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December 16, 2011

C 2011 American Chemical Society

Optically Induced Interaction ofMagnetic Moments in HybridMetamaterialsAndrey E. Miroshnichenko,†,* Boris Luk’yanchuk,‡ Stefan A. Maier,§ and Yuri S. Kivshar†

†Nonlinear Physics Centre and Centre for Ultra-high Bandwidth Devices for Optical Systems (CUDOS), Research School of Physics and Engineering, Australian NationalUniversity, Canberra ACT 0200, Australia, ‡Data Storage Institute, Agency for Science, Technology and Research, 117 608 Singapore, and §Department of Physics,Imperial College London, London SW7 2AZ, U.K.

Metamaterials offer many novelpossibilities to manipulate andcontrol the propagation of elec-

tromagnetic waves.1,2 Typical metamateri-als are created by a system of resonantsubwavelength elements for which electricand magnetic responses can be varied andmodified independently. The artificial mag-netism at high frequencies is based on theinductive response of normal metals,3 and itwas demonstrated for metal�dielectric“fishnet” structures3 and arrays of split-ringresonators.4 In such cases, the optically in-duced “ferromagnetic” (FM) ordering of in-ducedmagnetization of thewhole structureis utilized in order to achieve the negativemagnetic response required for left-handedmetamaterials.5�9 We would like to notethat such terminology borrowed from solid-state physics should be used with cau-tion. The difference between static andoptically induced magnetization hasbeen already discussed in the literature.10

However, many novel magnetic phenom-ena are expected in systems with staggeredmagnetization, i.e., with “antiferromagnetic”

(AFM) ordering.11 One of the well-knownexamples of such novel phenomena isrelated to the existence of giant magneto-resistance.12�14

A canonical subwavelength artificial mag-netic “meta-atom” is represented by a famil-iar split-ring resonator (SRR) that consists ofan inductive metallic ring with a gap. Sucha SRR can support an eigenmode with acircular current that gives rise to a localmagnetic dipole moment,15 which can beconsidered as amagnetic counterpart for anelectric dipole. This is one of the simplestexamples demonstrating that a nonmagneticmaterial can possess a nonzero inducedmag-netization. By nonmagneticmaterial wemeanamaterial whose bulk magnetization is zero

in the presence of a static magnetic field.The case of time-dependent fields becomestrickier since due to Maxwell's equationsboth electric and magnetic componentsintervene. There are other classical exam-ples where the magnetization can be in-duced by electric field only,16 including theinverse Faraday effect.17 The latter allowscreating static magnetization from rectifiedcircular polarized light. The SRR-based designwas the first and most frequently used formetamaterials, with unprecedented valuesof negative magnetic permeability at highfrequencies.18,19 However, the performanceof SRR-based metamaterials at optical fre-quencies is limited by high conduction lossesof the constitutive metallic elements and

* Address correspondence toaem124@physics.anu.edu.au.

Received for review November 9, 2011and accepted December 16, 2011.

Published online10.1021/nn204348j

ABSTRACT

We propose a novel type of hybrid metal�dielectric structures composed of silicon

nanoparticles and split-ring resonators for advanced control of optically induced magnetic

response. We reveal that a hybrid “metamolecule”may exhibit a strong distance-dependent

magnetic interaction that may flip the magnetization orientation and support

“antiferromagnetic” ordering in a hybrid metamaterial created by a periodic lattice of such

metamolecules. The propagation of magnetization waves in the hybrid structures opens new

ways for manipulating artificial “antiferromagnetic” ordering at high frequencies.

