Burger, S.

Dolling, G.

Linden, S.

Schädle, A.

Wegener, M.

English

KITopen

Observation of magnetization waves in negative-indexphotonic metamaterialsG. Dollinga and M. WegenerInstitut für Angewandte Physik, Universität Karlsruhe (TH), D-76131 Karlsruhe, Germanyand DFG-Center for Functional Nanostructures (CFN), Universität Karlsruhe (TH),D-76131 Karlsruhe, GermanyA. Schädle and S. BurgerZuse Institute Berlin, Takustrasse 7, D-14195 Berlin, Germanyand DFG Forschungszentrum Matheon, Technische Universität Berlin, D-10623 Berlin, GermanyS. LindenInstitut für Nanotechnologie, Forschungszentrum Karlsruhe in der Helmholtz-Gemeinschaft,D-76021 Karlsruhe, GermanyReceived 4 October 2006; accepted 26 October 2006; published online 8 December 2006By measuring and calculating oblique-incidence transmittance spectra of a negative-index photonicmetamaterial layer, the authors infer the in-plane dispersion relation of the magnetization wave.From the geometry and the dispersion shape, the authors conclude that coupling is predominantlyvia magnetic dipoles for wave propagation along the magnetic-dipole direction. Thesemagnetization waves are the classical analog of magnon excitations of quantum-mechanicalspins. © 2006 American Institute of Physics. DOI: 10.1063/1.2403905In the years 2005 and 2006, metamaterials with a nega-tive index of refraction n have become available in the opti-cal regime.1–4 The negative index results from tailoring theproperties of nanostructured metallic building blocks, “pho-tonic atoms,” the metamaterial is composed of. In particular,electric atoms with an electric-dipole moment leading to anegative electric permittivity5  and magnetic atoms with amagnetic-dipole moment leading to a negative magneticpermeability6,7  have to be combined and densely packedinto an effective material.8 To a first approximation, the op-tical properties of the metamaterial are the result of the op-tical properties of the individual photonic atoms. In otherwords, interactions among the different unit cells can beneglected.One step beyond that, the electric dipole-dipole andmagnetic dipole-dipole interactions between next-nearestneighbors and beyond lead to a coupling of the photonicatoms, thus modifying the material optical properties. Theaim of this letter is to measure the resulting dispersion rela-tions and to clarify the nature of the corresponding waves.In the present context, one can distinguish between fourdifferent types of couplings: On the one hand, electric eand magnetic m dipole-dipole interactions, respectively; onthe other hand, transverse  and longitudinal  waves,respectively. Here, transverse and longitudinal refer to thedipole orientation with respect to the in-plane wave vectork = kx ,ky of the excitation. The theory for all four cases hasbeen evaluated.9,10Intuitively, the interaction between equally orienteddipoles—be it electric or magnetic—is attractive repulsivefor a spatially homogeneous longitudinal transverse ar-rangement. Thus, the energetic minimum maximum of thedispersion relation Ek occurs for zero in-plane momentumk, and finite wave numbers increase decrease the energy.Note that electric and magnetic dipoles are usually perpen-dicular to each other. This means that, e.g., the electric trans-verse case competes with the magnetic longitudinal case.Thus, the photon energy increases or decreases with wavenumber, depending on whether electric or magnetic dipole-dipole interactions dominate. The question which onedominates cannot easily be answered a priori. Investigatingthis question is interesting in itself and also a sensitive test ofour current understanding of negative-index photonicmetamaterials.Here, we study the negative-index metamaterial designthat has led to the best performance in the optical regime sofar. It is shown in Fig. 1. The sample investigated here isclosely similar but not identical to that used in Ref. 4. Thesample parameters are given in the caption of Fig. 1. Cou-pling to the above mentioned modes is accomplished by ob-lique incidence of light onto the metamaterial layer. Thisgeometry leads to a modulus of the in-plane wave number kgiven byk =2sin , 1with the incident vacuum wavelength of light . By choos-ing different axes of rotation, different principal directions inreciprocal space can be investigated. The insets in Fig. 2illustrate the two cases k =k1,0 and k0,1. For thesample parameters given in Fig. 1b, for =40° angle ofincidence with respect to the surface normal and for =1500 nm, the modulus of the in-plane wave vector reachesa