Abstract

A compact analytical formula up to the order of k3, where k is a wave vector, is derived for the depolarization field Ed of a spheroidal particle by performing explicitly the steps of the recipe outlined by Meier and Wokaun [Opt. Lett. 8, 581 (1983)] . For the static component of Ed a general electrostatic formula valid for a particle of a general shape is rederived within the Meier and Wokaun framework. The dynamic k2-dependent depolarization component of Ed is shown to depend on dynamic geometrical factors, which can be expressed in terms of the standard geometrical factors of electrostatics. The Meier and Wokaun recipe itself is shown to be equivalent to a long-wavelength limit of the Green’s function technique. The resulting Meier and Wokaun long-wavelength approximation is found to exhibit a redshift compared against exact T-matrix results. At least for a sphere, it is possible to get rid of the redshift by assuming a weak nonuniformity of the field Eint inside a particle, which can be fully accounted for by a renormalization of the dynamic geometrical factors. My results may be relevant for various plasmonic, or nanoantenna, applications of spheroidal particles with a dominant electric dipole scattering, whenever it is necessary to go beyond the Rayleigh approximation and to capture the essential size-dependent features of scattering, local fields, SERS, hyper-Raman and second-harmonic-generation enhancements, decay rates, and photophysics of dipolar arrays.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
  2. R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  8. E. J. Zeman and G. C. Schatz, “Electromagnetic theory calculations for spheroids: an accurate study of particle size dependence of SERS and hyper-Raman enhancements,” in Dynamics on Surfaces, Proceedings of the 17th Jerusalem Symposium on Quantum Chemistry and Biochemistry, B.Pullman, ed. (Reidel, 1984), pp. 413-424.
    [CrossRef]
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    [CrossRef]
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    [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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2008 (4)

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

H. Kang and G. W. Milton, “Solutions to the Polya-Szego conjecture and the weak Eshelby conjecture,” Arch. Ration. Mech. Anal. 188, 93-116 (2008).
[CrossRef]

A. Moroz, “Electron mean-free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

2006 (1)

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

2005 (1)

2003 (2)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

2001 (1)

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

1996 (1)

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

1995 (1)

W.-H. Yang, G. C. Schatz, and R. P. van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

1987 (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634-643 (1987).
[CrossRef]

1985 (2)

1983 (1)

1980 (2)

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

1965 (1)

1953 (1)

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143-1151 (1953).
[CrossRef]

1909 (1)

R. Gans and H. Happel, “Zur Optik Kolloidaler Metallösungen,” Ann. Phys. 29, 277-300 (1909).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1973).

Agarwal, A.

Ahmadi, T. S.

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Cao, Y.

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

Chew, H.

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

Doyle, W. T.

El-Sayed, M. A.

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

Esumi, K.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

Gans, R.

R. Gans and H. Happel, “Zur Optik Kolloidaler Metallösungen,” Ann. Phys. 29, 277-300 (1909).
[CrossRef]

Green, T. C.

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

Happel, H.

R. Gans and H. Happel, “Zur Optik Kolloidaler Metallösungen,” Ann. Phys. 29, 277-300 (1909).
[CrossRef]

Henglein, A.

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1998).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999).

Jin, R.

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

Kang, H.

H. Kang and G. W. Milton, “Solutions to the Polya-Szego conjecture and the weak Eshelby conjecture,” Arch. Ration. Mech. Anal. 188, 93-116 (2008).
[CrossRef]

Kellogg, O. D.

O. D. Kellogg, Foundations of Potential Theory (Dover, 1953).

Kelly, K. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

Kerker, M.

Kuwata, H.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Landau, L.

L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Liao, P. F.

Lifschitz, E. M.

L. Landau and E. M. Lifschitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Liz-Marzán, L. M.

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

Mata-Osoro, G.

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

Meier, M.

Milton, G. W.

H. Kang and G. W. Milton, “Solutions to the Polya-Szego conjecture and the weak Eshelby conjecture,” Arch. Ration. Mech. Anal. 188, 93-116 (2008).
[CrossRef]

Mirkin, C. A.

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

Miyano, K.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

Moroz, A.

