dentcho a. genov, shuang zhang and xiang zhang- mimicking celestial mechanics in metamaterials

8/3/2019 Dentcho A. Genov, Shuang Zhang and Xiang Zhang- Mimicking celestial mechanics in metamaterials

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ARTICLESPUBLISHED ONLINE: 20 JULY 2009 | DOI: 10.1038/NPHYS1338

Mimicking celestial mechanics in metamaterialsDentcho A. Genov1,2, Shuang Zhang1 and Xiang Zhang1,3*

Einsteins general theory of relativity establishes equality between matterenergy density and the curvature of spacetime. Asa result, light and matter follow natural paths in the inherent spacetime and may experience bending and trapping in a specificregion of space. So far, the interaction of light and matter with curved spacetime has been predominantly studied theoreticallyandthrough astronomicalobservations.Here, we propose to link thenewly emerged field of artificialoptical materialsto that ofcelestial mechanics, thus opening the way to investigate light phenomena reminiscent of orbital motion, strange attractors andchaos, in a controlled laboratory environment. The opticalmechanical analogy enables direct studies of critical light/matterbehaviouraround massive celestial bodies and,on the otherhand, points towardsthe design of novel optical cavitiesand photontraps for application in microscopic devices and lasers systems.

The possibility to preciselycontrol theflowof light by designingthe microscopic response of an artificial optical material hasattracted great interest in the field of optics. This ongoing

revolution, facilitated by the advances in nanotechnology, hasenabled the manifestation of exciting effects such as negativerefraction1,2, electromagnetic invisibility devices35 and microscopywith super-resolution1,6. The basis for some of these phenomenais the equivalence between light propagation in curved space andin a locally engineered optical material. Interestingly, a similarbehaviour exists in the general theory of relativity, where thepresence of matterenergy densities results in curved spacetimeand complex motion of both matter and light7,8. In the classicalinterpretation, this fundamental behaviour is known as the opticalmechanical analogy and is revealed through the least-actionprinciples in mechanics, determining how a particle moves in anarbitrary potential9, and the Fermat principle in optics, describingthe ray propagation in an inhomogeneous media10.

The opticalmechanical analogy provides a rather useful linkbetween the study of light propagation in inhomogeneous mediaandthe motionof massive bodiesor light in gravitational potentials.Surprisingly, a direct mapping of the celestial phenomena by ob-serving photon motion in a controlled laboratory environment hasnot been studied so far in an actual experiment. Here, we proposerealistic optical materials that facilitate periodic, quasi-periodicand chaotic light motion inherent to celestial objects subjectedto complex gravitational fields. This dense optical media (DOM)could be readily achieved with the current technology, withinthe framework of artificial optical metamaterials. Furthermore,we introduce a new class of specifically designed DOM in theform of continuous-index photon traps (CIPTs) that can serveas novel broadband, radiation-free and thus perfect cavities. The

dynamics of the electromagnetic energy confinement in the CIPTsis demonstrated through full-wave calculations in the linear andnonlinear regimes. Possible mechanisms for coupling light into theCIPTs and specific composite media are discussed.

Optical attractors and photonic black holes

One of the most fascinating predictions of the general theory ofrelativity is the bending of light that passes near massive celestialobjects such as stars, nebulas or galaxies. This effect constituted

1NSF Nanoscale Science and Engineering Center, University of California, Berkeley, California 94720, USA, 2College of Engineering and Science, Louisiana

Tech University, Ruston, Louisiana 71272, USA, 3Material Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA.

*e-mail: [email protected].

one of the first pieces of evidence for the validity of Einsteinstheory and provided a glimpse into the interesting predictionsthat were to follow. As shown, first by Schwarzschild and then by

others, when the mass densities of celestial bodies reach a certaincritical value (for example, through gravitation collapse), objectstermed black holes form in space. Those entities fall under a moregeneral class of dynamic systems, where a particular point, curveor manifold in space acts as an attractor of both matter and light.Such systems are of great interest for science not only in terms ofthe fundamental studies of lightmatter interactions but also forpossible applications in optical devices that control, slow and traplight. It is thus important to investigate the associated phenomenaunder controlled laboratory conditions.

