Abstract

Light scattering of individual coated nonmagnetic spherical particles and effective parameters for a collection of such inclusions are studied at terahertz frequencies, with the emphasis on the conditions of achieving resonant light scatterings, induced magnetic resonance, and negative refraction. Moreover, the prediciton of those critical conditions is proposed. Different core–shell combinations aiming at inducing magnetic resonances are investigated, including plasmonic metamaterials and polar crystals. It is shown that the resonant scattering is controllable with fine adjustment of the core–shell ratio, varying from enhancement to suppression in the scattering drastically. Embedding identical coated nanospheres in a matrix, the polarizability and effective medium parameters of the bulk are examined to give better understanding of unusual scatterings and their predictions. The resonances in electric and magnetic dipole approximation for such a bulk medium are presented.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (3)

B. S. Luk'yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92, 773-776 (2008).

C. W. Qiu and B. S. Luk'yanchuk, “Peculiarities in light scattering of spherical particles with radial anisotropy,” J. Opt. Soc. Am. A 25, 1623-1628 (2008).
[CrossRef]

C. W. Qiu, S. Zouhdi, and Y. L. Geng, “Shifted resonances in coated metamaterial cylinders: enhanced backscattering and near-field effects,” Phys. Rev. E 77, 046604 (2008).
[CrossRef]

2007 (5)

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[CrossRef]

2006 (9)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

M. I. Tribelsky and B. S. Luk'yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[CrossRef]

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

B. S. Luk'yanchuk and V. Ternovsky, “Light scattering by a thin wire with a surface-plasmon resonance: bifurcations of the Poynting vector field,” Phys. Rev. B 73, 235432 (2006).
[CrossRef]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

2005 (1)

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717-3714 (2005).
[CrossRef]

2004 (1)

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

2003 (3)

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

2002 (2)

S. B. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720-5728 (2002).
[CrossRef]

2000 (1)

1998 (1)

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

1996 (2)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

1994 (1)

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

1989 (1)

A. H. Sihvola and I. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagn. Waves Appl. 3, 37-60 (1989).
[CrossRef]

1987 (1)

A. H. Sihvola, Effective Permittivity of Multiphase Mixtures: The Scatterer as a Two-Layer Sphere, Rep. 19 (Helsinki University of Technology, 1987).

1984 (1)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics, Vol. 8 (Pergamon, 1984).

1983 (1)

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

1982 (1)

A.D.Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982), Chap. 9.

1980 (1)

D. A. Weitz, T. J. Gramila, and A. Z. Genack, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of surface-enhanced Raman scattering,” Phys. Rev. Lett. 45, 355-358 (1980).
[CrossRef]

1975 (1)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

1974 (1)

G. Burns and E. Burstein, “Dirty displacive ferroelectrics,” Ferroelectrics 7, 297-299 (1974).
[CrossRef]

1954 (1)

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, 1954).

1941 (1)

R. H. Lyddane, R. G. Sachs, and E. Teller, “On the polar vibrations of alkali halides,” Phys. Rev. 59, 673-676 (1941).
[CrossRef]

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432-457 (1937).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Alù, A.

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Bergman, D. J.

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

Bohren, C. F.

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

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

Born, M.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, 1954).

Burns, G.

G. Burns and E. Burstein, “Dirty displacive ferroelectrics,” Ferroelectrics 7, 297-299 (1974).
[CrossRef]

Burokur, S. N.

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

Burstein, E.

G. Burns and E. Burstein, “Dirty displacive ferroelectrics,” Ferroelectrics 7, 297-299 (1974).
[CrossRef]

Chang, C. C.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007).
[CrossRef]

Chern, R. L.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007).
[CrossRef]

Chong, T. C.

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432-457 (1937).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Engheta, N.

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

Fan, S.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Genack, A. Z.

D. A. Weitz, T. J. Gramila, and A. Z. Genack, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of surface-enhanced Raman scattering,” Phys. Rev. Lett. 45, 355-358 (1980).
[CrossRef]

Geng, Y. L.

C. W. Qiu, S. Zouhdi, and Y. L. Geng, “Shifted resonances in coated metamaterial cylinders: enhanced backscattering and near-field effects,” Phys. Rev. E 77, 046604 (2008).
[CrossRef]

Gramila, T. J.

