Abstract

We determine the amount of light absorbed and scattered by metallic spheres in the presence of a substrate. The analysis is restricted to the spheres whose radius is small compared to the wavelength of the light such that the substrate-particle interactions are adequately described by the electrostatics limit. Results are presented for the absorption and scattering coefficients for: (i) the case when the electric permittivity of the spheres is described by the Drude model and (ii) for specific metals (silver, gold, and copper) for which the data on electrical permittivity as a function of the wavelength are available in the literature. It is found that it is possible to significantly increase the photovoltaic energy collected by a silicon substrate by depositing silver nanospheres on its surface. Mechanisms responsible for this increase are explored in detail in the electrostatics limit. Numerical results for the scattering are also used to derive an approximate formula that can be used to estimate the fractional increase in the photovoltaic energy. The increase predicted by this formula is qualitatively consistent with the literature data on the measured increase in the photocurrent by deposited silver nanospheres.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007).
    [Crossref]
  2. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
    [Crossref]
  3. S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
    [Crossref]
  4. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
    [Crossref]
  5. R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013).
    [Crossref]
  6. E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
    [Crossref]
  7. K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006).
    [Crossref]
  8. M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
    [Crossref]
  9. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).
  10. P. Drude, The Theory of Optics (Courier Corporation, 1925).
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    [Crossref]
  12. M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016).
    [Crossref]
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    [Crossref]
  15. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (CUP Archive, 1931).
  16. A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996).
    [Crossref]
  17. D. Shanks, “Non-linear Transformations of Divergent and Slowly Convergent Sequences,” Stud. Appl. Math. 34(1-4), 1–42 (1955).
    [Crossref]
  18. M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info .
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    [Crossref]
  20. P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
    [Crossref]
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    [Crossref]

2016 (1)

M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016).
[Crossref]

2014 (1)

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

2013 (2)

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013).
[Crossref]

2011 (1)

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
[Crossref]

2010 (2)

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref]

2007 (2)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007).
[Crossref]

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

2006 (3)

I. Romero, J. Aizpurua, G. W. Bryant, and F. J. G. De Abajo, “Plasmons in nearly touching metallic nanoparticles: singular response in the limit of touching dimers,” Opt. Express 14(21), 9988–9999 (2006).
[Crossref]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref]

K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006).
[Crossref]

2004 (2)

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

1996 (1)

A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996).
[Crossref]

1977 (1)

1955 (1)

D. Shanks, “Non-linear Transformations of Divergent and Slowly Convergent Sequences,” Stud. Appl. Math. 34(1-4), 1–42 (1955).
[Crossref]

Adato, R.

R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013).
[Crossref]

Aizpurua, J.

Altug, H.

R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013).
[Crossref]

Amey, J.

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref]

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007).
[Crossref]

Barnard, E. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

Bonnet, C.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Brongersma, M. L.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Broyer, M.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Bryant, G. W.

Cai, W.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Catchpole, K. R.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006).
[Crossref]

Christopher, P.

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
[Crossref]

Cong, T.

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

Cottancin, E.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

De Abajo, F. J. G.

Drude, P.

P. Drude, The Theory of Optics (Courier Corporation, 1925).

Green, M. A.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

Hilger, A.

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

Hobson, E. W.

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (CUP Archive, 1931).

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

Ingram, D. B.

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
[Crossref]

Israelowitz, M.

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

Jun, Y. C.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Kreibig, U.

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

Kunz, R. E.

Lermé, J.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Li, K.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Linic, S.

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
[Crossref]

Lukosz, W.

Manchon, D.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Mo, G.

A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996).
[Crossref]

Mosallaei, H.

M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016).
[Crossref]

Nordlander, P.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Oubre, C.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref]

Pellarin, M.

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Pillai, S.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006).
[Crossref]

Pinchuk, A.

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

Polman, A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref]

Polyanskiy, M. N.

M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info .

Prodan, E.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Romero, I.

Salary, M. M.

M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016).
[Crossref]

Sangani, A. S.

A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996).
[Crossref]

Schuller, J. A.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Shanks, D.

D. Shanks, “Non-linear Transformations of Divergent and Slowly Convergent Sequences,” Stud. Appl. Math. 34(1-4), 1–42 (1955).
[Crossref]

Stockman, M. I.

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Sureshkumar, R.

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

Trupke, T.

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

von Plessen, G.

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

White, J. S.