In the general theory of relativity, a complex particle/photonmotionis observed whenever theinherent spacetime is described bya metric gij that depends on the spatial coordinates x= {x1,x2,x3}and universal time t. In such curved, non-static spacetimes, the

propagation of matter and light rays follows the natural geodesiclines and is described by the Lagrangian

L= 12

g00(x,t)t2 1

2gij (x,t)x

i xj (1)

where the derivatives are taken over an arbitrary affine parameterand natural units have been adopted. Here, we investigate theprospect to mimic the effect of curved spacetimes (equation (1)) onmatter andlight in thelaboratory.To achieve this, we recognize thatcomplex light motion is also observed in metamaterials describedby inhomogeneous permittivities and permeabilities and could berelated to light dynamics in curved space through the invarianceof Maxwells equations under coordinate transformations4,5,11.

Metamaterials exhibiting complex electric and magnetic responseshave been the focus of recent efforts to create negative-refractive-index media1,2, electromagnetic cloaking24 and as concentratorsof light12. The above systems, however, suffer from high intrinsicloss and a narrow frequency range of operation and thus cannotbe considered as prospective media to simulate light motionin a curved spacetime vacuum. Although the introduction ofnew metamaterials13 may improve the overall performance, it isimportant to rely only on optical materials that are non-dissipative

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ARTICLES NATURE PHYSICS DOI: 10.1038/NPHYS1338

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Figure 1 | Optical attractors and PBHs. a, Curved space corresponding to a PBH with g00 = 1 showing regions of spatial stretching and compression.Depending on their initial angular momentum (impact parameter s), the rays approaching the PBH will be deflected (s> b), attracted (s< b) or settle

(s= b), on an unstable orbit. b,c, A set of such critical photon trajectories that fall into the singularity for a=0 (b) or approach the unstable orbit at a= 1(c) for an impact parameter b= 2. d,e, The FDFD calculation of the magnetic field density (the incident magnetic field is |H0| = 1), corresponding to thecritical trajectories with b= 1.5a= 15 m (d) and b=0.5a= 5 m (e). In the calculations, very high energy densities are obtained close to the centralattractor with the maximum value restricted only by the finite length scale that could be numerically investigated. The incident wavelength is

=0.5 m and the Gaussian beam width is w=4.

and non-dispersive in general. For that, we consider a class ofcentrally symmetric metrics that can be written under coordinatetransformation in an isotropic form (see the Methods section). Forsuch curved spacetimes, light behaves in space similarly as in anoptical media with an effective refraction index

n= (g/g00)1/2 (2)

where g = g11 = g22 = g33. The equivalence manifested inequation (2) is the basis for the studies of celestial phenomena in anactual table-top experiment. It enables the use of non-dissipativeand non-dispersive dielectric materials. To demonstrate this, wepropose a two-parameter family of isotropic centrally symmetric

metrics g=g00((b/r)2

+ (1 a/r)2

), where r= |x| is the radialcoordinate and a and b are constants; constraint limrg00 = 1

is imposed to recover the flat space at large distances. The choiceof this particular spacetime, which is schematically represented inFig. 1a, has a number of important rationales. It exhibits a physicalsingularity at r= 0 and as shown below unstable photon orbitsare supported at r= a, defining a photon sphere for the system14.These characteristics are reminiscent of a gravitational black holein the general theory of relativity although the proposed metricsare not solutions to the Einstein field equations in a vacuum.Notwithstanding, here we refer to the considered system as aphotonic black hole (PBH), or a spacetime where a particular pointin space acts as an attractorof electromagnetic radiation.

Figure 1b,c demonstrates photon trajectories incident on thePBH with an impact parameter s, representing the minimal distance

to the centre provided the photon is not deflected, that is equalto a critical value b. The Lagrangian formalism based on thecurved spacetime equation (1) predicts that the impinging rays willinevitably reach their destiny at the central attractor/singularity(Fig. 1b), or asymptotically approach the photon sphere at r= a(Fig. 1c). The spiral trajectories represent a critical point in thePBH phase space (see Supplementary Information) such that allphotons approaching with impact factors less than or equal tothe critical parameter b will never escape the system. Figure 1d,eshows full-wave simulations of Gaussian beams incident on aDOM with a refraction index profile described by equation (2)that maps the PBH space. The incident beams have propagationdirections and initial positions set on the PBH critical trajectories