value of greater than k =0.5 /a. This means that a sub-stantial fraction of the first Brillouin zone can be investigatedalong these lines.Experimental transmittance spectra are shown in Fig.2a for the polarization and geometry indicated on the top.The transmittance dips directly arise from the magnetic-dipole resonance, and the featureless electric response is thatof a diluted Drude metal with Re0 for the entirespectral region shown also see Ref. 8. For both oblique-aElectronic mail: gunnar.dolling@physik.uni-karlsruhe.deAPPLIED PHYSICS LETTERS 89, 231118 20060003-6951/2006/8923/231118/3/$23.00 © 2006 American Institute of Physics89, 231118-1Downloaded 19 Mar 2009 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspFirst published in:EVA-STAR (Elektronisches Volltextarchiv – Scientific Articles Repository) http://digbib.ubka.uni-karlsruhe.de/volltexte/1000010974incidence geometries shown in Fig. 2, the transmittancedip shifts towards higher photon energies with increasingangle .To further analyze these findings, we plot the photon-energy position of the transmittance dips see vertical dashedlines in Fig. 2 versus in-plane wave number k. The lattercan easily be calculated from the angle  via Eq. 1. Theresulting measured dispersion relations are shown in Fig. 3.It has been thoroughly discussed for the structure under in-vestigation that the electric dipoles are oriented along the xdirection and that the magnetic-dipole moments are orientedalong the y direction4 also see Fig. 1c. Thus, the spectralshape of the dispersion relations indicates that longitudinalelectric dipole-dipole coupling is dominant for propagationalong k1,0, whereas longitudinal magnetic dipole-dipolecoupling is dominant for propagation along k0,1 other-wise the photon energy in Fig. 3 should decrease rather thanincrease with increasing modulus of the in-plane wave num-ber. Waves resulting from magnetic dipole-dipole interac-tions have recently been discussed theoretically and havebeen called “magnetoinductive” waves.9 These magnetiza-tion waves are the classical analog of magnons for interact-ing quantum-mechanical spins. So far, these waves have onlybeen observed experimentally at megahertz frequencies.11Figure 3 also shows propagation directions other than thetwo principal directions k =k1,0 and k =k0,1. Here,the behavior is more complex and the photon energy versusin-plane momentum is a nonmonotonous function. We inter-pret this behavior in terms of the nontrivial competition be-tween the four types of interactions see discussion above:electric/magnetic and transverse/longitudinal.The spectral behavior at larger photon energies notshown is rather complex and dominated by various Ray-leigh anomalies, which are not at all related to the magnetic-dipole resonance. To verify that all spectral features dis-cussed in Figs. 2 and 3 are actually directly related to themagnetic-dipole resonance, we have fabricated another set ofsamples for which the upper of the two gold layers see Fig.1 is absent. Indeed, the transmittance dips shown in Fig. 2completely disappear and near-unity transmittance is ob-tained for all photon energies and angles  in the range ofFig. 2. In contrast, the various Rayleigh anomalies at higherphoton energies hardly change at all.To further test our interpretation, we compare themeasured data with numerical calculations based on an ad-vanced frequency-domain finite-element code. Bloch-periodic boundary conditions are applied in the x and ydirections12 on a square lattice with lattice constant a to al-low for oblique incidence of light. In the ±z direction weFIG. 1. Color a Scheme of negative-index metamaterial investigated. bMagnified view of one unit cell. Constituent materials are gold yellow andMgF2 green. Parameters are a=605 nm, wx=324 nm, wy =100 nm,t=30 nm, and s=32 nm. c The “magnetic atom” blue and the “electricatom” red in one unit cell are highlighted. The blue and red arrows sche-matically indicate a snapshot of the magnetic- and electric-dipole vectors,respectively.FIG. 2. Color a and b Measured oblique-incidence transmittancespectra for different angles  with respect to the surface normal, from 0° to40° in steps of 5° for clarity, the color alternates between black and red.The geometry is indicated on the top. c and d Calculated spectra cor-responding to a and b, respectively.FIG. 3. Color Photon energies of transmittance dips see vertical dashedlines in Fig. 2 vs in-plane wave number k as determined from the angle .a Experiment, b theory. Propagation directions are indicated