KEYWORDS: optical metamaterials . artificial magnetism . light-inducedcoupling

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the kinetic inductance of the electrons. Another exam-ple of plasmonic nanostructures that supports strongmagnetic resonances is perforated semishells.20 Thesestructures can be viewed as 3D analogues of SRRs.To improve the overall performance of metamaterials

at optical frequencies, one can employ dielectric particleswith low or negligible losses in the visible spectrum. Byusing the Mie theory21 it can be analytically demon-strated that the scattering of electromagnetic waves by asmall nonmagnetic dielectric particle may exhibit astrongmagnetic resonance. This resonance occurs whenthe wavelength inside the particle λS = λ0/nS approxi-mately equals its diameter 2RS, where λ0 is the incidentwavelengthandnS andRS are the refractive indexand theradius of the particle, respectively. A nonzero magneticdipolemoment (coefficient b1 inMie theory21) arises dueto circulation of the displacement currents. The effect issimilar to the appearance of the induced nonzero mag-netization of SRRs, with the only difference that displace-ment currents do not contribute to Joule's heating.At such resonant conditions the tangential componentsof the electric field polarization are antiparallel on thedifferent sides of the sphere along the propagationdirection, resulting in a strong overlap with themagneticmode. Thus, a dielectric particle can be considered asanother example of a magnetic-dipole scatterer of anincident electromagnetic wave at the resonant wave-length. This yields an increased magnetic field in thenear-field region around the particle. For the case of asilicon sphere with the radius in the range R = 40�100 nm the magnetic resonance can be located in thevisible spectrum.22 This makes such particles the bestcandidates to achieve low-loss magnetic response athigh frequencies. In the case of magnetic moments ofdifferent origin the overall magnetic response dependson the geometry of the system.In this paper, we demonstrate that optically induced

magnetic moments of a dielectric sphere and SRR caneffectively interact with each other when they are placedin a closeproximity (seeFigure1). An importantfindingofour study is thedemonstrationof the ability for staggeredpatterns to produce a distribution of the magnetic mo-ments associated with optically induced AFM-like re-sponse. Although, these are dynamically excited states,at any given moment of time mutual magnetization ofthe neighboring elements is antiparallel, which allows usto call such a state AFM-like, in analogy with the spinordering in solids. In addition, one- and two-dimensionallattices of such hybrid elements support novel types ofmagnetization waves. The AFM-like response of thehybrid metamaterials in the THz regime may offer arange of useful applications, including sensing of mag-neticfields.Wewould like to stress that there is aprincipaldifference between anoptically inducedmagnetic dipoleand a static magnet (compass needle). Namely, the staticmagnetic field does not influence the optically inducedmagnetic dipole. However, it can be used in the dynamic

regime foroptically inducedmagnetization,directobserva-tion of a propagating spin waves,23 and all-opticalmagnetic recording with the use of magneto-opticalmaterials.24 Also the effective generation of magneti-zation waves is similar to spin waves, which may carryelectrical signals through a dielectric medium.25

RESUTLS AND DISCUSSION

We start our theoretical analysis by employinga coupled-dipole approach. In the long-wavelengthlimit, when the characteristic sizes of a dielectric sphereand SRR are much smaller than the wavelength of theincident light, namely, RS,SRR , λ0, their optical proper-ties can be described by the effective dipoles

p1 ¼ ε0R1E0, m1, 2 ¼ χ1, 2H0 (1)

wherep1 andm1 are the induced electric andmagneticdipoles of the dielectric sphere, m2 is the magneticdipole of the SRR, RS, χS, and χSRR are the electric andmagnetic polarizabilities, and E0 andH0 are the incidentelectric andmagnetic fields. Since the electric responseof the SRR is very weak, it can be neglected.The effective polarizabilities of the sphere can be

obtained from the corresponding scattering dipolecoefficients a1 and b1 of the exact Mie solution21

R1 ¼ 6πia1k3, χ1 ¼ 6πi

b1k3

(2)

where k = 2π/λ0 is a wavenumber and the magneticpolarizability of SRR can be approximated as15 χ2 =Fω2/(ω0

2 � ω2), where ω0 is the resonant frequency ofa single SRR and F is a geometric factor.To study the optical response of an interacting

dielectric sphere and a SRR, we employ the methodof coupled electric and magnetic dipoles26�28 andwrite the following system of equations:

p1 ¼ R1 ε0E0 � d

c(n�m2)

� �,

m1 ¼ χ1fH0 þ am2 þ b(n 3m2)ng,m2 ¼ χ2fH0 þ am1 þ b(n 3m1)n � dc(n� p1)g

(3)

Figure 1. Schematic of a hybrid metal�dielectric structureconsisting of silicon spheres coupled to copper split-ringresonators.

