## Multipolar effects on the dipolar polarizability of magneto-electric antennas |

Optics Express, Vol. 21, Issue 14, pp. 16444-16454 (2013)

http://dx.doi.org/10.1364/OE.21.016444

Acrobat PDF (922 KB)

### 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

## 1. Introduction

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] .

7. 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] .

*n*≈ 3.5) have attracted keen interest since they possess both magnetic and electric polarizabilities [8

8. 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. Express **19**, 4815–4826 (2011) [CrossRef] [PubMed] .

10. 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] .

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

16. W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband Unidirectional Scattering by Magneto-Electric Core-Shell Nanoparticles,” ACS Nano **6**, 5489–5497 (2012) [CrossRef] [PubMed] .

17. 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. B **85**, 245432 (2012) [CrossRef] .

18. 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. B **86**, 125102 (2012) [CrossRef] .

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

22. A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express **19**, 2754–2772 (2011) [CrossRef] [PubMed] .

*α̿*of scatterers. This method aims at determining the various components of the polarizability tensor by illuminating the object with a superposition of plane waves, in such a way that the resulting excitations take the form of purely electric or magnetic fields polarized along one of three orthogonal directions (the principal axes of the scatterer), thereby permitting the determination of the corresponding column of the polarizability tensor.

23. J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Multipole approach to metamaterials,” Phys. Rev. A **78**, 043811 (2008) [CrossRef] .

25. D. Chigrin, C. Kremers, and S. Zhukovsky, “Plasmonic nanoparticle monomers and dimers: from nanoantennas to chiral metamaterials,” Appl. Phys. B **105**, 81–97 (2011) [CrossRef] .

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

27. I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B **83**, 245102 (2011) [CrossRef] .

25. D. Chigrin, C. Kremers, and S. Zhukovsky, “Plasmonic nanoparticle monomers and dimers: from nanoantennas to chiral metamaterials,” Appl. Phys. B **105**, 81–97 (2011) [CrossRef] .

28. 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] .

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

30. F. Capolino, *Theory and Phenomena of Metamaterials* (CRC, 2009) [CrossRef] .

31. 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. Express **18**, 14454–14466 (2010) [CrossRef] [PubMed] .

32. A. Chipouline, C. Simovskib, and S. Tretyakovb, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials **6**, 77–120 (2012) [CrossRef] .

31. 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. Express **18**, 14454–14466 (2010) [CrossRef] [PubMed] .

## 2. Principle

*α̿*of a dipolar particle illuminated by an electromagnetic field (

**E**

^{0},

**H**

^{0}) is generally cast [34]: where

*α̿*and

^{ee}*α̿*describe direct electric and magnetic effects while

^{mm}*α̿*and

^{em}*α̿*refer to the electromagnetic and magneto-electric polarizability tensors and render the possible bi-anisotropic character of the scatterer. Equation (1) predicts a linear dependence on the local incident field components, a property which is true when moments of a given angular momentum order are only induced by excitation fields of the same order. This situation is usually satisfied for highly symmetric particles, but proves questionable in full multipolar theory for particles of arbitrary form, wherein spatial derivatives of the excitation field can also contribute to the dipolar responses through [33, 35

^{me}35. P. Mazur and B. Nijboer, “On the statistical mechanics of matter in an electromagnetic field. I,” Physica XIX , 971 (1953) [CrossRef] .

*i*,

*j*, and

*k*corresponds to one space coordinate

*x*,

*y*, or

*z*, and where the dipolar (resp. quadrupolar) coupling terms are given by the tensors

*α̿*,

^{ee}*α̿*,

^{mm}*α̿*, and

^{em}*α̿*(resp.

^{me}*a̿*,

*b̿*,

*c̿*, and

*d̿*). Equation (2) introduces a dependence on the wavevector orientation appearing along any non-symmetric direction of the particle or nano-cluster.

*λ*/10. We proceed by a systematic inspection of the induced dipolar electric and magnetic moments for different incidence directions and polarizations. We perform the computations in a cartesian coordinate system that matches the principal axes of the resonator to obtain simplified (presumably diagonal) expressions of the direct electric and magnetic tensors

*α̿*and

_{ee}*α̿*in Eq. (1) and we place the origin at the center of mass of the resonator [37

_{mm}37. C. Rockstuhl, T. Zentgraf, E. Pshenay-Severin, J. Petschulat, A. Chipouline, J. Kuhl, T. Pertsch, H. Giessen, and F. Lederer, “The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach,” Opt. Express **15**, 8871–8883 (2007) [CrossRef] [PubMed] .

38. Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B **81**, 075116 (2010) [CrossRef] .

39. 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] .

**J**

*(*

_{vol}**r**) is readily deduced from the total field inside the scatterer. The induced electric and magnetic dipole moments are finally computed according to the following standard definitions [40]: where the integration is performed over the entire volume

*V*of the scatterer. The induced dipolar moments could also be computed through a multipolar decomposition [41

_{s}41. P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. **14**, 093033 (2012) [CrossRef] .

