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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 14 — Jul. 15, 2013
  • pp: 16444–16454
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Multipolar effects on the dipolar polarizability of magneto-electric antennas

S. Varault, B. Rolly, G. Boudarham, G. Demésy, B. Stout, and N. Bonod  »View Author Affiliations


Optics Express, Vol. 21, Issue 14, pp. 16444-16454 (2013)
http://dx.doi.org/10.1364/OE.21.016444


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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

Light-matter interactions have traditionally been investigated from the electric field viewpoint on account of the fact that electric dipole transition rates of quantum emitters are typically five orders of magnitude higher than magnetic dipole transitions. Nevertheless, by structuring non magnetic materials at a subwavelength scale, it is possible to create artificial ‘atoms’ which can resonantly interact with incident light via strong electric and/or magnetic polarizabilities. Split Ring Resonators (SRR) and U-shaped scatterers are certainly the most emblematic photonic components for creating artificial magnetism [1

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

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] .

]. More recently, dielectric Mie resonators of moderate refractive index (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

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] .

] which in turn lead to interesting scattering properties [11

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

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] .

]. Mie resonators have also been shown to specifically enhance either electric or magnetic local density of states (LDOS) and they could serve as efficient optical antennas for increasing the magnetic transition rates of trivalent lanthanide ions [17

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

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] .

]. These recent advances concerning the scattering and LDOS properties of dielectric Mie resonators were facilitated by an accurate knowledge of their complex dipolar and multipolar response functions. It should prove interesting to extend these findings to the case of metallic magneto-electric scatterers. However, although the calculation of induced polarization moments is well known for spherical particles, it can prove tricky for the case of highly asymmetric particles. The computation of a polarizability tensor for arbitrarily shaped particles is therefore a crucial step in the design and understanding of both metamaterials and optical antennas.

The method of counter-propagating waves [19

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

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] .

] is often proposed to numerically compute the polarizability tensor 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.

We study in this paper how such multipolar effects contribute to the creation of induced electric and magnetic dipoles for non-centrosymmetric planar scatterers illuminated from far-field region, and we derive a correction to the classical dipolar polarizability tensor that takes into account such contributions. We show that their full profile can only be explained by taking into account spatial derivatives of the excitation field according to the multipolar theory [33

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

]. As a typical example, we calculate the dipolar moments of a single U-shaped resonator and propose a method to determine an effective dipolar polarizability tensor that takes into account the multipolar contributions of the illuminating field in the case of a far field illumination. We calculate for arbitrary incidences the polarizability tensor derived with our method. We finally compare the results with full vector finite element numerical calculations to demonstrate the accuracy of this method.

2. Principle

The polarizability tensor α̿ of a dipolar particle illuminated by an electromagnetic field (E0,H0) is generally cast [34

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

]:
[pm]=[α¯¯eeα¯¯emα¯¯meα¯¯mm]α¯¯[E0H0]
(1)
where α̿ee and α̿mm describe direct electric and magnetic effects while α̿em and α̿me 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

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

, 35

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

]:
pi=αijeeEj0+aijkkEj0+αijemHj0+bijkkHj0+mi=αijmeEj0+cijkkEj0+αijmmHj0+dijkkHj0+
(2)
where each subscript 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 α̿ee, α̿mm, α̿em, and α̿me (resp. 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.

In the following, we aim to illustrate how multipolar couplings affect the induced dipolar moments for U-shaped resonators described in Fig. 1, even for particle sizes as small as 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 α̿ee and α̿mm in Eq. (1) and we place the origin at the center of mass of the resonator [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] .

,38

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] .

]. The field scattered by the object illuminated by a plane wave can be computed using an appropriate formulation of the Finite Element Method (FEM). This 3D vector formulation relies on the use of second order edge elements and perfectly matched layers [39

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] .

]. The current density Jvol(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

40. J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).

]:
p=+1jωVsJvol(r)dvm=12Vsr×Jvol(r)dv
(3)
where the integration is performed over the entire volume Vs of the scatterer. The induced dipolar moments could also be computed through a multipolar decomposition [41

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

] of the field radiated by the scatterer. This method can yield more accurate results for scatterers that poorly verify the quasi-static approximation. However, a comparison between the two approaches shows that Eq. (3) provides accurate results for the scatterer under study.

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.

3. Expression of the corrected polarizability tensor

For the scatterer of Fig. 1, we first plot the extinction cross-section in the case of normal incidence in Fig. 2. In the spectral domain considered here, we found the first three resonances to occur at λ1=1375nm, λ2=790nm, and λ3=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] .

]. From here on, we focus on the resonance λ1, 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] .

].

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.