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

J. Appl. Phys. (2)

K. R. Catchpole and S. Pillai, “Absorption enhancement due to scattering by dipoles into silicon waveguides,” J. Appl. Phys. 100(4), 044504 (2006).
[Crossref]

S. Pillai, K. R. Catchpole, T. Trupke, and M. A. Green, “Surface plasmon enhanced silicon solar cells.,” J. Appl. Phys. 101(9), 093105 (2007).
[Crossref]

J. Nanomater. (1)

M. Israelowitz, J. Amey, T. Cong, and R. Sureshkumar, “Spin coated plasmonic nanoparticle interfaces for photocurrent enhancement in thin film Si solar cells,” J. Nanomater. 2014, 1–9 (2014).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. Chem. C (1)

J. Lermé, C. Bonnet, M. Broyer, E. Cottancin, D. Manchon, and M. Pellarin, “Optical properties of a particle above a dielectric interface: cross sections, benchmark calculations, and analysis of the intrinsic substrate effects,” J. Phys. Chem. C 117(12), 6383–6398 (2013).
[Crossref]

Nano Lett. (1)

P. Nordlander, C. Oubre, E. Prodan, K. Li, and M. I. Stockman, “Plasmon hybridization in nanoparticle dimers,” Nano Lett. 4(5), 899–903 (2004).
[Crossref]

Nanotechnology (1)

A. Pinchuk, A. Hilger, G. von Plessen, and U. Kreibig, “Substrate effect on the optical response of silver nanoparticles,” Nanotechnology 15(12), 1890–1896 (2004).
[Crossref]

Nat. Commun. (1)

R. Adato and H. Altug, “In-situ ultra-sensitive infrared absorption spectroscopy of biomolecule interactions in real time with plasmonic nanoantennas,” Nat. Commun. 4(1), 2154 (2013).
[Crossref]

Nat. Mater. (3)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010).
[Crossref]

S. Linic, P. Christopher, and D. B. Ingram, “Plasmonic-metal nanostructures for efficient conversion of solar to chemical energy,” Nat. Mater. 10(12), 911–921 (2011).
[Crossref]

J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. 9(3), 193–204 (2010).
[Crossref]

Opt. Express (1)

Phys. Fluids (1)

A. S. Sangani and G. Mo, “An O (N) algorithm for Stokes and Laplace interactions of particles,” Phys. Fluids 8(8), 1990–2010 (1996).
[Crossref]

Phys. Rev. B (1)

M. M. Salary and H. Mosallaei, “Tailoring optical forces for nanoparticle manipulation on layered substrates,” Phys. Rev. B 94(3), 035410 (2016).
[Crossref]

Sci. Am. (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296(4), 56–62 (2007).
[Crossref]

Science (1)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311(5758), 189–193 (2006).
[Crossref]

Stud. Appl. Math. (1)

D. Shanks, “Non-linear Transformations of Divergent and Slowly Convergent Sequences,” Stud. Appl. Math. 34(1-4), 1–42 (1955).
[Crossref]

Other (4)

M. N. Polyanskiy, “Refractive index database,” https://refractiveindex.info .

E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics (CUP Archive, 1931).

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 2008).

P. Drude, The Theory of Optics (Courier Corporation, 1925).

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Figures (14)

