Abstract

Discrete-dipole approximation (DDA), which is used for computing scattering and absorption by particles of arbitrary geometry and material, is extended to the case of a rectangular cuboidal point lattice using an accurate, analytical expression of the polarizability of each cuboidal element at optical frequencies of up to 100 nm in size. This polarizability formulation (cuboidal lattice with depolarization or CLD) is shown to be more accurate in the computation of the extinction, scattering, and absorption cross sections when simulating dielectrics compared to other available and commonly used expressions of the polarizability. This can be used to reduce the number of dipoles N used, and therefore, the computation time while achieving the same accuracy of other formulations. The CLD formulation was applied to the Mie scattering problem and the results were compared to results from other DDA formulations, as well as to the Mie analytical solution for metal and dielectric spheres. Metal cubes were also simulated and different formulations compared.

© 2013 Optical Society of America

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References

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  1. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
    [CrossRef]
  2. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
    [CrossRef]
  3. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [CrossRef]
  4. M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
    [CrossRef]
  5. J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
    [CrossRef]
  6. C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. V. Giannini, J. A. Sánchez-Gil, O. L. Muskens, and J. G. Rivas, “Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced fluorescence,” J. Opt. Soc. Am. B 26, 1569–1577 (2009).
    [CrossRef]
  16. V. Giannini, A. Berrier, S. A. Maier, J. A. Sánchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
    [CrossRef]
  25. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).
  26. E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
    [CrossRef]
  27. M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112, 2234–2247 (2011).
    [CrossRef]
  28. M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E 82, 036703 (2010).
    [CrossRef]
  29. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
    [CrossRef]

2013

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[CrossRef]

M. A. Yurkin and M. Kahnert, “Light scattering by a cube: accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 123, 176–183 (2013).
[CrossRef]

2011

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112, 2234–2247 (2011).
[CrossRef]

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

2010

2009

2007

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

2006

2004

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

1998

N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

1994

1993

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

1991

1989

C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
[CrossRef]

1988

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1980

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

1975

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

1973

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Albella, P.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Berrier, A.

Chang, Y.-C.

Chaumet, P. C.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

de la Osa, R. A.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Draine, B. T.

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[CrossRef]

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” arXiv astro-ph/0403082 (2004).

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Fernández-Domínguez, A. I.

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

Flatau, P. J.

Fuchs, R.

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

Giannini, V.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[CrossRef]

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

V. Giannini, A. Berrier, S. A. Maier, J. A. Sánchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010).
[CrossRef]

V. Giannini, J. A. Sánchez-Gil, O. L. Muskens, and J. G. Rivas, “Electrodynamic calculations of spontaneous emission coupled to metal nanostructures of arbitrary shape: nanoantenna-enhanced fluorescence,” J. Opt. Soc. Am. B 26, 1569–1577 (2009).
[CrossRef]

González, F.

Goodman, J. J.

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

J. J. Goodman, B. T. Draine, and P. J. Flatau, “Application of fast-Fourier-transform techniques to the discrete-dipole approximation,” Opt. Lett. 16, 1198–1200 (1991).
[CrossRef]

Gutkowicz-Krusin, D.

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” arXiv astro-ph/0403082 (2004).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).

Heck, S. C.

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

Hoekstra, A. G.

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112, 2234–2247 (2011).
[CrossRef]

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E 82, 036703 (2010).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

Kahnert, M.

M. A. Yurkin and M. Kahnert, “Light scattering by a cube: accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 123, 176–183 (2013).
[CrossRef]

Maier, S. A.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[CrossRef]

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

V. Giannini, A. Berrier, S. A. Maier, J. A. Sánchez-Gil, and J. G. Rivas, “Scattering efficiency and near field enhancement of active semiconductor plasmonic antennas at terahertz frequencies,” Opt. Express 18, 2797–2807 (2010).
[CrossRef]

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

Martin, O. J. F.

N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Massa, E.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[CrossRef]

Min, M.

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E 82, 036703 (2010).
[CrossRef]

Mittra, R.

C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
[CrossRef]

Moreno, F.

Moroz, A.

Muskens, O. L.

Ng, M.-Y.

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Peterson, A.

C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
[CrossRef]

Piller, N. B.

N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

Rahmani, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Rivas, J. G.

Saiz, J. M.

Sánchez-Gil, J. A.

Sentenac, A.

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

Smith, C.

C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
[CrossRef]

Smith, D. A.

Stokes, K. L.

Xie, H.-Y.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Yurkin, M. A.

M. A. Yurkin and M. Kahnert, “Light scattering by a cube: accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 123, 176–183 (2013).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112, 2234–2247 (2011).
[CrossRef]

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E 82, 036703 (2010).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

M. A. Yurkin, “Computational approaches for plasmonics,” in Handbook of Molecular Plasmonics, F. Della Sala and S. D’Agostino, eds. (Pan Stanford, 2013), pp. 83–135.

Astrophys. J.

B. T. Draine and J. J. Goodman, “Beyond Clausius–Mossotti: wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophys. J. 405, 685–697 (1993).
[CrossRef]

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705–714 (1973).
[CrossRef]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Chem. Rev.

V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011).
[CrossRef]

IEEE Trans. Antennas Propag.