D. A. Weitz, T. J. Gramila, and A. Z. Genack, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of surface-enhanced Raman scattering,” Phys. Rev. Lett. 45, 355-358 (1980).
[CrossRef]

Hendi, A.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

Henn, J.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

Holden, A. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Holloway, C. L.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Hong, M. H.

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Huang, K.

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, 1954).

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, 2nd ed. (Wiley, 1975).

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Kabos, P.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

Kuester, E. F.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics, Vol. 8 (Pergamon, 1984).

Leonhardt, U.

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

Li, K.

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

Li, L. W.

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics, Vol. 8 (Pergamon, 1984).

Lin, Y.

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Lindell, I. V.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

A. H. Sihvola and I. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagn. Waves Appl. 3, 37-60 (1989).
[CrossRef]

Liu, X. X.

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007).
[CrossRef]

Luk'yanchuk, B. S.

C. W. Qiu and B. S. Luk'yanchuk, “Peculiarities in light scattering of spherical particles with radial anisotropy,” J. Opt. Soc. Am. A 25, 1623-1628 (2008).
[CrossRef]

B. S. Luk'yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92, 773-776 (2008).

M. I. Tribelsky and B. S. Luk'yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[CrossRef]

B. S. Luk'yanchuk and V. Ternovsky, “Light scattering by a thin wire with a surface-plasmon resonance: bifurcations of the Poynting vector field,” Phys. Rev. B 73, 235432 (2006).
[CrossRef]

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Lyddane, R. H.

R. H. Lyddane, R. G. Sachs, and E. Teller, “On the polar vibrations of alkali halides,” Phys. Rev. 59, 673-676 (1941).
[CrossRef]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717-3714 (2005).
[CrossRef]

O'Brien, S. B.

S. B. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

Ouchetto, O.

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

Papernov, S.

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720-5728 (2002).
[CrossRef]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

S. B. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics, Vol. 8 (Pergamon, 1984).

Popa, B. I.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

Qiu, C. W.

B. S. Luk'yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92, 773-776 (2008).

C. W. Qiu and B. S. Luk'yanchuk, “Peculiarities in light scattering of spherical particles with radial anisotropy,” J. Opt. Soc. Am. A 25, 1623-1628 (2008).
[CrossRef]

C. W. Qiu, S. Zouhdi, and Y. L. Geng, “Shifted resonances in coated metamaterial cylinders: enhanced backscattering and near-field effects,” Phys. Rev. E 77, 046604 (2008).
[CrossRef]

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

Razek, A.

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

Sachs, R. G.

R. H. Lyddane, R. G. Sachs, and E. Teller, “On the polar vibrations of alkali halides,” Phys. Rev. 59, 673-676 (1941).
[CrossRef]

Schmid, A. W.

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720-5728 (2002).
[CrossRef]

Schurig, D.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Shin, H.

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

Sihvola, A. H.

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

A. H. Sihvola and I. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagn. Waves Appl. 3, 37-60 (1989).
[CrossRef]

A. H. Sihvola, Effective Permittivity of Multiphase Mixtures: The Scatterer as a Two-Layer Sphere, Rep. 19 (Helsinki University of Technology, 1987).

Silveirinha, M. G.

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[CrossRef]

Smith, D. R.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Stockman, M. I.

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

Teller, E.

R. H. Lyddane, R. G. Sachs, and E. Teller, “On the polar vibrations of alkali halides,” Phys. Rev. 59, 673-676 (1941).
[CrossRef]

Ternovsky, V.

B. S. Luk'yanchuk and V. Ternovsky, “Light scattering by a thin wire with a surface-plasmon resonance: bifurcations of the Poynting vector field,” Phys. Rev. B 73, 235432 (2006).
[CrossRef]

Tretyakov, S. A.

S. A. Tretyakov and A. J. Viitanen, “Plane waves in regular arrays of dipole scatterers and effective-medium modeling,” J. Opt. Soc. Am. A 17, 1791-1797 (2000).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Tribelsky, M. I.

M. I. Tribelsky and B. S. Luk'yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[CrossRef]

Viitanen, A. J.