Fig. 1.
Fig. 1. A sketch showing the incident, reflected, and transmitted waves and the nomeclature used in the analysis. The substrate with an electric permittivity ${\alpha _s}{\varepsilon _m}$ occupies the space ${x_1} < 0$, the medium for ${x_1} > 0$ contains spheres of radius a with permittivity $\alpha {\varepsilon _m}$ and the surrounding medium with permittivity ${\varepsilon _m}$.
Fig. 2.
Fig. 2. The absorption coefficient ${Q_n}$ for a sphere with $\alpha = - 2 + 0.1\,\mbox{i}$ placed near a substrate with ${\alpha _s} = 10$ as a function of the number of multipoles ${N_s}$ used in the computations. (a) $h = 1$; (b)$h = 1.1.$ The open circles are the computed values and the stars are the estimates obtained by the repeated application of the Shanks transformation as explained in the text.
Fig. 3.
Fig. 3. The scattering coefficient ${S_n}$ for a sphere with $\alpha = - 2 + \,0.1\,\mbox{i}$ placed near a substrate with ${\alpha _s} = 10$ as a function of the number of multipoles used in the computations. (a) $h = 1$; (b) $h = 1.1$. See Fig. 2 for the legends.
Fig. 4.
Fig. 4. Absorption and scattering coefficients for the Drude model with $\delta = 0.1$. The open circles are the results of the computations and the dashed line is obtained by using the asymptotic formula for the dipole as given by (34). $h = 1.2$ and ${\alpha _s} = 11.3$.
Fig. 5.
Fig. 5. Absorption and scattering coefficients for $\delta = 0.05$, $h = 1.2$, and ${\alpha _s} = 11.3$. The solid lines are computed using the asymptotic result (34) while the dashed lines correspond to the estimates obtained by ignoring the sphere-substrate interaction altogether.
Fig. 6.
Fig. 6. Absorption and scattering coefficients for $h = 1$, $\delta = 0.1$, and ${\alpha _s} = 11.3$. The open circles correspond to the computed results while the dashed line corresponds to the asymptote (34).
Fig. 7.
Fig. 7. Absorption and scattering coefficients for the electric field parallel to the substrate for the Drude model. The open circles correspond to the computed results while the dashed line corresponds to the asymptote (36).
Fig. 8.
Fig. 8. Absorption and scattering coefficients for the electric field parallel to the surface of the substrate for $h = 1$ and ${\alpha _s} = 11.3$. The open circles represent numerical results while the dashed lines represent the estimates based on the dipole moment estimated using (36).
Fig. 9.
Fig. 9. The real and imaginary parts of the complex relative permittivity $\alpha $ as functions of the wavelength for silver (solid); gold (dashed); and copper (dashed-dotted). Data taken from Ref. 18.
Fig. 10.
Fig. 10. Absorption and scattering coefficients as functions of the wavelength for a silver sphere near a substrate with $h = 1.2$ and ${\alpha _s} = 11.2$. The solid lines correspond to the electric field normal to the substrate while the dashed lines are for the field parallel to the substrate.
Fig. 11.
Fig. 11. The absorption and scattering coefficients for gold (solid line) and copper (dashed line) for $h = 1.2$ and ${\alpha _s} = 11.2$.
Fig. 12.
Fig. 12. The energy-spectrum weighted average absorption and scattering coefficients as functions of h for silver spheres. The open circles correspond to the electric field normal to the substrate and the stars correspond to the electric field parallel to the substrate surface.
Fig. 13.
Fig. 13. Absorption and scattering coefficients for gold (stars) and copper (open circles) spheres as functions of the distance from the substrate. The solid lines joining the computed results correspond to the electric field normal to the substrate.
Fig. 14.
Fig. 14. Scattering coefficients ${S_n}$ (circles), ${S_t}$ (squares), and ${S_{ta}}$ (stars) for a pair of silver spheres near the substrate.

Tables (1)

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Table 1. Shanks transformation applied four times to the sequence of estimates of the dipole generated using Ns up to 9.

Equations (43)