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where c = 1/(ε0μ0)1/2 is the speed of light, n is a unit

vector pointing from the sphere to the SRR, and thecoefficients a, b, and d describe the interaction be-tween dipoles.29

a ¼ eikR

4πRk2 � 1

R2þ ik

R

� �,

b ¼ eikR

4πR� k2 þ 3

R2� 3ik

R

� �,

d ¼ eikR

4πRk2 þ ik

R

� �(4)

where R is the distance between the sphere and SRR.For a sphere and SRR being aligned along the x-axis

[n = (1,0,0)] and the incident light propagating alongthe z-direction [k= (0,0,k)], polarizedwith themagneticfield along the y-axis [H0 = (0,Hy,0)], the solution can bewritten as follows:

m1, y ¼ (1þ aχ2 þ d2R1χ2)(1 � a2χ1χ2 þ d2R1χ2)

χ2Hy,

m2, y ¼ (1þ aχ1)(1 � a2χ1χ2 þ d2R1χ2)

χ2Hy,

p1, x ¼ ε0R1Ex, p1, z ¼ � d

c

� �R1m1, y

with all other components vanishing. This simple solu-tion gives us a useful insight, and it provides theinformation about the optical response of the hybridsystem in different limiting cases. For the case of a farseparated sphere and SRR, their magnetization will beidentical to the uncoupled system since a,d f 0 withR f ¥, resulting in m(1,2),y = χ1,2Hy. However, in theopposite limit when the sphere and SRR are placedvery close to each other (Rf 0), the situation changesdramatically. The electric dipole moment of the spherecontributes strongly to the magnetization of thesphere itself since the cross-coupling coefficient d2 �R�4 becomes larger in the leading order compared tothe self-coefficient a � R�3 for very small distancesRf 0. This self-action of the longitudinal electric dipoleon the sphere magnetization may result in the oppo-site magnetic response compared to the case of theuncoupled pair, depending on the relative signs ofR1 and χ2. Thus, by placing a sphere and SRR in closeproximity, we may expect the appearance of an AFM-

like response of the whole structure.On the basis of the general analysis presented

above, we consider a small silicon sphere interactingwith a set of identical copper SRRs. To enhance themagnetic response of the SRR, we use amultistack withfive layers. We optimize the parameters to ensurestrong interaction between these two elements. Inparticular, the decoupled dielectric sphere and SRRhave close magnetic resonances ωS = 281 THz andωSRR = 250 THz, respectively (see Figure 2a). In ournumerical simulations, we confirm directly the exis-tence of a strong optically induced magnetic coupl-ing in the hybrid metamolecule. We observe that,

in addition to red-shifting independent magnetic re-sonances of the sphere and SRRs where the magneticresponse is dominated by one or another component,there appears a novel, AFM-like response with antipar-allel magnetization of both the elements of compar-able strength in the vicinity of the SRR's magneticresonance ωAFM = 257 THz. The longitudinal electricdipolemoment of the sphere induced by SRRs plays animportant role in the flipping the sign of magnetiza-tion, when two elements are placed in close proximity.Indeed, by studying the dependence of the magneti-zation on the distance between two elements, assummarized in Figure 2b, we clearly observe that bothelements exhibit a FM-like response for larger dis-tances R > Rcr, which changes to the AFM-like responsefor smaller distances R< Rcr, with staggeredmagnetiza-tion. The distance-dependent magnetic response is aunique feature of a hybrid structure. Such an effect doesnot exist in systems consisting of either silicon spheres22

or SRRs30,31 only, and it provides the opportunity

Figure 2. Magnetic response of a silicon sphere coupled toamultistack of SRRs. (a) Reflection curves of single elements(single silicon sphere and multistack of SRRs) and of thecoupled structure. Two peaks of the coupled system corre-spond to the magnetic resonances of isolated elements.Resonant dip corresponds to the AFM-like excitation withstaggered magnetic polarizabilities. (b) Dependence of theinduced magnetization of the sphere and SRRs on thedistance between at the resonant frequency ωAMF = 257THz.BelowthecriticaldistanceRcr≈1μmthe ferromagnetic-likemagnetization switches to antiferromagnetic. Inset shows thedistribution of the induced magnetic field at the AFM-likeresonance.