S. A. Tretyakov and A. J. Viitanen, “Plane waves in regular arrays of dipole scatterers and effective-medium modeling,” J. Opt. Soc. Am. A 17, 1791-1797 (2000).
[CrossRef]

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

Wang, Z. B.

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Weitz, D. A.

D. A. Weitz, T. J. Gramila, and A. Z. Genack, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of surface-enhanced Raman scattering,” Phys. Rev. Lett. 45, 355-358 (1980).
[CrossRef]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

Yannopapas, V.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717-3714 (2005).
[CrossRef]

Yao, H. Y.

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

Yeo, T. S.

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Zouhdi, S.

C. W. Qiu, S. Zouhdi, and Y. L. Geng, “Shifted resonances in coated metamaterial cylinders: enhanced backscattering and near-field effects,” Phys. Rev. E 77, 046604 (2008).
[CrossRef]

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

Am. J. Phys. (1)

C. F. Bohren, “How can a particle absorb more than the light incident on it?” Am. J. Phys. 51, 323-327 (1983).
[CrossRef]

Appl. Phys. A (1)

B. S. Luk'yanchuk and C. W. Qiu, “Enhanced scattering efficiencies in spherical particles with weakly dissipating anisotropic materials,” Appl. Phys. A 92, 773-776 (2008).

Ferroelectrics (1)

G. Burns and E. Burstein, “Dirty displacive ferroelectrics,” Ferroelectrics 7, 297-299 (1974).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antennas Propag. 51, 2596-2603 (2003).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558-2571 (2003).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

O. Ouchetto, C. W. Qiu, S. Zouhdi, L. W. Li, and A. Razek, “Homogenization of 3D periodic bianisotropic metamaterials,” IEEE Trans. Microwave Theory Tech. 54, 3893-3898 (2006).
[CrossRef]

IEICE Trans. Commun. (1)

C. W. Qiu, H. Y. Yao, S. N. Burokur, S. Zouhdi, and L. W. Li, “Electromagnetic scattering properties in a multilayered metamaterial cylinder,” IEICE Trans. Commun. E90-B, 2423-2429 (2007).
[CrossRef]

J. Appl. Phys. (1)

S. Papernov and A. W. Schmid, “Correlations between embedded single gold nanoparticles in SiO2 thin film and nanoscale crater formation induced by pulsed-laser radiation,” J. Appl. Phys. 92, 5720-5728 (2002).
[CrossRef]

J. Electromagn. Waves Appl. (1)

A. H. Sihvola and I. V. Lindell, “Polarizability and effective permittivity of layered and continuously inhomogeneous dielectric spheres,” J. Electromagn. Waves Appl. 3, 37-60 (1989).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys.: Condens. Matter (2)

S. B. O'Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys.: Condens. Matter 14, 4035-4044 (2002).
[CrossRef]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717-3714 (2005).
[CrossRef]

Phys. Rev. (1)

R. H. Lyddane, R. G. Sachs, and E. Teller, “On the polar vibrations of alkali halides,” Phys. Rev. 59, 673-676 (1941).
[CrossRef]

Phys. Rev. B (5)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Backward waves in magnetoelectrically chiral media: propagation, impedance, and negative refraction,” Phys. Rev. B 75, 155120 (2007).
[CrossRef]

C. W. Qiu, H. Y. Yao, L. W. Li, S. Zouhdi, and T. S. Yeo, “Routes to left-handed materials by magnetoelectric couplings,” Phys. Rev. B 75, 245214 (2007).
[CrossRef]

B. S. Luk'yanchuk and V. Ternovsky, “Light scattering by a thin wire with a surface-plasmon resonance: bifurcations of the Poynting vector field,” Phys. Rev. B 73, 235432 (2006).
[CrossRef]

Z. B. Wang, B. S. Luk'yanchuk, M. H. Hong, Y. Lin, and T. C. Chong, “Energy flow around a small particle investigated by classical Mie theory,” Phys. Rev. B 70, 035418 (2004).
[CrossRef]

Phys. Rev. E (4)

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. B. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 036621 (2006).
[CrossRef]

M. G. Silveirinha, A. Alù, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. E 75, 036603 (2007).
[CrossRef]

C. W. Qiu, S. Zouhdi, and Y. L. Geng, “Shifted resonances in coated metamaterial cylinders: enhanced backscattering and near-field effects,” Phys. Rev. E 77, 046604 (2008).
[CrossRef]