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W a b s = 4 z Im ( α 1 α + 2 ) π a 2 I 0 , W s c = ( 8 / 3 ) z 4 | | α 1 α + 2 | | 2 π a 2 I 0
α = 1 ω p 2 ω 2 + i ω / τ = 1 1 ω r 2 + δ 2 + i δ ( ω r 2 + δ 2 ) ω r
k = k ( cos θ i e 1 + sin θ i e 2 )
E i n = E 0 [ sin θ i sin ψ i e 1 + cos θ i sin ψ i e 2 + cos ψ i e 3 ] e i ( k x ω t )
E r = E 0 [ sin θ i sin ψ i R T E e 1 + cos θ i sin ψ i R T E e 2 + cos ψ i R T M e 3 ] e i ( k r x ω t )
R T E = cos θ t α s 1 / 2 cos θ i cos θ t + α s 1 / 2 cos θ i and R T M = cos θ i α s 1 / 2 cos θ t cos θ i + α s 1 / 2 cos θ t
I 0 = E 0 2 2 c μ 0 and I t = I 0 [ sin 2 ψ i T T E 2 + cos 2 ψ i T T M 2 ]
T T E = 2 cos θ i cos θ t + α s 1 / 2 cos θ i and T T M = 2 cos θ i cos θ i + α s 1 / 2 cos θ t
× E = i z H , × H = i z ε r ( x ) E , E = 0 , H = 0
E s = i z × ( r Ψ ) + × × ( r Φ )
H s = i z ε r × ( r Φ ) + × × ( r Ψ )
E i m s = × × ( r i m Φ i m ) , H i m s = i z × ( r i m Φ i m ) ( for x 1 0 )
E a p p s = × × ( r Φ a p p ) , H i m s = i α s z × ( r Φ a p p ) ( for x 1 0 )
Φ = n = 1 ( 1 / n ) ϕ n 1 with ϕ n 1 = r n 1 m = 0 n [ A n m Y n m ( θ , φ ) + A ~ n m Y ~ n m ( θ , φ ) ]
E s = ϕ E with ϕ E = n = 1 ϕ n 1
A i m , n m = ( 1 ) n m + 1 α s 1 α s + 1 A n m , A a p p , n m = 2 α s + 1 A n m
E = ϕ t o t a l with ϕ t o t a l = n = 1 m = 0 n [ ( A n m r n 1 + C n m r n ) Y n m + ( A ~ n m r n 1 + C ~ n m r n ) Y ~ n m ]
G 1 = sin θ i sin ψ i ( 1 R T E ) , G 2 = cos θ i sin ψ i ( 1 + R T E ) , G 3 = cos ψ i R T M
t n A n m + C n m = 0 with t n = α n + n + 1 n ( α 1 )
Q a b s = 1 π Re r = 1 ( E × H ) n d A
E = G + k × × ( r k Φ k ) , H = i z [ ( 1 / 2 ) × ( r G r ) + k × ( r k Φ k ) ]
Q a b s = Im [ z π r = 1 { G × k × ( r k Φ k ) + ( 1 / 2 ) × ( r G r ) × k × × ( r k Φ k ) } n d A ]
Q a b s = z π r < 1 G k r k 2 Im ( Φ k ) d V
Q a b s = 4 z Im ( G D )
Q s c = Re [ 1 π H ( E s × H s ) n d A ]
E s = × × ( r Φ a p ) with Φ a p = D a p ( exp ( i z s r ) r ) and D a p = 2 α s + 1 k D k
Q s c = 16 α s z z s 3 3 ( α s + 1 ) 2 D D = 16 z 4 α s 5 / 2 3 ( α s + 1 ) 2 D D
D = D G 1 e 1 + D ( G 2 e 2 + G 3 e 3 )
Q a b s = 4 z [ Q n G 1 2 + Q t ( G 2 2 + G 3 2 ) ]
Q s c = 16 z 4 α s 5 / 2 3 ( α s + 1 ) 2 [ S n G 1 2 + S t ( G 2 2 + G 3 2 ) ]
t n A n 0 + δ n 1 + β s p = 1 ( 1 2 h ) n + p + 1 ( n + p ) ! n ! p ! A p 0 = 0 , n = 1 , 2 ,
β s = α s 1 α s + 1
D ( n , 1 ) = D ( n + 1 , 0 ) D ( n 1 , 0 ) ( D ( n , 0 ) ) 2 D ( n + 1 , 0 ) + D ( n 1 , 0 ) 2 D ( n , 0 )
D = [ t 1 2 β s R 3 9 β s 2 R 8 t 2 6 β s R 5 16 β s 2 R 10 t 3 20 β s R 7 + O ( R 12 ) ] 1 ( R 2 h )
t n A n 1 δ n 1 + β s p = 1 ( 1 2 h ) n + p + 1 ( n + p ) ! ( n + 1 ) ! ( p 1 ) ! A p 1 = 0
D = [ t 1 β s R 3 3 β s 2 R 8 t 2 4 β s R 5 6 β s 2 R 10 t 3 15 β s R 7 + O ( R 12 ) ] 1 ( R 2 h )
E ( λ ) = c 1 λ 5 [ exp ( c 2 λ T ) 1 ]
< S n >= λ 1 λ 2 ( λ 0 / λ ) 4 S n ( λ ) E ( λ ) d λ λ 1 λ 2 E ( λ ) d λ , < Q n >= λ 1 λ 2 ( λ 0 / λ ) Q n ( λ ) E ( λ ) d λ λ 1 λ 2 E ( λ ) d λ
Q s c = 16 z 4 α s 5 / 2 3 ( α s + 1 ) 2 [ S n G 1 2 + S t ( G t 2 ( G m ) 2 ) + S t a ( G m ) 2 ]
D t = [ t 1 λ R 3 + O ( R 8 ) ] 1 where λ = β s ( 1 1 4 2 ) + 2
S n = 19.9 , S t = 16.8 , and S t a = 30.9
I s c I t ϕ a 4 α s 5 / 2 ( 1 + α s 1 / 2 ) 2 3 ( α s + 1 ) 2 ( 2 π a λ 0 ) 4 < S >
I s c / I t 1.77 × 10 5 ϕ a a 4 ( a in nm )

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