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to control the overall effective magnetic response.Thus, the role of SRRs in our hybrid metamaterial is tocontrol the relative phase.Next, we consider a two-dimensional lattice of the

hybrid metamolecules that corresponds to a novelhybridmetamaterial. Such periodic lattices of sphericalparticles can be fabricated by laser-induced transfertechnique.32 Due to spherical symmetry the inducedmagnetic dipole of the spheres is always parallel to theexciting magnetic field component. Thus, we consideran in-plane TM polarized wave at normal incidence.The AFM-like response of a single element implies thatthe periodic structure may also exhibit a staggeredmagnetization in the transverse direction, which is of aresonant origin, and it is closely related to the AFM-likeresponse. However, in periodic structures we concen-trate on staggered magnetization in the longitudinaldirection along the propagation. In Figure 3a we plotthe dispersion diagram of a two-dimensional lattice ofcoupled hybrid metamolecules with a period Dz =350 nm in the z-direction and compare it with thedispersions of the corresponding “monatomic” latticesof silicon spheres and SRRs, with the same period. Weobserve clearly the appearance of a new branch in thecoupled system corresponding to the staggered AFM-like magnetization in the transverse direction. It isinteresting that in the vicinity of the band edge onecan see a staggered magnetic response in the longi-tudinal direction too, i.e., a checkerboard pattern of thetotal two-dimensional magnetization. There could beseveral reasons for that, including (i) a staggered

induced magnetic field along the propagation at theband edge and (ii) interaction of induced longitudinalelectric dipoles of dielectric spheres. Both of theseconditions are met for the AFM branch near the fre-quency ω = 260 THz, which corresponds to the AFM-like resonance of a singlemetamolecule (see Figure 2a)and the band edge of the two-dimensional structure(see Figure 3a). The numerical results for the excitationat this particular frequency are shown in Figure 3c.Moreover, this frequency corresponds to the crossingof two dispersion curves, where mostly dielectricspheres are excited, leading to strong localization ofthe AFM-like response in a low-loss dielectric material.It also results in a stronger magnetic interaction be-tween spheres and SRRs, where the resonantly excitedlongitudinal dielectric dipoles of the spheres play anessential role in magnetization along the propagation.In this way, the control of the mutual phase of magne-tization becomes possible. We find that a robuststaggered magnetization pattern of dielectric spheresin the vicinity of the AFM-like resonance of a singlehybrid element depends weakly on the periodicityin the z-direction up to the values of Dz = 600 nm(see Figure 2a). For larger periods, the dispersion dia-gram of the coupled systems changes significantly,and the AFM branch shifts away from the resonantfrequency.To estimate the propagation length of the induced

magnetization waves in such a structure, we employthe method recently suggested in ref 33. This methodallows us to extract complex propagation constants of

Figure 3. (a) Comparison of the dispersion diagrams of two-dimensional lattices of spheres, SRRs, and hybrid sphere�SRRsystemwith theperiodicity in the z-directionDz=350nm. (b) Calculated propagation length andeffective refractive indexof afinite structure. (c) Magnetic field distribution Hy of a two-dimensional structure at the resonant frequencyω = 260 THz. Thisgives a typical example of the staggered two-dimensional checkerboard magnetization of the hybrid structure.

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the Bloch modes based on a finite number of samplepointsper period. The results are summarized in Figure 3c,where the propagation length is inversely proportional tothe imaginary part of the corresponding wavevector,∼Im(k)�1. It decays exponentially with frequency, andthe AFM-like resonance ωAFM = 257 THz corresponds tosix periods of the structure. At the same time, the effectiverefractive index, calculatedas a ratio between the real partof the wavevector and its free space counterpart neff =Re(khybrid)/k, becomes relatively large; neff ≈ 10. It corre-sponds to the regime with a slow phase velocity.

