Abstract

We present a theoretical study of electric field scattering by wavelength-sized spheroids. The incident, internal, and scattered fields are computed analytically by a spheroidal coordinate separation-of-variables solution, assuming axially incident monochromatic illumination. The main sources of possible numerical errors are identified and an additional point-matching procedure is implemented to provide a built-in test of the validity of the results. Numerical results were obtained for prolate and oblate particles with particular aspect ratios and sizes, and a refractive index of 1.33 relative to the surrounding medium. Special attention is paid to the characteristics of the near-field in close proximity to the spheroids. It is shown that particles with sizes close to the incident wavelength can produce high field enhancements whose spatial location and extension can be controlled by the particle geometry.

© 2010 Optical Society of America

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    [CrossRef]
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    [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).
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    [CrossRef]
  21. L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
    [CrossRef]
  22. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
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    [CrossRef]
  24. W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing (Cambridge Univ. Press, 1992).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. V. V. Somsikov and N. V. Voshchinnikov, “On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared,” Astron. Astrophys. 345, 315–320 (1999).
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    [CrossRef]
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    [CrossRef]

2009

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

L. Boyde, K. J. Chalut, and J. Guck, “Interaction of Gaussian beam with near-spherical particle: an analytic-numerical approach for assessing scattering and stresses,” J. Opt. Soc. Am. A 26, 1814–1826 (2009).
[CrossRef]

2008

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

2007

2006

V. G. Farafonov and V. B. Il’in, “Scattering of light by axially symmetric particles: modification of the point-matching method,” Opt. Spectrosc. 100, 437–447 (2006).
[CrossRef]

2005

2004

L. A. Blanco and F. J. García de Abajo, “Spontaneous emission enhancement near nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 89, 37–42 (2004).
[CrossRef]

2003

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wavefunctions,” J. Phys. A 36, 5477–5495 (2003).
[CrossRef]

2001

1999

V. V. Somsikov and N. V. Voshchinnikov, “On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared,” Astron. Astrophys. 345, 315–320 (1999).

1998

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

1996

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

1992

A. R. Sebak and B. P. Sinha, “Scattering by a conducting spheroidal object with dielectric coating at axial incidence,” IEEE Trans. Antennas Propag. 40, 268–274 (1992).
[CrossRef]

1990

K. S. Joo and M. F. Iskander, “A new procedure of point-matching method for calculating the absorption and scattering of lossy dielectric objects,” IEEE Trans. Antennas Propag. 38, 1483–1490 (1990).
[CrossRef]

1986

A. C. Ludwig, “A comparison of spherical wave boundary-value matching versus integral-equation scattering solutions for a perfectly conducting body,” IEEE Trans. Antennas Propag. 34, 857–865 (1986).
[CrossRef]

M. P. Cline, P. W. Barber, and R. K. Chang, “Surface-enhanced electric intensities on transition- and noble-metal spheroids,” J. Opt. Soc. Am. B 3, 15–21 (1986).
[CrossRef]

1984

M. Nishimura, S. Takamatsu, and H. Shigesawa, “A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process,” Electron. Commun. Jpn. 67, 75–81 (1984).
[CrossRef]

1982

J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

1977

B. P. Sinha and R. H. Macphie, “Electromagnetic scattering by prolate spheroids for plane-waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

1975

1974

J. A. Morrison and M. J. Cross, “Scattering of a plane electromagnetic-wave by axisymmetric raindrops,” Bell Syst. Tech. J. 53, 955–1019 (1974).

1973

T. Oguchi, “Attenuation and phase rotation of radio-waves due to rain—calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31–38 (1973).
[CrossRef]

1970

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

1908

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. (Berlin) 25, 377–445 (1908).
[CrossRef]

Abbott, P. C.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wavefunctions,” J. Phys. A 36, 5477–5495 (2003).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Asano, S.

Barber, P. W.

Barton, J. P.

Blanco, L. A.

L. A. Blanco and F. J. García de Abajo, “Spontaneous emission enhancement near nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 89, 37–42 (2004).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

Boyde, L.

Cai, X. S.

Chalut, K. J.