(as in Fig. 1b,c), and the light is concurrently torn apart, withhalf the beams being scattered into infinity following hyperbolicalpaths, whereas the rest spirals into the central attractor. This raybehaviour is consistent with the expected ray trajectories in theoriginal curved spacetime, but demonstrated in the equivalentDOM. In addition to the predicted ray behaviour, we observeelectromagneticenergy density fluctuations (jets) leaving the systemin radialdirections (see Fig. 1e). This effect is due tothe wavenatureof light interaction with the inhomogeneous refraction index, andaffects the electromagnetic rays that approach the PBH with impactfactors slightly above the critical value. At the centre of the system,owing to the spacial singularity inherent to the PBH metrics,the effective refractive index approaches infinity. The singularlycorresponds to an increasing stretching of space (see Fig. 1a), whichacts as an effective potential well attracting light. Concurrently, the

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optical attractor can be consideredas an effective cavity, decreasingboth the group velocity and wavelength, which in turns gives rise toexceptionally high local energy densities.

Although the singularity in the refractive index profile of theideal PBH is of fundamental interest for science, it is probablynot feasible in practice. Such index profiles or the associated lowgroup velocities have been discussed in the literature and specificsystems have been suggested such as high-velocity vortex flow 15,16,microstructured nonlinear fibres17, metamaterial waveguides18,

metal nanoparticles19 and BoseEinstein condensates20. Further-more, close to the centre singularity, the ray picture fails and onecannottrust the predictions as per Fig. 1b,c. However, experimentalobservation of the main optical phenomena related to systemssuch as PBH or gravitational black holes does not necessitateexceptionally large refraction indices, or extremely low phase/groupvelocities. In fact, next we show that theunderlying electromagneticphenomena can be demonstrated using DOM in the form of simplebinary metaldielectric or pure dielectric composites.

Figure 2 shows finite refractive index profiles n mapped tothat of a PBH, obtained by varying the volume fraction p ofcopper metal nanoparticles suspended in air (Fig. 2a) or air holesin a GaAs substrate (Fig. 2b). These composite materials can benano-engineered through controlled deposition of metal particles

in a dielectric host or growth of deep subwavelength in size airpockets in high-index materials. The electromagnetic phenomenaassociated with these realistic systems are shown in Fig. 2c,d,respectively. Despite the finite indices, the aircopper mixturecan provide a complete picture of the far-field scattering profileinherent to a PBH (Fig. 2c), and photon trapping into theunstableorbit (at the photon sphere) as well as critical behaviour at lowerimpact factors can be demonstrated in the airGaAs composite(Fig. 2d). Specifically, at sufficiently large distances from the PBH(rmax(a,b)), the light rays are deflected by an angle =2a/s(Fig. 2c) that has the same dependence on the impact parameter s asthat of the Einstein angle for gravitational lensing, but with reversedsign8. This results in a diverging energy flux at large distances withmaximum field intensity observed at ym = (8a|x|)1/2. As metalsare highly dispersive, the aircopper system will have a narrowoperational bandwidth around the plasma resonance wavelengthp = 0.27m, and the incident light will suffer energy loss, whichalthough much lower compared with current metamaterials is stillconsiderable (the imaginary part of the effective index is shown inFig. 2a). To overcome these obstacles, one can instead design a puredielectric media based on airGaAs composites. Owing to the largerrefractive index of the semiconductor (nmax =nGaAs = 3.25; ref. 21),for this system, it is even possible to follow the light propagationin the region of space below the photon sphere with significantlyreduced intrinsic losses and for a wide range of frequencies (belowthe GaAs bandgap). This is shown in Fig. 2d, where four Gaussianbeams are sent into the system with initial angular moments andpositions centred on critical trajectories. All beams asymptotically

approach the photon sphere where they are ripped apart, with halfthe energy being scattered into the far-field, and the rest collapsestowards the centre. The electromagnetic density fluctuations orjets of energy leaving the photon sphere are clearly seen. Toguarantee that this new phenomenon is not due to the truncatedregion below the photon sphere, an index-matched absorption coreis used. The absorption core has the role of the central attractorand assures that all scattering and wave phenomena are properlyrepresented by the DOM.

Overall, the DOM as per Fig. 2c provides the means to studyelectromagnetic wave interactions with critical points in spacetime(in particular the photon sphere) in a relatively simple table-top experiment. Such critical (saddle) points have an importantrole in establishing the overall system dynamics and can serveas a source of instability and chaos. The full range of DOMs

00

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Figure 2 | Mimicking the PBHelectromagnetic phenomenon in the

laboratory. ad, Mixtures of low-concentration copper metal nanoparticles

in air (a) and airGaAs composite media (b) can be used to provide the

far-field scattering of the PBH with a scattering angle = tan1(ym/|x|) (c)and critical behaviour in the vicinity of the photon sphere ( d), respectively.

