Abstract

We show the important role played by the multipolar coupling between the illuminating field and magneto-electric scatterers even in the small particle limit (λ/10). A general multipolar method is presented which, for the case of planar non centrosymmetric particles, generates a simple expression for the polarizability tensor that directly links the dipolar moment to the incident field. The relevancy of this approach is demonstrated by comparing thoroughly the dipolar moments predicted by the method with full numerical calculations.

© 2013 OSA

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2013 (2)

S. Person, M. Jain, Z. Lapin, J. J. Saenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett.13 (4), pp 1806–1809 (2013).
[PubMed]

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Luk’yanchuk, “Directional visible light scattering by silicon nanoparticles,” Nat. Commun.4 (2013).
[CrossRef]

2012 (11)

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband Unidirectional Scattering by Magneto-Electric Core-Shell Nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

C. M. Dodson and R. Zia, “Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths,” Phys. Rev. B86, 125102 (2012).
[CrossRef]

N. Guth, B. Gallas, J. Rivory, J. Grand, A. Ourir, G. Guida, R. Abdeddaim, C. Jouvaud, and J. de Rosny, “Optical properties of metamaterials: Influence of electric multipoles, magnetoelectric coupling, and spatial dispersion,” Phys. Rev. B85, 115138 (2012).
[CrossRef]

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

P. Spinelli, M. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on sub-wavelength surface mie resonators,” Nat. Commun.3(2012).
[CrossRef]

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express20, 20376–20386 (2012).
[CrossRef] [PubMed]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3(2012).
[CrossRef]

M. Liu, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Optical activity and coupling in twisted dimer meta-atoms,” Appl. Phys. Lett.100, 111114 (2012).
[CrossRef]

A. Chipouline, C. Simovskib, and S. Tretyakovb, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials6, 77–120 (2012).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

2011 (7)

I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B83, 245102 (2011).
[CrossRef]

A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express19, 2754–2772 (2011).
[CrossRef] [PubMed]

D. Chigrin, C. Kremers, and S. Zhukovsky, “Plasmonic nanoparticle monomers and dimers: from nanoantennas to chiral metamaterials,” Appl. Phys. B105, 81–97 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express19, 4815–4826 (2011).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B84, 235429 (2011).
[CrossRef]

Y. Terekhov, A. Zhuravlev, and G. Belokopytov, “The polarizability matrix of split-ring resonators,” Moscow Univ. Phys. Bull.66, 254–259 (2011).
[CrossRef]

D. Morits and C. Simovski, “Isotropic negative effective permeability in the visible range produced by clusters of plasmonic triangular nanoprisms,” Metamaterials5, 162–168 (2011).
[CrossRef]

2010 (4)

G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett.105, 255501 (2010).
[CrossRef]

C. Simovski and S. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonic. Nanostruct.8, 254–263 (2010). Tacona Photonics 2009.
[CrossRef]

J. Petschulat, J. Yang, C. Menzel, C. Rockstuhl, A. Chipouline, P. Lalanne, A. Tüennermann, F. Lederer, and T. Pertsch, “Understanding the electric and magnetic response of isolated metaatoms by means of a multipolar field decomposition,” Opt. Express18, 14454–14466 (2010).
[CrossRef] [PubMed]

Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B81, 075116 (2010).
[CrossRef]

2009 (2)

G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett.34, 2216–2218 (2009).
[CrossRef] [PubMed]

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett.103, 213902 (2009).
[CrossRef]

2008 (2)

J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Multipole approach to metamaterials,” Phys. Rev. A78, 043811 (2008).
[CrossRef]

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photonics2, 614–617 (2008).
[CrossRef]

2007 (2)

2006 (2)

2005 (1)

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95, 203901 (2005).
[CrossRef] [PubMed]

2003 (1)

A. Ishimaru, S.-W. Lee, Y. Kuga, and V. Jandhyala, “Generalized constitutive relations for metamaterials based on the quasi-static lorentz theory,” IEEE Trans. Antennas Propag.51, 2550–2557 (2003).
[CrossRef]

1998 (1)

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys.70, 447–466 (1998).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

1953 (1)

P. Mazur and B. Nijboer, “On the statistical mechanics of matter in an electromagnetic field. I,” Physica XIX, 971 (1953).
[CrossRef]

Abdeddaim, R.