Chang, R. K.

Cline, M. P.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Cross, M. J.

J. A. Morrison and M. J. Cross, “Scattering of a plane electromagnetic-wave by axisymmetric raindrops,” Bell Syst. Tech. J. 53, 955–1019 (1974).

Dereux, A.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Falloon, P. E.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wavefunctions,” J. Phys. A 36, 5477–5495 (2003).
[CrossRef]

Farafonov, V. G.

V. G. Farafonov and V. B. Il’in, “Scattering of light by axially symmetric particles: modification of the point-matching method,” Opt. Spectrosc. 100, 437–447 (2006).
[CrossRef]

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, 1957).

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

French, R. H.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” in 99th Annual Meeting of the American-Ceramic-Society (American Ceramic Society, 1997), pp. 469–479.

García de Abajo, F. J.

L. A. Blanco and F. J. García de Abajo, “Spontaneous emission enhancement near nanoparticles,” J. Quant. Spectrosc. Radiat. Transf. 89, 37–42 (2004).
[CrossRef]

Girard, C.

C. Girard and A. Dereux, “Near-field optics theories,” Rep. Prog. Phys. 59, 657–699 (1996).
[CrossRef]

Gouesbet, G.

Grehan, G.

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and Atoms—Introduction to Quantum Electrodynamics (Wiley-Interscience, 1997).

Guck, J.

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).

Hodge, D. B.

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley-VCH, 2004).

Il’in, V. B.

V. G. Farafonov and V. B. Il’in, “Scattering of light by axially symmetric particles: modification of the point-matching method,” Opt. Spectrosc. 100, 437–447 (2006).
[CrossRef]

Iskander, M. F.

K. S. Joo and M. F. Iskander, “A new procedure of point-matching method for calculating the absorption and scattering of lossy dielectric objects,” IEEE Trans. Antennas Propag. 38, 1483–1490 (1990).
[CrossRef]

Joo, K. S.

K. S. Joo and M. F. Iskander, “A new procedure of point-matching method for calculating the absorption and scattering of lossy dielectric objects,” IEEE Trans. Antennas Propag. 38, 1483–1490 (1990).
[CrossRef]

Kang, X. -K.

L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Kattawar, G. W.

Keller, O.

O. Keller, “Near-field photon wave mechanics in the Lorenz gauge,” Phys. Rev. A 76, 062110 (2007).
[CrossRef]

Kooi, P. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

Leong, M. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

Leong, M. -S.

L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Li, C. H.

Li, L. W.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

Li, L. -W.

L.-W. Li, X.-K. Kang, and M.-S. Leong, Spheroidal Wave Functions in Electromagnetic Theory (Wiley, 2002).

Ludwig, A. C.

A. C. Ludwig, “A comparison of spherical wave boundary-value matching versus integral-equation scattering solutions for a perfectly conducting body,” IEEE Trans. Antennas Propag. 34, 857–865 (1986).
[CrossRef]

Luque, A.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

Macphie, R. H.

B. P. Sinha and R. H. Macphie, “Electromagnetic scattering by prolate spheroids for plane-waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

Marti, A.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

Mazeron, P.

J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

Mendes, M. J.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

Mie, G.

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. (Berlin) 25, 377–445 (1908).
[CrossRef]

Morrison, J. A.

J. A. Morrison and M. J. Cross, “Scattering of a plane electromagnetic-wave by axisymmetric raindrops,” Bell Syst. Tech. J. 53, 955–1019 (1974).

Nishimura, M.

M. Nishimura, S. Takamatsu, and H. Shigesawa, “A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process,” Electron. Commun. Jpn. 67, 75–81 (1984).
[CrossRef]

Novotny, L.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2006).

Oguchi, T.

T. Oguchi, “Attenuation and phase rotation of radio-waves due to rain—calculations at 19.3 and 34.8 GHz,” Radio Sci. 8, 31–38 (1973).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Ravey, J. C.

J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

Ren, K. F.

Sebak, A. R.

A. R. Sebak and B. P. Sinha, “Scattering by a conducting spheroidal object with dielectric coating at axial incidence,” IEEE Trans. Antennas Propag. 40, 268–274 (1992).
[CrossRef]

Shigesawa, H.