The effective refractive indices of the composites are presented with solid

and short-dashed lines for the real and imaginary parts, respectively, and

long-dashed lines are used for the metal/air volume fractions. The shaded

areas in a and b represent the region of space where the refraction indices

are truncated. In the FDFD calculations, we have set c, a=0.25 m,b=

1.5a and d, a

=30

m, b

=50

m. The incident wavelengths and beam

widths are =0.24m (w= 20) and = 1.55 m (w=8), respectively.For the airGaAs composite, the absorption coefficient is fixed at 5 cm1

(the intrinsic semiconductor absorption for frequency below the

bandgap)21 everywhere in space except within the truncation region where

it is increased to 100 cm1 to simulate the energy sink represented by thecentral attractor.

applications including concentrating and storing electromagneticenergy are discussed next.

Photon traps, orbital motion and high-Q-factor cavities

The equivalence between light ray motion around massive celestialobjects and in an artificially engineered optical medium hasstrong implications as the general formalism, represented by the

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Figure 3 | CIPTs based on airGaInAsP composite media. a,b Refractive index profile (a) and volume fraction (b) of the GaInAsP. c, FDFD simulations of

bound and stable light propagation in the form of A, a circular orbit with period 2 and B, a Rosetta type of orbit with period 4. In the calculations, a

linear source with width w=6 and wavelength = 1.55 m is used. df, Poincare maps (PT) corresponding to all possible photon trajectories for linear ( d)and nonlinear (e,f) refractive indices. In the linear case, the photon motion follows stable periodic and quasi-periodic trajectories for all initial conditions.

With the introduction of nonlinearity, the stability of the system is disturbed with some trajectories passing through the cavity boundaries and leaving the

system. f, For above-critical values of the nonlinear index nNL/nmax=0.2, chaotic behaviour is observed similarly to the three-body problem in celestialmechanics. In generating the Poincare maps, the sc ale-invariant parameters r/a and r/a are sampled at times tm = 2m/NL that are an integer number mof the nonlinear period (or T-period map), and the derivative is taken with respect to the azimuth angle .

Lagrangian equation (1), also describes restricted motion of matterin potential fields. Namely, under certain conditions the dynamicsof matter in complex gravitational potentials can be replicated bythe propagation of light in DOM. Here, we introduce an interestingprospective, with significant practical importance, which is tomimic planetary motion in the form of a stable and confinedphoton trajectory.

According to Bertrands theorem

22

, a general particle orbitin classical mechanics is stable and closed under any perturba-tions only in two types of central potential: Keplers problemU=m/r< 0 and the harmonic oscillator U=mr2 0, where mis the mass. The theorem, however, demands further restrictions inoptics where it can be shown that Keplers potential no longer sup-ports stable and closed circular or elliptical orbits for photons. Thisconclusion follows from the kinematic analogy10, which casts thedynamic eikonal (ray) equation in a classical mechanical form witheffective potential and kinetic energies U=K=n2/2directlyre-lated to the refraction index. Thus, in contrast to a particle that mayhave positive or negative total energyU+K, photons are equivalentto zero energy states, which also follows from the null geodeticcondition L0, or the finite speed of light. As a consequence, onlyparabolic solutionsare supported in this particular case.

Although Bertrands theorem states that bound and exponen-tially stable photon orbits that are closed under any perturbationsmay not exist in centrally symmetric media, it does not precludebound and stable motion that is not necessarily closed underperturbations; that is, the perturbed trajectory may cover a finitevolume of space. Here, we propose a universal family of opticalsystems that exhibit those characteristics. The effective refractive

indices can be directly obtained from the stability condition of theLagrangian (1) resulting in

n(r)= Ar

exp

(r)

rdr

(3)

where(r) is an arbitrarymonotonously decreasing function andAis a constant (see Supplementary Information). The above family ofrefractive indices, which we refer to as CIPTs,support stablecircularand complex ray motion.