A. Moroz, “Electron mean-free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

A. Moroz, “Improvement of Mishchenko's T-matrix code for absorbing particles,” Appl. Opt. 44, 3604-3609 (2005).
[CrossRef] [PubMed]

Mulvaney, P.

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

Newton, R. G.

R. G. Newton, Scattering Theory of Waves and Particles (Springer, 1982).

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Pecharromán, C.

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

Penninkhof, J. J.

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

Pérez-Juste, J.

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

Polman, A.

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

Schatz, G. C.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

W.-H. Yang, G. C. Schatz, and R. P. van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634-643 (1987).
[CrossRef]

E. J. Zeman and G. C. Schatz, “Electromagnetic theory calculations for spheroids: an accurate study of particle size dependence of SERS and hyper-Raman enhancements,” in Dynamics on Surfaces, Proceedings of the 17th Jerusalem Symposium on Quantum Chemistry and Biochemistry, B.Pullman, ed. (Reidel, 1984), pp. 413-424.
[CrossRef]

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1973).

Stevenson, A. F.

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143-1151 (1953).
[CrossRef]

Tamaru, H.

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, 2002).

van Blaaderen, A.

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

van Duyne, R. P.

W.-H. Yang, G. C. Schatz, and R. P. van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

Wang, D.-S.

Wang, Z. L.

T. S. Ahmadi, Z. L. Wang, T. C. Green, A. Henglein, and M. A. El-Sayed, “Shape-controlled synthesis of colloidal platinum nanoparticles,” Science 272, 1924-1926 (1996).
[CrossRef] [PubMed]

Waterman, P. C.

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3, 825-839 (1971).
[CrossRef]

Wokaun, A.

A. Wokaun, “Surface enhancement of optical fields mechanism and applications,” Mol. Phys. 56, 1-33 (1985).
[CrossRef]

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

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

A. Wokaun, “Surface-enhanced electromagnetic processes,” in Solid State Physics, H.Ehrenreich, D.Turnbull, and F.Seitz, eds. (Academic, 1984), Vol. 38, pp. 223-294.
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 2002).

Yaghjian, A. D.

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

Yang, W.-H.

W.-H. Yang, G. C. Schatz, and R. P. van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

Zeman, E. J.

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634-643 (1987).
[CrossRef]

E. J. Zeman and G. C. Schatz, “Electromagnetic theory calculations for spheroids: an accurate study of particle size dependence of SERS and hyper-Raman enhancements,” in Dynamics on Surfaces, Proceedings of the 17th Jerusalem Symposium on Quantum Chemistry and Biochemistry, B.Pullman, ed. (Reidel, 1984), pp. 413-424.
[CrossRef]

Zhao, L. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

Zheng, J. G.

R. Jin, Y. Cao, C. A. Mirkin, K. L. Kelly, G. C. Schatz, and J. G. Zheng, “Photoinduced conversion of silver nanospheres to nanoprisms,” Science 294, 1901-1903 (2001).
[CrossRef] [PubMed]

Ann. Phys. (1)

R. Gans and H. Happel, “Zur Optik Kolloidaler Metallösungen,” Ann. Phys. 29, 277-300 (1909).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. 83, 4625-4627 (2003).
[CrossRef]

Arch. Ration. Mech. Anal. (1)

H. Kang and G. W. Milton, “Solutions to the Polya-Szego conjecture and the weak Eshelby conjecture,” Arch. Ration. Mech. Anal. 188, 93-116 (2008).
[CrossRef]

J. Appl. Phys. (1)

A. F. Stevenson, “Electromagnetic scattering by an ellipsoid in the third approximation,” J. Appl. Phys. 24, 1143-1151 (1953).
[CrossRef]

J. Chem. Phys. (1)

W.-H. Yang, G. C. Schatz, and R. P. van Duyne, “Discrete dipole approximation for calculating extinction and Raman intensities for small particles with arbitrary shapes,” J. Chem. Phys. 103, 869-875 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. Chem. (1)