M. Nishimura, S. Takamatsu, and H. Shigesawa, “A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process,” Electron. Commun. Jpn. 67, 75–81 (1984).
[CrossRef]

Sinha, B. P.

A. R. Sebak and B. P. Sinha, “Scattering by a conducting spheroidal object with dielectric coating at axial incidence,” IEEE Trans. Antennas Propag. 40, 268–274 (1992).
[CrossRef]

B. P. Sinha and R. H. Macphie, “Electromagnetic scattering by prolate spheroids for plane-waves with arbitrary polarization and angle of incidence,” Radio Sci. 12, 171–184 (1977).
[CrossRef]

Somsikov, V. V.

V. V. Somsikov and N. V. Voshchinnikov, “On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared,” Astron. Astrophys. 345, 315–320 (1999).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Takamatsu, S.

M. Nishimura, S. Takamatsu, and H. Shigesawa, “A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process,” Electron. Commun. Jpn. 67, 75–81 (1984).
[CrossRef]

Tan, K. Y.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Thiele, E. S.

E. S. Thiele and R. H. French, “Light-scattering properties of representative, morphological rutile titania particles studied using a finite-element method,” in 99th Annual Meeting of the American-Ceramic-Society (American Ceramic Society, 1997), pp. 469–479.

Tobias, I.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in FORTRAN 77: The Art of Scientific Computing (Cambridge Univ. Press, 1992).

Voshchinnikov, N. V.

V. V. Somsikov and N. V. Voshchinnikov, “On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared,” Astron. Astrophys. 345, 315–320 (1999).

Wang, J. B.

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wavefunctions,” J. Phys. A 36, 5477–5495 (2003).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th ed. (Cambridge Univ. Press, 1999).
[PubMed]

Xu, F.

Yamamoto, G.

Yang, P.

Yeo, T. S.

L. W. Li, M. S. Leong, T. S. Yeo, P. S. Kooi, and K. Y. Tan, “Computations of spheroidal harmonics with complex arguments: a review with an algorithm,” Phys. Rev. E 58, 6792–6806 (1998).
[CrossRef]

Zhai, P. W.

Ann. Phys. (Berlin)

G. Mie, “Articles on the optical characteristics of turbid tubes, especially colloidal metal solutions,” Ann. Phys. (Berlin) 25, 377–445 (1908).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

M. J. Mendes, A. Luque, I. Tobias, and A. Marti, “Plasmonic light enhancement in the near-field of metallic nanospheroids for application in intermediate band solar cells,” Appl. Phys. Lett. 95, 071105 (2009).
[CrossRef]

Astron. Astrophys.

V. V. Somsikov and N. V. Voshchinnikov, “On the applicability of the Rayleigh approximation for coated spheroids in the near-infrared,” Astron. Astrophys. 345, 315–320 (1999).

Bell Syst. Tech. J.

J. A. Morrison and M. J. Cross, “Scattering of a plane electromagnetic-wave by axisymmetric raindrops,” Bell Syst. Tech. J. 53, 955–1019 (1974).

Electron. Commun. Jpn.

M. Nishimura, S. Takamatsu, and H. Shigesawa, “A numerical analysis of electromagnetic scattering of perfect conducting cylinders by means of discrete singularity method improved by optimization process,” Electron. Commun. Jpn. 67, 75–81 (1984).
[CrossRef]

IEEE Trans. Antennas Propag.