N. Guth, B. Gallas, J. Rivory, J. Grand, A. Ourir, G. Guida, R. Abdeddaim, C. Jouvaud, and J. de Rosny, “Optical properties of metamaterials: Influence of electric multipoles, magnetoelectric coupling, and spatial dispersion,” Phys. Rev. B85, 115138 (2012).
[CrossRef]

Aizpurua, J.

Albani, M.

Albella, P.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3(2012).
[CrossRef]

Bebey, B.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

Belokopytov, G.

Y. Terekhov, A. Zhuravlev, and G. Belokopytov, “The polarizability matrix of split-ring resonators,” Moscow Univ. Phys. Bull.66, 254–259 (2011).
[CrossRef]

Bidault, S.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

Bonod, N.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express20, 20376–20386 (2012).
[CrossRef] [PubMed]

Boudarham, G.

G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett.105, 255501 (2010).
[CrossRef]

Bozhevolnyi, S. I.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95, 203901 (2005).
[CrossRef] [PubMed]

Busch, K.

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photonics2, 614–617 (2008).
[CrossRef]

Capolino, F.

Chantada, L.

Chichkov, B. N.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B84, 235429 (2011).
[CrossRef]

Chigrin, D.

D. Chigrin, C. Kremers, and S. Zhukovsky, “Plasmonic nanoparticle monomers and dimers: from nanoantennas to chiral metamaterials,” Appl. Phys. B105, 81–97 (2011).
[CrossRef]

Chipouline, A.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Chui, S. T.

L. Zhou and S. T. Chui, “Magnetic resonances in metallic double split rings: Lower frequency limit and bian-isotropy,” Appl. Phys. Lett.90, 041903 (2007).
[CrossRef]

Commandré, M.

de Lange, O. L.

R. E. Raab and O. L. de Lange, Multipole Theory in Electromagnetism: Classical, Quantum, and Symmetry Aspects, with Applications (Oxford University, 2005).

de Rosny, J.

N. Guth, B. Gallas, J. Rivory, J. Grand, A. Ourir, G. Guida, R. Abdeddaim, C. Jouvaud, and J. de Rosny, “Optical properties of metamaterials: Influence of electric multipoles, magnetoelectric coupling, and spatial dispersion,” Phys. Rev. B85, 115138 (2012).
[CrossRef]

de Vries, P.

P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys.70, 447–466 (1998).
[CrossRef]

Demésy, G.

Dineen, C.

Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B81, 075116 (2010).
[CrossRef]

Dodson, C. M.

C. M. Dodson and R. Zia, “Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: Calculated emission rates and oscillator strengths,” Phys. Rev. B86, 125102 (2012).
[CrossRef]

Enkrich, C.

M. W. Klein, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, “Single-slit split-ring resonators at optical frequencies: limits of size scaling,” Opt. Lett.31, 1259–1261 (2006).
[CrossRef] [PubMed]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic metamaterials at telecommunication and visible frequencies,” Phys. Rev. Lett.95, 203901 (2005).
[CrossRef] [PubMed]

Eriksen, R. L.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

Etrich, C.

Evlyukhin, A. B.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B84, 235429 (2011).
[CrossRef]

Eyraud, C.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3(2012).
[CrossRef]

Feth, N.

G. Boudarham, N. Feth, V. Myroshnychenko, S. Linden, J. García de Abajo, M. Wegener, and M. Kociak, “Spectral imaging of individual split-ring resonators,” Phys. Rev. Lett.105, 255501 (2010).
[CrossRef]

M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nat. Photonics2, 614–617 (2008).
[CrossRef]

Frimmer, M.