R. L. Chern, X. X. Liu, and C. C. Chang, “Particle plasmons of metal nanospheres: application of multiple scattering approach,” Phys. Rev. E 76, 016609 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

H. Shin and S. Fan, “All-angle negative refraction for surface plasmon waves using a metal-dielectric-metal structure,” Phys. Rev. Lett. 96, 073907 (2006).
[CrossRef] [PubMed]

A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. 97, 073902 (2006).
[CrossRef] [PubMed]

M. I. Tribelsky and B. S. Luk'yanchuk, “Anomalous light scattering by small particles,” Phys. Rev. Lett. 97, 263902 (2006).
[CrossRef]

D. A. Weitz, T. J. Gramila, and A. Z. Genack, “Anomalous low-frequency Raman scattering from rough metal surfaces and the origin of surface-enhanced Raman scattering,” Phys. Rev. Lett. 45, 355-358 (1980).
[CrossRef]

K. Li, M. I. Stockman, and D. J. Bergman, “Self-similar chain of metal nanospheres as an efficient nanolens,” Phys. Rev. Lett. 91, 227402 (2003).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432-457 (1937).
[CrossRef]

Science (2)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977-980 (2006).
[CrossRef] [PubMed]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

Other (8)

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

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, 1975).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd ed., Course of Theoretical Physics, Vol. 8 (Pergamon, 1984).

A.D.Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982), Chap. 9.

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

I. V. Lindell, A. H. Sihvola, S. A. Tretyakov, and A. J. Viitanen, Electromagnetic Waves in Chiral and Bi-Isotropic Media (Artech House, 1994).

A. H. Sihvola, Effective Permittivity of Multiphase Mixtures: The Scatterer as a Two-Layer Sphere, Rep. 19 (Helsinki University of Technology, 1987).

M. Born and K. Huang, Dynamical Theory of Crystal Lattices (Clarendon, 1954).

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Figures (10)

Fig. 1
Fig. 1

Geometry for plane wave scattering by the conventional dielectric sphere covered by a plasmonic shell, which can be interpreted by dipole moments. The permeabilities in all regions are assumed to be identical to that of free space, and the permittivities are relative values.

Fig. 2
Fig. 2

Constructive scattering of a plasmonic nanosphere versus its permittivity when the materials in the core and shell are identical, i.e., n 2 = n 3 . Solid curve, b = λ 100 ; dashed curve, b = λ 10 . The inset illustrates the variation of scattering in a fine region in the vicinity of plasmon resonance.

Fig. 3
Fig. 3

Constructive scattering dominated by 2 m multipoles for the plasmonic coated nanosphere when the core sphere has ϵ 2 = 4 and the plasmonic shell has ϵ 1 = 3 . The outer radius is fixed at b = 0.01 λ .

Fig. 4
Fig. 4

Constructive scattering dominated by 2 m multipoles for the plasmonic coated nanosphere when the core sphere has ϵ 2 = 4 and the plasmonic shell has ϵ 1 = 3 . The outer radius is fixed at b = 0.1 λ .

Fig. 5
Fig. 5

Normalized backscattered cross section with the inset to show the enhancement and suppression for the coated plasmonic nanosphere. The parameters in the core and shell are ϵ 2 = 4 and ϵ 1 = 3 , respectively.

Fig. 6
Fig. 6

Drude materials in the coating with different dielectric cores at the ratio a b = 0.5 . The outer radius b is 10 μ m , and the plasmon frequency ω p = 3 THz .

Fig. 7
Fig. 7

Three-dimensional plot of the backscattering of the dielectric core ϵ 2 = 4 coated by a Drude material whose outer radius b is 2 μ m and the plasmon frequency is ω p = 30 THz .

Fig. 8
Fig. 8

Backscattering versus radii ratio for the coated sphere in which the permittivity of the core is markedly larger than that of the shell. Both layers are dielectrics and the outer radius is b = 0.01 λ .

Fig. 9
Fig. 9

Effective parameters of the coated spheres with a Drude shell. b = 2 μ m , a = 1.6 μ m , and ω p = 30 THz . Solid (dashed) curve corresponds to the real (imaginary) part. The filling fraction is 0.5, and the dimensionless damping factor is 0.01.