by the in-plane wave vector k =kcos , sin and illustrated at the top. =0 cor-responds to Figs. 2a and 2c and = /2 to Figs. 2b and 2d.231118-2 Dolling et al. Appl. Phys. Lett. 89, 231118 2006Downloaded 19 Mar 2009 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jspapply transparent boundary conditions based on the perfectlymatched layer method.13,14 We discretize a unit cell with anunstructured mesh of about 600 prisms. Maxwell’s equationsare discretized using vectorial finite elements Whitney ele-ments of second order, leading to a sparse matrix equationwith about 130 000 unknowns.Geometrical parameters are as in Fig. 1. For the permit-tivity of gold we assume a frequency dependence accordingto the Drude model, with a plasma frequency of pl=1.33	1016 s−1 and a collision frequency of c=1.15	1014 s−1.The relative permittivity of the glass substrate is taken as2.25 and that of the MgF2 spacer layer as 1.90.Calculated results are shown in Figs. 2 and 3. Obviously,very good agreement between experiment and theory is ob-tained. In particular, the calculated dispersion relations quan-titatively agree with the experimental ones and reproduce theobserved anisotropy regarding propagation along the twoprincipal directions k =k1,0 and k =k0,1 as well as forthe other three directions.Finally, we address the question which parts of thenegative-index structure are important for the observed cou-plings. If, for example, the thin metal double wires markedred in Fig. 1c are completely eliminated in the calcula-tions, the transmittance dips in Fig. 2 hardly show any influ-ence on  not shown. This observation indicates that thethin metal double wires mediate both the above-mentionedstrong electric and magnetic dipole-dipole couplings. Thisobservation also unambiguously shows that the observedspectral shifts are not a property of the isolated magneticatoms.In conclusion, we have investigated the dispersion rela-tion of in-plane waves in negative-index photonic metamate-rials arising from the coupling of the photonic atoms themetamaterial is composed of. The longitudinal magneticwaves are the classical analog of quantum-mechanical spinwaves, well known as magnon excitations. Regarding experi-ments in the optical regime, such magnetization waves areobserved here for the first time to the best of our knowledge.For other propagation directions, we find a complex interplaybetween magnetization and polarization waves.The authors acknowledge support by the DeutscheForschungsgemeinschaft DFG and the State of Baden-Württemberg through the DFG-Center for Functional Nano-structures CFN within subproject A1.5. The authors ac-knowledge support through the DFG-Center Mathematics forKey Technologies Matheon. The research of one of theauthors S.L. is further supported through a “Helmholtz-Hochschul-Nachwuchsgruppe” VH-NG-232.1S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J.Brueck, Phys. Rev. Lett. 95, 137404 2005.2V. M. Shalaev, W. Cai, U. K. Chettiar, H. Yuan, A. K. Sarychev, V. P.Drachev, and A. V. Kildishev, Opt. Lett. 30, 3356 2005.3G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden,Science 312, 892 2006.4G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Opt.Lett. 31, 1800 2006.5J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, Phys. Rev. Lett.76, 4773 1996.6J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, IEEE Trans.Microwave Theory Tech. 47, 2075 1999.7V. A. Podolskiy, A. K. Sarychev, E. E. Narimanov, and V. M. Shalaev, J.Opt. A, Pure Appl. Opt. 7, 32 2005.8S. Zhang, W. Fan, K. J. Malloy, S. R. J. Brueck, N. C. Panoiu, and R. M.Osgood, Opt. Express 13, 4922 2005.9E. Shamonina and L. Solymar, J. Magn. Magn. Mater. 300, 38 2006.10M. L. Brongersma, J. W. Hartman, and H. A. Atwater, Phys. Rev. B 62,R16356 2000.11M. C. K. Wiltshire, E. Shamonina, I. R. Young, and L. Solymar, Electron.Lett. 39, 215 2003.12S. Burger, R. Klose, A. Schädle, F. Schmidt, and L. Zschiedrich, Proc.SPIE 5728, 164 2005.13L. Zschiedrich, R. Klose, A. Schädle, and F. Schmidt, J. Comput. Appl.Math. 188, 12 2006.14L. Zschiedrich, S. Burger, B. Kettner, and F. Schmidt, Proc. SPIE 6115,611515 2006.231118-3 Dolling et al. Appl. Phys. Lett. 89, 231118 2006Downloaded 19 Mar 2009 to 129.13.72.198. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp

Observation of magnetization waves in negative-index photonic metamaterials

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Observation of magnetization waves in negative-index photonic metamaterials

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