*θ*. This new formulation of the polarizability tensor will then be applied to Eq. (1) to obtain the induced dipolar moments for a given incidence direction. It will finally be shown that this formulation, obtained under specific illumination conditions, is able to accurately predict the dipolar moment for arbitrary directions of incidence.

## 3. Expression of the corrected polarizability tensor

*λ*

_{1}=1375nm,

*λ*

_{2}=790nm, and

*λ*

_{3}=630nm. The first and third resonances correspond to odd modes (magnetic modes) and are excited by electric fields parallel to the gap (blue curve in Fig. 2), while the second resonance, associated with an even mode (electric mode), is excited with a polarization perpendicular to the gap (black curve in Fig. 2) [37

37. C. Rockstuhl, T. Zentgraf, E. Pshenay-Severin, J. Petschulat, A. Chipouline, J. Kuhl, T. Pertsch, H. Giessen, and F. Lederer, “The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach,” Opt. Express **15**, 8871–8883 (2007) [CrossRef] [PubMed] .

*λ*

_{1}, at which the magnetic behavior of U-shaped resonators first appears and for which scatterer response remains predominantly dipolar [31

31. 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. Express **18**, 14454–14466 (2010) [CrossRef] [PubMed] .

*θ*and

*ϕ*together with a polarization angle,

*ψ*, defined in Fig. 1. The electric field can be cast: and the magnetic field is simply obtained by

**H**=

_{0}**k**×

**E**, where

_{0}**k**is the incident wave vector. According to the highly subwavelength thickness of the resonator, we neglect all effects of the field gradients along this direction, implying

*t*≈ 0, with

_{ijy}*t*=(

*a*,

*b*,

*c*,

*d*). Furthermore, and as a consequence of the high symmetry (resp. asymmetry) of the structure along the

*yOz*(resp.

*xOy*) plane, we assume

*t*≪

_{ijx}*t*for plane wave incidences with a phase reference matching a symmetry axis of the structure.

_{ijz}*yOz*plane, and we rotate the incidence direction around the

*x*axis. The incident electric field is polarized along the

*x*axis (parallel to the gap) which consists in letting

*θ*vary while setting (

*ϕ*,

*ψ*)=(+

*π*/2, −

*π*/2). According to Eqs. (2) and (4), and up to the magnetic quadrupole order, the induced electric dipole moment is: while the other components

*p*and

_{y}*p*remain negligible (we carefully verified this result numerically).

_{z}*xOz*and

*yOz*allows the calculation of the parameters

*c*, and

_{yxz}*d*that appear in Eq. (9). Since we investigate here only the first resonance, we consider that the electric polarizability

_{yyz}*α*parallel to the arms of the U-shape is constant, as this element describes the effect of the second mode of the resonator which is not resonant at this frequency.

_{zz}*λ*

_{1}can be described by a polarizability tensor of the form:

## 4. Numerical determination of the polarizability tensor

*p*(

_{x}*θ*) (resp.

*p*(

_{z}*θ*)) is plotted in red (resp. blue), while the magnetic dipole moment is plotted in green. The full and dotted curves denote the real and imaginary parts of those quantities obtained from FEM computations. For comparison, we plot with black markers in the same graph the results deduced from Eqs. (7) and (8) for

*p*(

_{x}*θ*) (and similar equations for

*p*(

_{z}*θ*) and

*m*(

_{y}*θ*) which are not detailed here for clarity concerns).

*z*axis, the induced moments depend on the propagation direction,

*i.e.*the dipolar moment obtained at

*θ*= 0° or 180° are different. Actually, for such a case, the gap is standing either at the back or the front of the scatterer (with respect to the propagation direction) leading to different microscopic charge distributions and near field distributions. These deviations observed in the induced moments as a modification of the coupling mechanism with the excitation field even for particles as small as

*λ*/10 are fully taken into account by the spatial derivatives in Eqs. (2). One can note that for a wave propagating normal to the resonator plane (

*y*axis), or parallel to the gap (

*x*axis), no difference can be observed between the two counter propagating directions, that is a consequence of the high symmetry property of the scatterer under these illuminations. More precisely, it can be observed that

*m*and

_{z}*p*are respectively symmetric and anti-symmetric with respect to the

_{z}*θ*= 180° axis. This difference results from the fact that the incident magnetic field component

*H*is not modified when modifying

_{z}*θ*in the

*xOz*incidence plane, while the sign of the electric field component

*E*is. Those results tend to valid the assumptions made according to the most relevant spatial derivatives of the excitation field involved in the dipolar response of the particle.

_{z}*α̿*by the corrected tensor

*α̿*given in Eq. (9) (Fig. 4(a)–4(c)). As in Fig. 3, the colored curves represent the FEM results while black markers represent the semi-analytical derivations.