E. J. Zeman and G. C. Schatz, “An accurate electromagnetic theory study of surface enhancement factors for silver, gold, copper, lithium, sodium, aluminum, gallium, indium, zinc, and cadmium,” J. Phys. Chem. 91, 634-643 (1987).
[CrossRef]

J. Phys. Chem. B (1)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668-677 (2003).
[CrossRef]

J. Phys. Chem. C (2)

J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen, “Optical properties of spherical and oblate spheroidal gold shell colloids,” J. Phys. Chem. C 112, 4146-4150 (2008).
[CrossRef]

A. Moroz, “Electron mean-free path in a spherical shell geometry,” J. Phys. Chem. C 112, 10641-10652 (2008).
[CrossRef]

Mol. Phys. (1)

A. Wokaun, “Surface enhancement of optical fields mechanism and applications,” Mol. Phys. 56, 1-33 (1985).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

C. Pecharromán, J. Pérez-Juste, G. Mata-Osoro, L. M. Liz-Marzán, and P. Mulvaney, “Redshift of surface plasmon modes of small gold rods due to their atomic roughness and end-cap geometry,” Phys. Rev. B 77, 035418 (2008).
[CrossRef]

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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of the integration [Eq. (28)] over the surface of the volume V V δ . The surface normals at the intersection of each ray emanating from the origin with the surfaces of V δ and V, which are related by scaling, necessarily point in opposite directions.

Fig. 2
Fig. 2

Dynamic geometrical factors D and D for an electric field applied parallel and perpendicular to the spheroid rotational axis, respectively. As an illustration, for spheroids with the aspect ratio 2:1, 3:1, 4:1, 5:1, and 6:1 the eccentricity takes on the values of 0.866 , 0.943 , 0.968 , 0.98 , and 0.986 , respectively. For prolate spheroids, D , whereas for oblate spheroids D = 9 π 16 as e 1 .

Fig. 3
Fig. 3

Comparison of the extinction efficiency obtained in the Rayleigh limit, by MLWA, and MWLWA, against the exact T-matrix method results. The results are shown for an oblate silver spheroidal particle with the major to minor axis ratio of 5:1 and an equivalent-volume-sphere radius of 20 nm ( a = b 34.2 and c 6.84 nm ). Electric field is oriented perpendicular to the rotational symmetry axis.

Fig. 4
Fig. 4

Comparison of the extinction efficiency obtained in the Rayleigh limit, by MLWA, MWLWA, interpolated LWA, and MWLWA with a nonuniform P against the exact T-matrix method results. The results are shown for a prolate silver spheroidal particle with the major to minor axis ratio of 4:1 and an equivalent-volume-sphere radius of 40 nm ( a = b 25.2 nm and c 100.8 nm ). Electric field is oriented along the rotational symmetry axis.

Fig. 5
Fig. 5

Comparison of the interpolated LWA (dashed-dotted-dotted curves) against the exact T-matrix method results (solid curves) for prolate silver spheroidal particles. From left to right, spheroids with the major to minor axis ratios increasing from 2:1 to 6:1 and with ( r ev ; a = b ; c ) ( 60 ; 47.6 ; 95.2 ) , (50; 34.7; 104), (40; 25.2; 100.8), (30; 17.5; 87.7), and ( 20 ; 11 ; 66 ) nm . Electric field is oriented along the rotational symmetry axis.

Equations (82)