To illustrate this statement, we consider the most simplecase where d/dr=/a( 0). The corresponding refractionindices are then obtained from equation (3) and can be writtenas n = ns(a/r)1er/a, where a is the radius of the desiredcircular orbit and ns is an arbitrary scaling factor. Furthermore,

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we choose the = 2 case, and design the CIPT as an airGaInAsPcomposite. This specific material system has been selected toprovide operation at the infrared spectral range, using the highrefractive index of the GaInAsP at low absorption, which forfrequencies below the bandgap (hg = 1.2 eV) is less than 1cm1(ref. 21). The actual index and air volume fractions are shownin Fig. 3a,b. Full-wave simulations of two light trajectories forlinear sources positioned at rin = a and rin = 1.62a are shown inFig. 3c. According to the CIPT phase space shown in Fig. 3d, all

ray trajectories are confined in space and the photon trapping isachieved without involving index discontinuities at the boundariesbetween different media. These bulk type of cavities where thetrapped light satisfies the Lyapunov stability condition, constitutes asubstantial improvement compared with the conventional gradientrefraction index waveguides, and to the best of our knowledge, hasnot been studied in the literature. The CIPTs may provide newdirections for the development of optical devices and high-Q-factorcavities for electromagnetic energy storage with minimal radiationlosses. Specifically, coupling CIPTs with a light source, such asa quantum dot, could result in enhanced spontaneous emission,strong atomphoton coupling (cavity quantum electrodynamics),Raman lasing or enhanced high-order optical nonlinearities. Thesephenomena are due to the decreased photon decay rate, which

for CIPTs is = i + r i = Imn/Ren, where i and rstand for intrinsic dissipation and radiation loss, respectively.The CIPT quality factor Q = / is thus limited only bythe intrinsic loss and can be on par or larger than those ofphotonic crystals23 and disc microcavities24. In the case of theairGaInAsP composite, the quality factor can be as high as106 when taking into account the actual material absorptionin the semiconductor21. Furthermore, the Laypunovs stabilityof the photon trajectories implies that finite index variationsdue to fabrication imperfections will not lead to photon lossfrom the CIPT; the photons simply hop to another close-bybound trajectory (see Fig. 3d). This is another advantage of CIPTsin respect to conventional cavities and waveguides where lightscattering at surface or volume imperfections can lead to photons

escaping from the system.

Nonlinearity and chaos

Theonsetof chaos in dynamic systems is oneof themost fascinatingproblems in science and is observed in areas as diverse as molecularmotion25, population dynamics26 and optics27. In particular, aplanet around a star can undergo chaotic motion if a perturbation,suchas another large planet, is present. However, owing to the largespatial distances between the celestial bodies, and the long periodsinvolved in the study of their dynamics, the direct observationof chaotic planetary motion has been a challenge28. The use ofthe opticalmechanical analogy may enable such studies to beaccomplished in the laboratory. Furthermore, in the case of theCIPTs,where trapping of light occurs with unprecedented lifetimes,

theonsetof chaos could enableaccessto those closed raytrajectoriesfrom thefar-field. This is directly addressedby involvingthe analogywith the celestial mechanics where a third-body perturbation canbe modelled as a periodic nonlinear refraction index modulation;n(r, t) = nCIPT(r) + nNL sin2(NLt) introduced by an externalelectrical or electromagnetic source.

The Poincae maps of the photon motion under these settingsare shown in Fig. 3e,f. In contrast to the non-perturbed linear case(Fig. 3d), where all photon trajectories are closed and determinedby the first integral of the system, the introduction of nonlinearharmonic variations brings a fundamentally different behaviour.For nNL/nmax = 0.05 (Fig. 3e), and period of the modulationsT= 2/NL = 6c/a, we observe substantial expansion of the origi-nal phase space, the boundary of which is marked with a blue line.The original invariant curves have broken up into four separate

domains. As a result, photons maypass at distances r> rmax=1.74a,where the CIPT refractive index truncates to unity. Those light rayswill escape on reaching the outer boundary or concurrently, exter-nal light could be coupled into the otherwise perfect cavity (in thelinear sense). This is accomplished if the nonlinearity is switched offfor times t< a/c, leading to a fraction of the impinging light beingtrapped inside the system. Similar behaviour could be observedeven for moderate nonlinearities nNL/nmax


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9. de Maupertuis, P. L. M. Accord de diffrentes lois de la nature qui avaientjusquici paru incompatibles. Mm. As. Sc. Paris 417427 (1744).