I. Sersic, M. Frimmer, E. Verhagen, and A. F. Koenderink, “Electric and magnetic dipole coupling in near-infrared split-ring metamaterial arrays,” Phys. Rev. Lett.103, 213902 (2009).
[CrossRef]

Froufe-Pérez, L.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun.3(2012).
[CrossRef]

Froufe-Pérez, L. S.

Fu, Y. H.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the studied U-shaped ring resonator and definition of the referential cartesian coordinate system. The U-shaped resonator is made of gold [36] and embedded in air. The resonator has equal lateral dimensions lx and lz of 200nm and a thickness ly of 25nm, while the gap width is set to 60nm.

Fig. 2
Fig. 2

Extinction cross-section for a single U-shaped resonator at normal incidence for an electric field parallel (black) and normal (blue) to the gap, i.e along and perpendicular to êx according to the notations in Fig. 1.

Fig. 3
Fig. 3

Real (solid line) and imaginary (dashed line) parts of the induced dipolar moments px(θ) (red), pz(θ) (blue) and mz(θ) (green) calculated with Eq.3 for the two conditions of incidence used to determine the polarizabilities in Eq. (2). The dipolar moments obtained through the retrieval procedure are indicated with black markers. Planes of incidence yOz (a) and xOz (b).

Fig. 4
Fig. 4

Real (solid line) and imaginary (dashed line) parts of the induced dipolar moments as a function of incidence. (a) Test case 1 (θ = π/4, ψ = π). (b) Test case 2 (θ = π/6, ψ = π/3). (c) Test case 3 (ϕ = π/6, ψ = π/3).

Fig. 5
Fig. 5

Real (solid line) and imaginary (dashed line) parts of the induced dipolar moments as a function of incidence in the configuration of Fig. 4(c) by using the non-corrected polarizability tensor of Eq. (1)

Equations (9)

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[ p m ] = [ α ¯ ¯ e e α ¯ ¯ e m α ¯ ¯ m e α ¯ ¯ m m ] α ¯ ¯ [ E 0 H 0 ]
p i = α i j e e E j 0 + a i j k k E j 0 + α i j e m H j 0 + b i j k k H j 0 + m i = α i j m e E j 0 + c i j k k E j 0 + α i j m m H j 0 + d i j k k H j 0 +
p = + 1 j ω V s J vol ( r ) d v m = 1 2 V s r × J vol ( r ) d v
E 0 ( r ) = { + cos ψ cos θ cos ϕ sin ψ sin ϕ + cos ψ cos θ sin ϕ + sin ψ cos ϕ cos ψ sin θ
p x y O z = α x x e e + a x x z y O z cos θ + α x y e m cos θ + b x y z y O z cos 2 θ + b x z z y O z sin θ cos θ
p x x O z = α x x e e cos θ + a x x z x O z cos 2 θ + a x z z x O z sin θ cos θ + α x y e m + b x y z x O z
y O z { p x ( θ ) + p x ( θ + π ) = 2 ( α x x e e + b x y z y O z cos 2 θ ) p x ( θ ) p x ( θ + π ) = 2 ( α x y e m + a x x z y O z ) cos θ
x O z { p x ( θ ) + p x ( θ + π ) = 2 ( α x y e e + a x x z x O z cos 2 θ ) p x ( θ ) p x ( θ + π ) = 2 ( α x x e e + b x y z x O z ) cos θ
α ¯ ¯ cor = [ α x x e e + a x x z cos θ 0 0 0 α x y e m + b x y z cos θ 0 0 0 0 0 0 0 0 0 α z z e e 0 0 0 0 0 0 0 0 0 α y x m e + c y x z cos θ 0 0 0 α y y m m + d y y z cos θ 0 0 0 0 0 0 0 ]

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