A. C. Ludwig, “A comparison of spherical wave boundary-value matching versus integral-equation scattering solutions for a perfectly conducting body,” IEEE Trans. Antennas Propag. 34, 857–865 (1986).
[CrossRef]

K. S. Joo and M. F. Iskander, “A new procedure of point-matching method for calculating the absorption and scattering of lossy dielectric objects,” IEEE Trans. Antennas Propag. 38, 1483–1490 (1990).
[CrossRef]

A. R. Sebak and B. P. Sinha, “Scattering by a conducting spheroidal object with dielectric coating at axial incidence,” IEEE Trans. Antennas Propag. 40, 268–274 (1992).
[CrossRef]

J. Appl. Phys.

A. Luque, A. Marti, M. J. Mendes, and I. Tobias, “Light absorption in the near field around surface plasmon polaritons,” J. Appl. Phys. 104, 113118 (2008).
[CrossRef]

J. Math. Phys.

D. B. Hodge, “Eigenvalues and eigenfunctions of spheroidal wave equation,” J. Math. Phys. 11, 2308–2312 (1970).
[CrossRef]

J. Opt. (Paris)

J. C. Ravey and P. Mazeron, “Light-scattering in the physical optics approximation—application to large spheroids,” J. Opt. (Paris) 13, 273–282 (1982).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. A

P. E. Falloon, P. C. Abbott, and J. B. Wang, “Theory and computation of spheroidal wavefunctions,” J. Phys. A 36, 5477–5495 (2003).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

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Figures (10)

Fig. 1
Fig. 1

Coordinate system for scattering by a spheroid. The particles considered in this work are dielectric with an index of refraction of n r = 1.33 relative to the surrounding medium. The surfaces ξ = const are spheroids confocal with the particle, and ξ 0 coincides with the particle surface. The direction of illumination ( K 0 ) is collinear with the spheroid axis of symmetry ( z ) . A scattered wave in the z y plane is shown, making an angle θ with the negative z axis. It propagates along K S and its electric field ( E S ) can have a component longitudinal ( E S r ) and transverse ( E S θ ) relative to the propagation direction.

Fig. 2
Fig. 2

Computational time to calculate the angular harmonics S m n ( C , η ) and S = d S / d η , with m = n = 0 , η = 0.5 , and C = x + x i , as a function of x. The calculations were performed using the Mathematica 7.0 built-in routine and the routine used in this work. For all x values both codes give the same results for S and S up to the 14th decimal place. The inset shows the results for x = 50 , which match those presented in [3].

Fig. 3
Fig. 3

Electric field distribution in the y z plane of a sphere with n r = 1.33 . Left—Total field ( E t ) magnitude squared. For spheres the size parameter is C = 2 π R / λ , where R is the radius. The white circle is at the spot of maximum intensity. Center—Squared magnitude of the longitudinal component ( E t r ) . The polar plot depicts the ES potential magnitude ( | V | ) at the particle surface ( r = R ) . Right—Squared magnitude of the transverse component ( E t θ ) . The polar plot shows the far-field intensity function I θ [Eq. (14)]. In the plots of | V | and I θ the circular dots correspond to the values obtained with the ES (or Rayleigh) approximation [8, 14].

Fig. 4
Fig. 4

Electric field distributions in the y z plane of a n r = 1.33 oblate spheroid with size parameter C = 2 and aspect ratio b / a = 2 . In the I θ curve the circular dots correspond to the values obtained by Asano and Yamamoto [2], shown for comparison.

Fig. 5
Fig. 5

Same as Fig. 4 but for a prolate spheroid with aspect ratio a / b = 2 .

Fig. 6
Fig. 6

Electric field distributions in the y z plane of a n r = 1.33 oblate with size parameter C = 7 and aspect ratio b / a = 2 . In the I θ curves the circular dots correspond to the values obtained in [2].

Fig. 7
Fig. 7

Same as Fig. 6 but for a prolate with aspect ratio a / b = 2 .

Fig. 8
Fig. 8

Electric field distribution in the y z plane of a n r = 1.33 spheroid with size parameter C = 14.6 . This is an elongated prolate with aspect ratio a / b = 5 and the same R e q as the prolate in Fig. 7.

Fig. 9
Fig. 9

Angle θ (in degrees) between the incidence direction and the position of maximum field enhancement. In the plots of Figs. 3, 4, 5, 6, 7, 8 this is the angle between the white dashed line and the negative z axis. The schematic E S and K S vectors represent the polarization of the scattered field at the spot of maximum field intensity. From the dipole to the FS regime the polarization goes from being purely longitudinal to transverse, passing by a transition regime where it has both longitudinal and transverse components.