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E int = E 0 + E d ,
E d = 4 π 3 ε h ( 1 x 2 i 2 3 x 3 ) P ,
4 π P = ε h ( ε 1 ) E int = ε h ( ε 1 ) ( E 0 + E d ) ,
α MW = ε 1 ε + 2 ( ε 1 ) x 2 i 2 x 3 3 ( ε 1 ) a 3 .
α MW = V 4 π ε 1 1 + L eff ( ε 1 ) ,
L eff = L 1 3 x 2 i 2 9 x 3 ,
L eff = L k 2 3 ( 3 V 4 π ) 2 3 i k 3 V 6 π ,
α MW = α R 1 k 2 a α R i 2 k 3 3 α R ,
α R = ε 1 ε + 2 a 3 .
α R = V 4 π ε 1 1 + L ( ε 1 ) ,
α Mie = i 3 2 k 3 T E 1 = ε 1 ε + 2 ( 6 ε 12 ) x 2 10 i 2 x 3 3 ( ε 1 ) a 3 ,
α MW = α R 1 k 2 l E D α R i 2 k 3 3 α R ,
E r = 2 cos θ ( [ p ] r 3 + [ p ̇ ] c r 2 ) ,
E θ = sin θ ( [ p ] r 3 + [ p ̇ ] c r 2 + [ p ̈ ] c 2 r ) .
[ p ̇ ] = i ω [ p ] , [ p ̈ ] = ω 2 [ p ] ,
E = E r cos θ E θ sin θ ,
E = E r sin θ + E θ cos θ ,
d E d , = [ 1 r 3 ( 3 cos 2 θ 1 ) + k 2 2 r ( cos 2 θ + 1 ) + i 2 3 k 3 ] P d V ,
d E d , = [ 3 sin θ cos θ r 3 + k 2 sin θ cos θ 2 r ] P d V .
d E d = { 3 r ̂ ( P r ̂ ) P r 3 + r ̂ ( P r ̂ ) + P 2 r k 2 + i 2 k 3 3 P } d V ,
E d ; 1 r 3 = 4 π L ¯ P ,
L ¯ = 1 4 π V d S r ̂ r 2 = V r ̂ d S r 2 ,
L z = { 1 e 2 e 3 ( e + arctanh e ) prolate 1 e 2 ( 1 1 e 2 e arcsin e ) oblate } ,
arctanh e = 1 2 ln 1 + e 1 e ,
e 2 = { c 2 a 2 c 2 prolate a 2 c 2 a 2 oblate } .
L x + L y + L z = 1 .
d E d ; 1 r 3 = 3 r ̂ ( P r ̂ ) P r 3 d V = [ G ¯ 0 ( r ) P ] d V ,
G ¯ 0 ( r ) = ( 1 r ) = 3 r ̂ r ̂ 1 r 3 ,
E d ; 1 r 3 = V [ G ¯ 0 ( r ) P ] d V .
E d ; 1 r 3 = lim δ 0 V V δ [ G ¯ 0 ( r ) P ] d V 4 π L ¯ P ,
V V δ [ G ¯ 0 ( r ) P ] d V = V V δ ( P r ) d V = ( V V δ ) ( P d S r ) 0 ,
E d ; 1 r 3 = 4 π L ¯ P ,
d E d ; 1 r 3 = 3 r ̂ ( P r ̂ ) P r 3 d V = d V ( P r r 3 ) .
E d ; 1 r 3 = Φ = V P r r 3 d V ,
E d ; 1 r 3 = V d S P r ,
E d ; 1 r 3 = ( V r ̂ d S r 2 ) P = 4 π L ¯ P ,
V d E d , ; 1 r 0 ,
V d E d , ; 1 r = k 2 V cos 2 θ + 1 2 r d V = k 2 V l E D z ,
D z = 3 4 × { 1 + e 2 1 e 2 L z + 1 prolate ( 1 2 e 2 ) L z + 1 oblate } .
2 c a D + D z = 3 × { 1 e arctanh e prolate 1 e 2 e arcsin e oblate } .
D = D z { 1 2 e 2 5 prolate 1 + 2 e 2 5 oblate } ,
D + 2 c a D 3 × { 1 + e 2 3 + e 4 5 + O ( z 6 ) prolate 1 e 2 3 2 e 4 15 + O ( e 6 ) oblate } .