CONCLUSION

We have studied the optically induced interaction ofmagnetic moments of hybrid metal�dielectric struc-tures consisting of dielectric spheres and split-ringresonators. We have revealed that each element ofsuch a hybrid structure exhibits a strong magneticresponse in the THz frequency range, and coupling of

two different elements leads to a strong interaction ofmagneticmoments between the spheres and split-ringresonators. On the basis of the coupled-dipole model,we have demonstrated an important role of resonantlyinduced longitudinal electric dipoles in the formationof an AFM-like response with a staggered magnetiza-tion below a critical separation. Arranging the hybridmeta-atoms into a two-dimensional lattice allowscreating a hybrid metamaterial supporting staggeredmagnetization in the longitudinal direction due to themagnetic interaction between the dielectric spheres.Aweak dependence of the staggeredmagnetization ofthe dielectric spheres on the distance between them atthe AFM-like resonance of a single metamoleculeindicates a significant contribution of the inducedlongitudinal electric dipoles into the overall magneticinteraction. We believe this approach opens novelpossibilities for the manipulation and control of artifi-cial “antiferromagnetism” at optical frequencies.

METHODSMie Theory. The optical properties of a single dielectric

sphere can be analyzed in the framework of the Mie theory.21

It provides full analytical insight into the problem of the lightscattering by a sphere. By using this method, the frequencies ofthe electric andmagnetic resonances are calculated. In contrastto plasmonic particles, where the electromagnetic field isnegligible inside the particle and all resonances can be esti-mated by using the Mie scattering coefficients only, the situa-tion is completely different for dielectric particles. In the lattercase, the electromagnetic field is nonzero inside the particle,and the accurate position of the resonances should be calcu-lated based on the internal field enhancement for a particularmode. Our analysis reveals thatmagnetic resonance takes placewhen the wavelength inside the particle roughly equals theparticle's diameter.

Method of Coupled Electric and Magnetic Dipoles. We employ themethod of coupled electric and magnetic dipoles to estimatethe excitation states of the hybrid system consisting of adielectric sphere and SRR.26 The electric and magnetic polariz-abilities of the sphere can be obtained from the correspondingMie scattering coefficients. Although we are dealing with thedielectric nonmagnetic sphere, it may support resonantly ex-cited magnetic modes with nonzero magnetization, which willscatter light similarly to a magnetic dipole in the far field. Thus,in addition to electric polarizability, it allows us to assign aneffective magnetic polarizability to the dielectric sphere as well.Under the considered excitation conditions, the electronicresponse of the SRR can be neglected.

Numerical Simulations. All numerical results are obtainedusing CST Microwave Studio. In our simulations, we take a Sisphere with radius R = 150 nm and permittivity ε = 12.25. ForSRR we consider copper with an inner radius R1 = 70 nm, outerradius R2 = 75 nm, thickness h = 5 nm, and gap width g = 10 nm.The choice of the material might not be optimal, and forpractical realization it is better to use some alloys of weaklydissipating plasmonic materials, as was recently indicated inref 34.

The dispersion relations, the effective propagation length,and the refractive index were retrieved by using the Bloch-mode extraction method from near-field data in periodicarrays.33

Acknowledgment. The authors thank M. Decker, B. Ivanov,A. Slavin, H. Giessen, and M. I. Dyakonov for useful discussions.

The work was supported by the Australian Research Councilthrough Discovery, Future and Federation Fellowships, as wellas Centre of Excellence CUDOS in Australia and the LeverhulmeTrust in the UK. The research presented in this paper issupported by the Agency for Science, Technology and Research(A*STAR): SERC Metamaterials Program on Superlens, grant no.092 154 0099 and A*STAR TSRP Program, grant no. 102 1520018.

REFERENCES AND NOTES1. Electromagnetic Metamaterials: Physics and Engineering

Explorations; Engheta, N.; Ziolkowski, R. W., Eds.; Wiley-IEEE Press: New York, 2006.

