RdR,
Fjpm=Vp[j(kpr)]*Ym*(Ω)Ej0d3r=Vp{T0jeik0r+R0jeik0r}[j(kpr)]*Ym*(Ω)d3r=Vp{T0jeik0r+R0jeik0xx+ik0yyik0zz}[j(kpr)]*Ym*(Ω)d3r,
eik0r=4πmij(k0r)Ym*(Ωk0)Ym(Ω),
Fjpm=4πT0jeik0zamiYm*(Ωk0)Vp[j(kpr)]*j(k0r)Ym(Ω)Ym*(Ω)d3r+4πR0jeik0zamiYm*(πθk0,φk0)Vp[j(kpr)]*j(k0r)Ym(Ω)Ym*(Ω)d3r=4πi[T0jeik0zaYm*(Ωk0)+R0jeik0zaYm*(πθk0,φk0)]0a[j(kpr)]*j(k0r)r2dr.
VpVqeiknρsin(φ+φ˜n)e±qnzeqna[j(kpr)]*Ym*(Ω)eiknρsin(φ+φ˜n)e±qnzeqnaj(kqr)Ym(Ω)d3rd3r=12πVpeiknρsin(φ+φ˜n)e±qnzeqna[j(kpρ2+z2)]*P˜m(zρ2+z2)eimφd3r×12πVqeiknρsin(φ+φ˜n)e±qnzeqnaj(kqρ2+z2)P˜m(zρ2+z2)eimφd3r=2πeimφ˜naadz0a2z2ρdρe±qnzeqna[j(kpρ2+z2)]*P˜m(zρ2+z2)Jm(knρ)×2πeimφ˜naadz0a2z2ρdρe±qnzeqnaj(kqρ2+z2)P˜m(zρ2+z2)Jm(knρ)=eimφ˜naadze±qnzeqna[Inmp(z)]*×eimφ˜naadze±qnzeqnaInmq(z)Enmp±Bnmq±,
02πeiknρsin(φ+φ˜n)eimφdφ=2πeimφ˜nJm(knρ).
12πVqeikn[ρsin(φ+φ˜n)ρsin(φ+φ˜n)]eqn|zz|j(kqρ2+z2)P˜m(zρ2+z2)eimφd3r=2πaadz0a2z2ρdρeimφ˜neiknρsin(φ+φ˜n)eqn|zz|j(kqρ2+z2)P˜m(zρ2+z2)Jm(knρ)=eimφ˜neiknρsin(φ+φ˜n)[azdzeqn(zz)Inmq(z)+zadzeqn(zz)Inmq(z)].
Vp[j(kpr)]*Ym*(Ω)eimφ˜neiknρsin(φ+φ˜n)[azdzeqn(zz)Inmq(z)+zadzeqn(zz)Inmq(z)]d3r=eimφ˜neimφ˜naadz[Inmp(z)]*[azdzeqn(zz)Inmq(z)+zadzeqn(zz)Inmq(z)]Mnmmpq+.
Vp[j(kpr)]*Ym*(Ω)eimφ˜neiknρsin(φ+φ˜n)[azdzeqn(zz)Inmq(z)zadzeqn(zz)Inmq(z)]d3r=eimφ˜neimφ˜naadz[Inmp(z)]*[azdzeqn(zz)Inmq(z)zadzeqn(zz)Inmq(z)]Mnmmpq.
Vp[j(kpr)]*Ym*(Ω)eimφ˜neiknρsin(φ+φ˜n)Inmq(z)d3r=eimφ˜neimφ˜naadz[Inmp(z)]*Inmq(z)Mnmmpq0.

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