x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 ,
x = a x , y = b y , z = c z ,
( x ) 2 + ( y ) 2 + ( z ) 2 = 1 .
r 2 = a 2 ( x ) 2 + b 2 ( y ) 2 + c 2 ( z ) 2 ,
r 2 = a 2 ρ 2 + c 2 ( z ) 2 = a 2 [ ρ 2 + ( c 2 a 2 ) ( z ) 2 ] ,
d V = a 2 c ρ d ρ d z d φ .
cos 2 θ + 1 = 1 r 2 ( x j 2 + r 2 ) ,
V cos 2 θ + 1 2 r d V = 2 π a c 0 1 d z 0 1 ( z ) 2 ρ 2 + 2 ( c 2 a 2 ) ( z ) 2 [ ρ 2 + ( c 2 a 2 ) ( z ) 2 ] 3 2 ρ d ρ ,
E d = 4 π ( L ¯ k 2 V 4 π D ¯ i 2 k 3 3 V 4 π 1 ) P ,
E d = k 2 lim δ 0 V V δ [ G ¯ ( r ) P ] d V 4 π L ¯ P
G ¯ ( r , r ) = G ¯ ( R ) = ( 1 + k 2 ) e i k R R ,
G ¯ ( r ) = 1 k 2 [ 3 r ̂ r ̂ 1 r 3 + r ̂ r ̂ + 1 2 r k 2 + i 2 k 3 3 ] + O ( k 2 ) ,
D ̃ = D ̃ = 1 .
P c = P ( 1 + 1 2 k 2 r 2 sin 2 θ ) ,
k 2 5 cos 2 θ 3 cos 4 θ 2 r P = 5 r ̂ 2 3 2 r r ̂ ( P r ̂ ) k 2 .
E d = 4 π 3 ε h ( 1 4 5 x 2 i 2 3 x 3 ) P ,
D z ; r n = 3 4 × { ( 5 e 2 3 ) ( 1 e 2 ) e 2 L z + 1 e 2 prolate 1 e 2 e 2 [ ( 2 e 2 + 3 ) L z 1 ] oblate } .
D a v = 0.37 + 0.63 D .
1 V V E 0 cos ( k r ) d V E 0 ( 1 x 2 10 ) .
α M W ; m = ε 1 ε + 2 ( 9 ε 12 ) x 2 10 i 2 x 3 3 ( ε 1 ) a 3 .
T A l = m [ x j l ( x ) ] j l ( x s ) j l ( x ) [ x s j l ( x s ) ] m [ x h l ( x ) ] j l ( x s ) h l ( x ) [ x s j l ( x s ) ] ,
T E 1 = i 2 x 3 3 ( ϵ 1 ) [ 1 ( ϵ + 1 ) x 2 10 ] ϵ + 2 ( ϵ 1 ) ( ϵ + 10 ) x 2 10 i 2 x 3 3 ( ϵ 1 )
T E 1 = i 2 x 3 3 ϵ 1 ϵ + 2 ( 6 ϵ 12 ) x 2 10 i 2 x 3 3 ( ϵ 1 ) ,
T E 1 = i 2 x 3 3 ( ϵ 1 ) ( 1 x 2 10 ) ϵ + 2 ( 7 ϵ 10 ) x 2 10 i 2 x 3 3 ( ϵ 1 )
ϵ 2 12 x 2 5
ϵ = 2 3 x 2 .
ϵ = 2 24 + ( ϵ ) 2 10 x 2 2 ϵ 3 x 3 6 7 x 4 .
σ sca 6 π k 2 T E 1 2 ,
σ abs 3 π 2 k 2 ( 1 1 + 2 T E 1 2 ) ,
σ tot 6 π k 2 Re T E 1 .
S S * = ( 1 + 2 T ) ( 1 + 2 T * ) = 1 + 4 Re T + 4 T 2 = 1 + 4 T 2 > 1 ,
0 S S * 1 .
1 4 T 2 Re T T 2 0
2 k 3 3 α 2 Im α 2 k 3 3 α 2 + 3 8 k 3 ,
I = V f d V ,
lim δ 0 V v f d V ,
V d V r β , 0 < β < 3 ,
cos 2 n θ 2 r d V = π a 2 { 1 e arctanh e n = 0 1 1 e 2 L z n = 1 1 e 2 [ L z 1 e 2 1 3 ] n = 2 } ,
cos 2 n θ 2 r d V = π a 2 { 1 e 2 e arcsin e n = 0 ( 1 e 2 ) L z n = 1 1 e 2 3 e 2 [ 1 3 ( 1 e 2 ) L z ] n = 2 } .
cos 2 n θ 2 r d V = π a 2 2 n + 1 .

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