Fig. 10
Fig. 10

| E t | 2 profiles along the radial direction of maximum field ( θ MAX ) , represented by the white dashed line in the plots of Section 3, for particles with size close to λ. r is the distance to the particle center, in units of λ. The spheroids with aspect ratio 2 correspond to those in Figs. 6, 7. The prolate with a / b = 5 corresponds to Fig. 8. The oblate with b / a = 5 has the same R e q as the oblate with b / a = 2 . The gray crossed circles represent the intersection with the particle surface. The inset shows similar | E t | 2 profiles for spheres with radius R = λ but distinct refractive indices n r , computed with Mie theory. The curve for n r = 1.33 is compared with that (circular dots) calculated with the method described in this work using an almost spherical prolate ( a / b = 1.001 ) .

Equations (34)

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ψ m o e n ( i ) ( C ; η , ξ , ϕ ) = S m n ( C , η ) R m n ( i ) ( C , ξ ) × { cos ( m ϕ ) sin ( m ϕ ) , }
C = K d 2 ,
E 0 = 1 K 0 n = 0 A n M e 0 n x ( 1 ) ( C 0 ; η , ξ , ϕ ) ,     H 0 = i K 0 Z 0 n = 0 A n N e 0 n x ( 1 ) ( C 0 ; η , ξ , ϕ ) ,
E S = n = 0 [ α n M e 0 n + ( 4 ) ( C 0 ; η , ξ , ϕ ) + β n + 1 M e 1 , n + 1 z ( 4 ) ( C 0 ; η , ξ , ϕ ) ] ,
H S = i Z 0 n = 0 [ α n N e 0 n + ( 4 ) ( C 0 ; η , ξ , ϕ ) + β n + 1 N e 1 , n + 1 z ( 4 ) ( C 0 ; η , ξ , ϕ ) ] ,
M e 0 n + ( i ) ( C ; η , ξ , ϕ ) = 1 2 M e 0 n x ( i ) ( C ; η , ξ , ϕ )
N e 0 n + ( i ) ( C ; η , ξ , ϕ ) = 1 2 N e 0 n x ( i ) ( C ; η , ξ , ϕ ) .
E I = n = 0 [ γ n M e 0 n + ( 1 ) ( C I ; η , ξ , ϕ ) + δ n + 1 M e 1 , n + 1 z ( 1 ) ( C I ; η , ξ , ϕ ) ] ,
H I = i Z I n = 0 [ γ n N e 0 n + ( 1 ) ( C I ; η , ξ , ϕ ) + δ n + 1 N e 1 , n + 1 z ( 1 ) ( C I ; η , ξ , ϕ ) ] ,
N = integer ( K 0 α + 4 ) ,
Q A = R X ,
X = R 1 Q A .
A X = B ,
A ( X correct + X error ) = B + d B ,
A X error = d B .
X correct = X X error .
2 π R e q / λ 1 ,     | n r | 2 π R e q / λ 1 ,
I θ ( θ ) = | n = 0 i n π α n S 0 , n ( C 0 , cos   θ ) | 2 .
x = d 2 ( 1 η 2 ) ( ξ 2 ± 1 )   cos   ϕ ,
y = d 2 ( 1 η 2 ) ( ξ 2 ± 1 )   sin   ϕ ,