Fig. 10
Fig. 10

Effective refractive index of a class of TlCl spheres coated by Drude materials. The outer radius is b = 5.2 μ m , and ω p = 14.4 THz . The filling fraction is 0.5, and the dimensionless damping for both TlCl and Drude materials is 0.01.

Equations (18)

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a m = ψ m ( y ) [ ψ m ( n s y ) A m χ m ( n s y ) ] n s ψ m ( y ) [ ψ m ( n s y ) A m χ m ( n s y ) ] ξ m ( y ) [ ψ m ( n s y ) A m χ m ( n s y ) ] n s ξ m ( y ) [ ψ m ( n s y ) A m χ m ( n s y ) ] ,
b m = n s ψ m ( y ) [ ψ m ( n s y ) B m χ m ( n s y ) ] ψ m ( y ) [ ψ m ( n s y ) B m χ m ( n s y ) ] n s ξ m ( y ) [ ψ m ( n s y ) B m χ m ( n s y ) ] ξ m ( y ) [ ψ m ( n s y ) B m χ m ( n s y ) ] ,
A m = n s ψ m ( n s x ) ψ m ( n x ) n ψ m ( n s x ) ψ m ( n x ) n s χ m ( n s x ) ψ m ( n x ) n χ m ( n s x ) ψ m ( n x ) ,
B m = n s ψ m ( n x ) ψ m ( n s x ) n ψ m ( n x ) ψ m ( n s x ) n s ψ m ( n x ) χ m ( n s x ) n ψ m ( n x ) χ m ( n s x ) ,
Q sca = 2 π k 0 2 m = 1 ( 2 m + 1 ) ( a m 2 + b m 2 ) ,
σ = π k 0 2 m = 1 ( 1 ) m ( 2 m + 1 ) ( a m b m ) 2 .
α e = 4 π b 3 Δ { ( ϵ 1 1 ) ( ϵ 2 + 2 ϵ 1 ) ( a b ) 3 [ 2 ϵ 1 2 + ϵ 1 ( 1 2 ϵ 2 ) ϵ 2 ] } ,
α m = 4 π b 3 Δ { ϵ 1 ( 1 ϵ 1 ) ( a b ) 3 } ,
Δ = ( ϵ 1 + 2 ) ( ϵ 2 + 2 ϵ 1 ) 2 ( a b ) 3 ( ϵ 1 ϵ 2 ) ( ϵ 1 1 ) .
a b = ( ϵ 1 1 ) ( ϵ 2 + 2 ϵ 1 ) 2 ϵ 1 2 + ϵ 1 ( 1 2 ϵ 2 ) ϵ 2 3 ,
a b ( ϵ 1 + 2 ) ( ϵ 2 + 2 ϵ 1 ) 2 ( ϵ 1 ϵ 2 ) ( ϵ 1 1 ) 3 ,
ϵ Drude = 1 ω p 2 ω ( ω + i γ ) ,
a 1 = i 2 3 ( k 0 b ) 3 ( ϵ 1 1 ) ( ϵ 2 + 2 ϵ 1 ) ( a b ) 3 [ 2 ϵ 1 2 + ϵ 1 ( 1 2 ϵ 2 ) ϵ 2 ] ( ϵ 1 + 2 ) ( ϵ 2 + 2 ϵ 1 ) 2 ( a b ) 3 ( ϵ 1 ϵ 2 ) ( ϵ 1 1 ) .
a 1 = i 2 3 ( k 0 b ) 3 ( 1 ϵ 1 ) ( a b ) 3 ( 2 ϵ 1 + 1 ) ( 2 + ϵ 1 ) 2 ( a b ) 3 ( 1 ϵ 1 ) .
ϵ eff = k 0 3 + i 4 π N a 1 k 0 3 i 2 π N a 1 ,
μ eff = k 0 3 + i 4 π N b 1 k 0 3 i 2 π N b 1 ,
ϵ polar ( ω ) = ϵ + ϵ 0 ϵ 1 ( ω ω T ) 2 i ( γ ω T ) ( ω ω T ) ,
ω L 2 ω T 2 = ϵ 0 ϵ .

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