2. Metamaterials: Theory, Design, and Applications; Cui, T. J.;Smith, D.; Liu, R., Eds.; Springer-Verlag: Heidelberg, 2009.

3. Shalaev, V. M. Optical Negative Index Metamaterials. Nat.Photon 2007, 1, 41–48.

4. Lahiri, B.; McMeekin, S. G.; Khokhar, A. Z.; De la Rue, R. M.;Johnson, N. P. Magnetic Response of Split Ring Resonators(SRRs) at Visible Frequencies. Opt. Express 2010, 18, 3210–3218.

5. Liu, H.; Genov, D. A.; Wu, D. M.; Liu, Y. M.; Steele, J. M.; Sun,C.; Zhu, S. N.; Zhang, X. Magnetic Plasmon PropagationAlong a Chain of Connected Subwavelength Resonators atInfrared Frequencies. Phys. Rev. Lett. 2006, 97, 243902.

6. Wang, S. M.; Li, T.; Liu, H.; Wang, F. M.; Zhu, S. N.; Zhang, X.Magnetic PlasmonModes in Periodic Chains of Nanosand-wiches. Opt. Express 2008, 16, 3560–3565.

7. Liu, N.; Fu, L.; Kaiser, S.; Schweizer, H.; Giessen, H. PlasmonicBuilding Blocks for Magnetic Molecules in Three-Dimensional Optical Metamaterials. Adv. Mater. 2008,20, 3859–3865.

8. Liu, N.; Giessen, H. Three-Dimensional Optical Metamateri-als As Model Systems for Longitudinal and TransverseMagnetic Coupling. Opt. Express 2008, 16, 21233–21238.

9. Ghadarghadr, S.; Mosallaei, H. Coupled Dielectric Nano-particles Manipulating Metamaterials Optical Characteris-tics. IEEE Trans. Nanotechnology 2009, 8, 582–594.

10. Liu, N.; Giessen, H. Coupling Effects in Optical Metamater-ials. Angew. Chem. 2010, 49, 9838–9852.

11. Liu N.; Mukherjee S.; Bao K.; Brown L. V.; Dorfmüller J.;Nordlander P.; Halas N. J. Magnetic Plasmon Formationand Propagation in Artificial Aromatic Molecules. NanoLett. 2011, accepted.

ARTIC

LE

MIROSHNICHENKO ET AL . VOL. 6 ’ NO. 1 ’ 837–842 ’ 2012

www.acsnano.org

842

12. Grünberg, P.; Schreiber, R.; Pang, Y.; Brodsky, M. B.; Sowers,H. Layered Magnetic Structures: Evidence for Antiferro-magnetic Coupling of Fe Layers across Cr Interlayers. Phys.Rev. Lett. 1986, 57, 2442–2445.

13. Baibich, M. N.; Broto, J. M.; Fert, A.; Nguyen an Dau, F.;Petroff, F.; Etienne, P.; Creuzet, G.; Friederich, A.; Chazelas, J.Giant Magnetoresistance of (001)Fe/(001)Cr MagneticSuperlattices. Phys. Rev. Lett. 1988, 61, 2472–2475.

14. Fullerton E. E; Schuller I. K. The 2007 Nobel Prize in Physics:Magnetism and Transport at the Nanoscale. ACS Nano2007 1, 384�389.

15. Pendry, J. B.; Holden, A. J.; Robbins, D. J.; Stewart, W. J.Magnetism from Conductors, and Enhanced Non-LinearPhenomena. IEEE Trans. Microwave Theory Tech. 1999, 47,2075–2084.

16. Tamm, I. E. Basics of Theory of Electricity [in Russian];FizMatLit: Moscow, 2003; pp 539�543.

17. Stanciu, C. D.; Hansteen, F.; Kimel, A. V.; Kirilyuk, A.;Tsukamoto, A.; Itoh, A.; Rasing, Th. All-Optical MagneticRecording with Circularly Polarized Light. Phys. Rev. Lett.2007, 99, 047601.