_{cor}*ϕ*is varied (Fig. 4(a)–4(b)). Let us remind that angle

*ϕ*refers to the out-of-plane rotation around the scatterer, and according to our previous assumptions regarding the symmetry of the resonator together with incidence conditions, this linearity was expected. However, for rotations about the in-plane angle

*θ*, which bears the asymmetry information of the scatterer, such linearity does not hold and we can notice in Fig. 4(c) drastic changes in the behavior of the induced moments as a function of

*θ*.

## 5. Conclusion

## Acknowledgments

## References and links

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. |

2. | C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express |

3. | 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. |

4. | 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. Photonics |

5. | 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. |

6. | 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. B |

7. | 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. |

8. | 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. Express |

9. | A. B. Evlyukhin, C. Reinhardt, and B. N. Chichkov, “Multipole light scattering by nonspherical nanoparticles in the discrete dipole approximation,” Phys. Rev. B |

10. | 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. |

11. | P. Spinelli, M. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on sub-wavelength surface mie resonators,” Nat. Commun. |

12. | B. Rolly, B. Stout, and N. Bonod, “Boosting the directivity of optical antennas with magnetic and electric dipolar resonant particles,” Opt. Express |

13. | 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. |

14. | S. Person, M. Jain, Z. Lapin, J. J. Saenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. |

15. | 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. |

16. | W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband Unidirectional Scattering by Magneto-Electric Core-Shell Nanoparticles,” ACS Nano |

17. | 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. B |

18. | 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. B |

19. | Y. Terekhov, A. Zhuravlev, and G. Belokopytov, “The polarizability matrix of split-ring resonators,” Moscow Univ. Phys. Bull. |

20. | D. Morits and C. Simovski, “Isotropic negative effective permeability in the visible range produced by clusters of plasmonic triangular nanoprisms,” Metamaterials |

21. | 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. |

22. | A. Vallecchi, M. Albani, and F. Capolino, “Collective electric and magnetic plasmonic resonances in spherical nanoclusters,” Opt. Express |

23. | J. Petschulat, C. Menzel, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Multipole approach to metamaterials,” Phys. Rev. A |

24. | L. Zhou and S. T. Chui, “Magnetic resonances in metallic double split rings: Lower frequency limit and bian-isotropy,” Appl. Phys. Lett. |

25. | D. Chigrin, C. Kremers, and S. Zhukovsky, “Plasmonic nanoparticle monomers and dimers: from nanoantennas to chiral metamaterials,” Appl. Phys. B |

26. | P. de Vries, D. V. van Coevorden, and A. Lagendijk, “Point scatterers for classical waves,” Rev. Mod. Phys. |

27. | I. Sersic, C. Tuambilangana, T. Kampfrath, and A. F. Koenderink, “Magnetoelectric point scattering theory for metamaterial scatterers,” Phys. Rev. B |

28. | M. Liu, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Optical activity and coupling in twisted dimer meta-atoms,” Appl. Phys. Lett. |

29. | C. Simovski and S. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonic. Nanostruct. |

30. | F. Capolino, |

31. | 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. Express |

32. | A. Chipouline, C. Simovskib, and S. Tretyakovb, “Basics of averaging of the Maxwell equations for bulk materials,” Metamaterials |

33. | R. E. Raab and O. L. de Lange, |

34. | I. V. Lindell, A. Sihvola, and S. Tretyakov, |

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

36. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

37. | C. Rockstuhl, T. Zentgraf, E. Pshenay-Severin, J. Petschulat, A. Chipouline, J. Kuhl, T. Pertsch, H. Giessen, and F. Lederer, “The origin of magnetic polarizability in metamaterials at optical frequencies - an electrodynamic approach,” Opt. Express |

38. | Y. Zeng, C. Dineen, and J. V. Moloney, “Magnetic dipole moments in single and coupled split-ring resonators,” Phys. Rev. B |

39. | G. Demésy, F. Zolla, A. Nicolet, and M. Commandré, “Versatile full-vectorial finite element model for crossed gratings,” Opt. Lett. |

40. | J. D. Jackson, |

41. | P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(290.5850) Scattering : Scattering, particles

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 17, 2013

Revised Manuscript: June 2, 2013

Manuscript Accepted: June 3, 2013

Published: July 2, 2013

**Citation**

S. Varault, B. Rolly, G. Boudarham, G. Demésy, B. Stout, and N. Bonod, "Multipolar effects on the dipolar polarizability of magneto-electric antennas," Opt. Express **21**, 16444-16454 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16444

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### References

- 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]
- C. Rockstuhl, F. Lederer, C. Etrich, T. Zentgraf, J. Kuhl, and H. Giessen, “On the reinterpretation of resonances in split-ring-resonators at normal incidence,” Opt. Express14, 8827–8836 (2006). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
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