C. Smith, A. Peterson, and R. Mittra, “A conjugate gradient algorithm for the treatment of multiple incident electromagnetic fields,” IEEE Trans. Antennas Propag. 37, 1490–1493 (1989).
[CrossRef]

N. B. Piller and O. J. F. Martin, “Increasing the performance of the coupled-dipole approximation: a spectral approach,” IEEE Trans. Antennas Propag. 46, 1126–1137 (1998).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

M. A. Yurkin and A. G. Hoekstra, “The discrete-dipole-approximation code ADDA: capabilities and known limitations,” J. Quant. Spectrosc. Radiat. Transfer 112, 2234–2247 (2011).
[CrossRef]

M. A. Yurkin and A. G. Hoekstra, “The discrete dipole approximation: an overview and recent developments,” J. Quant. Spectrosc. Radiat. Transfer 106, 558–589 (2007).
[CrossRef]

M. A. Yurkin and M. Kahnert, “Light scattering by a cube: accuracy limits of the discrete dipole approximation and the T-matrix method,” J. Quant. Spectrosc. Radiat. Transfer 123, 176–183 (2013).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

New J. Phys.

E. Massa, S. A. Maier, and V. Giannini, “An analytical approach to light scattering from small cubic and rectangular cuboidal nanoantennas,” New J. Phys. 15, 063013 (2013).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. B

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732–1740 (1975).
[CrossRef]

Phys. Rev. E

M. A. Yurkin, M. Min, and A. G. Hoekstra, “Application of the discrete dipole approximation to very large refractive indices: filtered coupled dipoles revived,” Phys. Rev. E 82, 036703 (2010).
[CrossRef]

P. C. Chaumet, A. Sentenac, and A. Rahmani, “Coupled dipole method for scatterers with large permittivity,” Phys. Rev. E 70, 036606 (2004).
[CrossRef]

O. J. F. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909–3915 (1998).
[CrossRef]

Proc. IEEE

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[CrossRef]

Other

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University, 2012).

D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” arXiv astro-ph/0403082 (2004).

M. A. Yurkin and A. G. Hoekstra, “User manual for the discrete dipole approximation code ADDA 1.2,” 2013, http://a-dda.googlecode.com/svn/tags/rel_1.2/doc/manual.pdf .

S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

M. A. Yurkin, “Computational approaches for plasmonics,” in Handbook of Molecular Plasmonics, F. Della Sala and S. D’Agostino, eds. (Pan Stanford, 2013), pp. 83–135.

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Figures (5)

Fig. 1.
Fig. 1.

Comparison between the extinction cross sections, calculated using Mie theory and the DDA with CLD polarizability, of (a) gold (Au) and (b) dielectric (refractive index m=2) spheres of radii R=50, 100, and 150 nm placed in a vacuum (ϵB=1). The DDA calculations were done with a fixed number of dipoles per lambda (Dpl=100). The spectra for gold spheres show a well defined localized plasmon resonance.

Fig. 2.
Fig. 2.

Comparison of the averaged relative error (compared to Mie theory) in the extinction cross section (σext) for (a) a gold and (b) dielectric (refractive index m=2) sphere of radius R=150nm for different prescriptions of the polarizability with a changing number of dipoles per lambda (Dpl). The relative error was averaged across the entire range of wavelengths, i.e., from 300 to 800 nm. For dielectrics, the CLD prescription had the lowest relative error, followed by FCD and LDR, with the RRC last. For metals, all prescriptions showed similar behavior.

Fig. 3.
Fig. 3.

Comparison of the relative error (compared to Mie theory) in the extinction cross section (σext) for (a) a gold and (b) dielectric (refractive index m=2) sphere of radius R=150 nm for different prescriptions of the polarizability with a fixed low number of dipoles per lambda (Dpl=10). For dielectrics, the CLD prescription had the lowest relative error, followed by FCD and LDR, with the RRC last. For metals, all prescriptions showed similar behavior.

Fig. 4.
Fig. 4.

Comparison between the extinction cross sections, calculated using the DDA with different prescriptions for the polarizability, of (a) gold (Au) and (b) dielectric (refractive index m=2) cubes of effective radii Reff=50, 100, and 150 nm placed in a vacuum (ϵB=1). The DDA calculations were done with a fixed number of dipoles per lambda (Dpl=100). The spectra for gold cubes show a well defined localized plasmon resonance. The smaller peaks arose because of higher order modes than the main dipolar peak [29].

Fig. 5.
Fig. 5.

(a) Comparison between the absorption cross section of gold spheres of different radii (R=50, 100, 150 nm) calculated using the DDA with CLD polarizability and Mie theory. (b) Comparison between the absorption cross section of gold cubes of different effective radii (Reff=50, 100, 150 nm) calculated using different prescriptions of the polarizability. The calculations were done with a fixed number of dipoles per lambda (Dpl=100).

Equations (15)

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

Ei=Einc,i+j=1,jiNGi,jB·ΔϵjϵBEjVj+Mi·ΔϵiϵBEiL·ΔϵiϵBEi,i=1,,N,
Mi=limδV0ViδVdrGB(ri,r),
αi=ViΔϵϵB(I+(LiMi)ΔϵϵB)1,
Einc,i=αi1Pij=1,jiNGi,jPj,i=1,,N,
GB(ri,rj)=[kB(IRRR2)1ikBRR2(I3RRR2)]exp(ikBR)4πR,
Pi=ΔϵiϵBViEi.
αiCM=3d3ϵiϵBϵi+2ϵB,
αiRRC=αiCM116πikB3αCM.
α=8abcϵBϵϵB14π[2Ω+kB22β+163ikB3abc],
Ω=4arcsin(bc(a2+b2)(a2+c2)),
β=ccbbaa1x2+y2+z2(1+x2x2+y2+z2)dxdydz,
βcube=16[log(3+131)π6]a212.6937a2.
σext=kB|E0|2j=1NI(E¯inc,j·Pj).
σabs=kB|E0|2j=1N{I(E¯j·Pj)16πkB3P¯j·Pj},
ϵ(ω)=ϵωP2ω(ω+iγ),

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