To illustrate our approach to deduce the contributing polarizabilities in Eq. (2), we study the induced electric dipole moments for two different incidence conditions. The electric field is defined using the polar and azimuthal angles θ and ϕ together with a polarization angle, ψ, defined in Fig. 1. The electric field can be cast:
E0(r)={+cosψcosθcosϕsinψsinϕ+cosψcosθsinϕ+sinψcosϕcosψsinθ
(4)
and the magnetic field is simply obtained by H0 = k × E0, where 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 tijy ≈ 0, with 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 tijxtijz for plane wave incidences with a phase reference matching a symmetry axis of the structure.

We first consider the case of a plane wave incidence whose wavevector lies in the 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:
pxyOz=αxxee+axxzyOzcosθ+αxyemcosθ+bxyzyOzcos2θ+bxzzyOzsinθcosθ
(5)
while the other components py and pz remain negligible (we carefully verified this result numerically).

A similar approach regarding the magnetic dipole moments for incidence planes xOz and yOz allows the calculation of the parameters αyxme, αyymm, cyxz, and dyyz that appear in Eq. (9). Since we investigate here only the first resonance, we consider that the electric polarizability αzz 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.

We must now quantitatively determinate all the parameters included in this corrected polarizability tensor.

4. Numerical determination of the polarizability tensor

We present in Fig. 3 the dipolar moments numerically computed via Eq. (3) for the two incidence cases described previously. The electric dipole moment perpendicular (resp. parallel) to the gap px(θ) (resp. pz(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 px(θ) (and similar equations for pz(θ) and my(θ) which are not detailed here for clarity concerns).

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).

We observe in Fig. 3 that the dipolar moments obtained by calculating Eq. (3) with FEM follow a square cosine profile that cannot be predicted by considering the formulation of Eq. (1) which only involves scalar dipolar polarizabilities. This result highlights the quantitative influence of the spatial derivative of the fields on the dipolar moments through the square cosine functions appearing in Eqs. (7), (8). We can also observe that for a wave propagating along the 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 mz and pz are respectively symmetric and anti-symmetric with respect to the θ = 180° axis. This difference results from the fact that the incident magnetic field component Hz is not modified when modifying θ in the xOz incidence plane, while the sign of the electric field component Ez 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.

However, to verify the general nature of the corrected dipolar polarizability tensor in Eq. (9) that we obtained under specific illumination conditions, we need to verify its accuracy in arbitrary incidence conditions. For that purpose, we numerically compute the induced dipolar moments using Eq. (3) for oblique incidence that we compare to the induced dipolar moments directly predicted from Eq. (1) when substituting the usual polarizability tensor α̿ by the corrected tensor α̿cor 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.

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).

In order to quantify the influence of the multipolar effects on the dipolar moments, we finally compute the dipolar polarizabilities alone by considering a tensorial formulation analog to Eq. (1) and assuming a linear dependence between the induced moments and the excitation field. The results displayed in Fig. 5 confirm the mismatch with the FEM results when the higher order corrective parameters that describe the asymmetry of the resonator are neglected.

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)

5. Conclusion

We systematically studied the induced dipolar moments for a U-shaped resonator under far-field illumination and proposed a multipolar based correction to the classically employed dipolar polarizability tensor. We do not operate a multipolar decomposition on the field scattered by the structure, but we analyze how the multipolar coupling between the incident field and the resonator affects the dipolar response of non-centrosymmetric nanoparticles. The importance of the asymmetry on the multipolar coupling is confirmed in the case of a U-shaped planar resonator by the fact that the corresponding corrective terms only play a role when the in-plane rotation angle (which bears the asymmetry) is varied. The relevant quadrupolar tensors are linked to the direct quadrupole-quadrupole coupling, whose influence is found to be very small in the radiated field. Nevertheless, they still impact the scattering process thanks to the perturbation they provoke on the dipolar response: a purely dipolar-dipolar response would present identical induced moments for a backward or forward propagating wave relative to the gap position, contrary to what is observed in this study. We limited this study to the case of a far field incidence which allowed us to reveal multipolar effects on the dipolar response of the scatterer in a simple way. Nevertheless, we believe that this analysis opens the way to the case of near-field illuminations and calculation of local density of states of magneto-electric scatterers.

Acknowledgments

The authors acknowledge Redha Abdeddaim, Victor Grigoriev and Bruno Gallas for stimulating discussions. This research was funded by the French Agence Nationale de la Recherche under Contract No. ANR-11-BS10-002-02 TWINS.

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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] .

40.

J. D. Jackson, Classical Electrodynamics, 3rd ed. (John Wiley & Sons, New York, 1999).

41.

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

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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