18. Soukoulis, C. M.; Linden, S.; Wegener, M. Negative RefractiveIndex at Optical Wavelengths. Science 2007, 315, 47–49.

19. Soukoulis, C. M.; Wegener, M. Past Achievements andFuture Challenges in the Development of Three-Dimen-sional Photonic Metamaterials. Nat. Photon. 2011, 5, 523–530.

20. Mirin, N. A.; Ali, T. A.; Nordlander, P.; Halas, N. J. PerforatedSemishells: Far-Field Directional Control and Optical Fre-quency Magnetic Response. ACS Nano 2010, 4, 2701–2712.

21. Bohren, C. F.; Huffman, D. R. Absorption and Scattering ofLight by Small Particles; Wiley: New York, 1998; pp111�117.

22. Evlyukhin, A. B.; Reinhardt, C.; Seidel, A.; Luk'yanchuk, B.;Chichkov, B. N. Optical Response Features of Si Nanopar-ticle Arrays. Phys. Rev. B 2010, 82, 045404.

23. Madami, M.; Bonetti, S.; Consolo, G.; Tacchi, S.; Carlotti, G.;Gubbiotti, G.; Mancoff, F. B.; Yar, M. A.; Ekerman, J. DirectObservation of a Propagating SpinWave Induced by Spin-Transfer Torque. Nat. Nanotechnol. 2011, 6, 635–638.

24. Temnov, V. V.; Armelles, G.; Woggon, U.; Guzatov, D.;Cebollada, A.; Garcia-Martin, A.; Garcia-Martin, J.-M.;Thomay, T.; Leitenstorfer, A.; Bratschitsch, R. ActiveMagneto-Plasmonics in Hybrid Metal-Ferromagnet Struc-tures. Nat. Photon 2010, 4, 107–111.

25. Kajiwara, Y.; Harii, K.; Takahashi, S.; Ohe, J.; Uchida, K.;Mizuguchi, M.; Umezawa, H.; Kawai, H.; Ando, K.; Takanashi,K.; et al. Transmission of Electrical Signals by Spin-WaveInterconversion in a Magnetic Insulator. Nature 2010, 464,262–266.

26. Mulholland, G. W.; Bohren, C. F.; Fuller, K. A. Light Scatter-ing by Agglomerates: Coupled Electric and MagneticDipole Method. Langmuir 1994, 10, 2533–2546.

27. Garcá-Camara, B.; Moreno, F.; Gonzalez, F.; Martin, O. J. F.Light Scattering by an Array of Electric and MagneticNanoparticles. Opt. Express 2010, 18, 10001–10015.

28. Chaumeta, P. C.; Rahmani, A. Coupled-Dipole Method forMagnetic and Negative-Refraction Materials. J. Quant.Spectrosc. Radiat. Transfer 2009, 110, 22–29.

29. Jackson, J. D. Classical Electrodynamics; Wiley: New York,1975; pp 407�416.

30. Decker, M.; Linden, S.; Wegener, M. Coupling Effects inLow-Symmetry Planar Split-Ring Resonator Arrays. Opt.Lett. 2009, 34, 1579–1581.

31. Decker, M.; Burger, S.; Linden, S.; Wegener, M. Magnetiza-tion Waves in Split-Ring-Resonator Arrays: Evidence forRetardation Effects. Phys. Rev. B 2009, 80, 193102.

32. Kuznetsov, A. I.; Evlyukhin, A. B.; Goncalves, M. R.;Reinhardt, C.; Koroleva, A.; Arnedillo, M. L.; Kiyan, R.; Marti,O.; Chichkov, B. N. Laser Fabrication of Large-Scale Nano-particle Arrays for Sensing Applications. ACS Nano 2011, 5,4843–4849.

33. Ha, S.; Sukhorukov, A. A.; Dossou, K. B.; Botten, L. C.;de Sterke, C. M.; Kivshar, Yu. S. Bloch-Mode Extraction

from Near-Field Data in Periodic Waveguides. Opt. Lett.2009, 34, 3776–3778.

34. Boltasseva, A.; Atwater, H. A. Low-Loss Plasmonic Meta-materials. Science 2011, 331, 290–291.

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