z = d 2 η ξ ,
1 η 1 ,     ε ξ < ,     0 ϕ < 2 π ,
M e 0 n , η x ( i ) = 2 ( ξ 2 ± 1 ) 1 / 2 d ( ξ 2 ± η 2 ) 1 / 2 S 0 n d d ξ R 0 n ( i )   sin   ϕ ,
M e 0 n , ξ x ( i ) = 2 ( 1 η 2 ) 1 / 2 d ( ξ 2 ± η 2 ) 1 / 2 d d η S 0 n R 0 n ( i )   sin   ϕ ,
M e 0 n , ϕ x ( i ) = 2 d ( ξ 2 ± η 2 ) [ ξ ( 1 η 2 ) d d η S 0 n R 0 n ( i ) + η ( ξ 2 ± 1 ) S 0 n d d ξ R 0 n ( i ) ] cos   ϕ .
N e 0 n , η x ( i ) = 4 K d 2 ( ξ 2 ± η 2 ) 1 / 2 [ η S 0 n ξ ( ( ξ 2 ± 1 ) 3 / 2 ( ξ 2 ± η 2 ) d d ξ R 0 n ( i ) ) 1 ( ξ 2 ± 1 ) 1 / 2 d d η S 0 n R 0 n ( i ) + ( 1 η 2 ) d d η S 0 n ξ ( ξ ( ξ 2 ± 1 ) 1 / 2 R 0 n ( i ) ξ 2 ± η 2 ) ] cos   ϕ ,
N e 0 n , ξ x ( i ) = 4 K d 2 ( ξ 2 ± η 2 ) 1 / 2 [ ξ η ( ( 1 η 2 ) 3 / 2 ( ξ 2 ± η 2 ) d d η S 0 n ) R 0 n ( i ) + 1 ( 1 η 2 ) 1 / 2 S 0 n d d ξ R 0 n ( i ) + ( ξ 2 ± 1 ) η ( η ( 1 η 2 ) 1 / 2 S 0 n ξ 2 ± η 2 ) d d ξ R 0 n ( i ) ] cos   ϕ ,
N e 0 n , ϕ x ( i ) = 4 ( 1 η 2 ) 1 / 2 ( ξ 2 ± 1 ) 1 / 2 K d 2 ( ξ 2 ± η 2 ) [ 1 ( ξ 2 ± 1 ) 1 / 2 η ( ( 1 η 2 ) 1 / 2 d d η S 0 n ) R 0 n ( i ) + 1 ( 1 η 2 ) 1 / 2 S 0 n ξ ( ( ξ 2 ± 1 ) 1 / 2 d d ξ R 0 n ( i ) ) ] sin   ϕ .
M e 1 n , η z ( i ) = 2 η d ( ξ 2 ± η 2 ) 1 / 2 ( 1 η 2 ) 1 / 2 S 1 n R 1 n ( i )   sin   ϕ ,
M e 1 n , ξ z ( i ) = 2 ξ d ( ξ 2 ± η 2 ) 1 / 2 ( ξ 2 ± 1 ) 1 / 2 S 1 n R 1 n ( i )   sin   ϕ ,
M e 1 n , ϕ z ( i ) = 2 ( ξ 2 ± 1 ) 1 / 2 ( 1 η 2 ) 1 / 2 d ( ξ 2 ± η 2 ) [ η d d η S 1 n R 1 n ( i ) ξ S 1 n d d ξ R 1 n ( i ) ] cos   ϕ .
N e 1 n , η z ( i ) = 4 ( 1 η 2 ) K d 2 ( ξ 2 ± η 2 ) 1 / 2 [ η d d η S 1 n ξ ( ξ 2 ± 1 ξ 2 ± η 2 R 1 n ( i ) ) S 1 n ξ ( ξ ( ξ 2 ± 1 ) ξ 2 ± η 2 d d ξ R 1 n ( i ) ) + ξ ( 1 η 2 ) ( ξ 2 ± 1 ) S 1 n R 1 n ( i ) ] cos   ϕ ,
N e 1 n , ξ z ( i ) = 4 ( ξ 2 ± 1 ) 1 / 2 K d 2 ( ξ 2 ± η 2 ) 1 / 2 [ ξ η ( 1 η 2 ξ 2 ± η 2 S 1 n ) d d ξ R 1 n ( i ) η ( η ( 1 η 2 ) ξ 2 ± η 2 d d η S 1 n ) R 1 n ( i ) + η ( 1 η 2 ) ( ξ 2 ± 1 ) S 1 n R 1 n ( i ) ] cos   ϕ ,
N e 1 n , φ z ( i ) = 4 ( 1 η 2 ) 1 / 2 ( ξ 2 ± 1 ) 1 / 2 K d 2 ( ξ 2 ± η 2 ) [ ξ ξ 2 ± 1 d d η S 1 n R 1 n ( i ) + η 1 η 2 S 1 n d d ξ R 1 n ( i ) ] sin   ϕ .

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