10. Evans, J. The ray form of Newtons law of motion. Am. J. Phys. 61,347350 (1993).

11. Plebanski, J. Electromagnetic waves in gravitational fields. Phys. Rev. 118,13961408 (1960).

12. Kildishev, A. V. & Shalaev, V. M. Engineering space for light via transformationoptics. Opt. Lett. 33, 4345 (2008).

13. Valentine, J. et al. Three dimensional optical metamaterial exhibiting negativerefractive index. Nature 455, 376379 (2008).

14. Shapiro, S. L. & Teukolsky, S. A. White Dwarfs, and Neutron Stars: The Physicsof Compact Objects (Wiley, 1983).

15. Leonhardt, U. & Piwnicki, P. Relativistic effects of light in moving media withextremely low group velocity. Phys. Rev. Lett. 84, 822825 (2000).

16. Visser, M. Comment on Relativistic effects of light in moving media withextremely low group velocity. Phys. Rev. Lett. 85, 52525252 (2000).

17. Philbin, T. G. et al. Fiber-optical analog of the event horizon. Science 319,13671370 (2008).

18. Tsakmakidis, K. L., Boardman, A. D. & Hess, O. Can light be stopped inrealistic metamaterials? Nature 455, E11E12 (2008).

19. Bashevoy, M., Fedotov, V. & Zheludev, N. Optical whirlpool on an absorbingmetallic nanoparticle. Opt. Express 13, 83728379 (2005).

20. Hau, L. V. et al. Light speed reduction to 17 m s1 in an ultracold atomic gas.Nature 397, 594598 (1999).

21. Levinstein, M., Rumyantsev, S. & Shur, M. (eds) Handbook Series onSemiconductor Parameters Vol. 1, 2 (World Scientific, 1996, 1999).

22. Bertrand, J. Mecanique analytique. C. R. Acad. Sci. 77, 849853 (1873).23. Painter, O. et al. Two-dimensional photonic band-gap defect mode laser.

Science 284, 18191821 (1999).24. Armani, D. K. et al. Ultra-high-Q toroid microcavity on a chip. Nature 421,

925928 (2003).25. Wu, G. Nonlinearity and Chaos in Molecular Vibrations (Elsevier, 2005).26. Costantino, R. F., Desharnais, R. A., Cushing, J. M. & Dennis, B. Chaotic

dynamics in an insect population. Science 275, 389391 (1997).

27. Zheludev, N. I. Polarization instability and multistability in nonlinear optics.Usp. Fiz. Nauk 157, 683717 (1989).

28. Malhotra, R., Holman, M. & Ito, T. Chaos and stability of the solar system.Proc. Natl Acad. Sci. USA 8, 12342 (2001).

29. Islam, M. N., Ippen, E. P., Burkhardt, E. G. & Bridges, T. J. Picosecondnonlinear absorption and four-wave mixing in GaInAsP. Appl. Phys. Lett. 47,10421044 (1985).

30. Lucchetti, L. et al. Colossal optical nonlinearity in dye doped liquid crystals.Opt. Commun. 233, 417424 (2004).

31. Brzozowski, L. et al. Direct measurements of large near-band edge nonlinearindex change from 1.48 to 1.55m in InGaAs/InAlGaAs multiquantum wells.

Appl. Phys. Lett. 82, 44294431 (2003).32. Bruggeman, D. A. G. Berechnung verschiedener physikalischer Konstanten

von heterogenen Substanzen. Ann. Phys. (Leipzg) 24, 636679 (1935).33. Maxwell-Garnett, J. C. Colours in metal glasses and in metallic films. Phil.

Trans. R. Soc. Lond. 203, 385420 (1904).

AcknowledgementsThis work has been supported by US Army Research Office ARO MURI program

50432PH-MUR, the NSF Nano-scale Science and Engineering Center (NSEC) under

Grant No. CMMI-0751621 and Louisiana Board of Regents under contract number

LEQSF (2007-12)-ENH-PKSFI-PRS-01. We would also like to thank G. Bartal and D. Pile

for important discussions and assistance.

Author contributionsD.A.G. conceived and implemented the theory and numerical simulations, designed

the DOM and CIPT media and prepared the manuscript; S.Z. and X.Z. contributed

extensivelyin the dataanalyses and conceptualization,and editedthe manuscript.

Additional informationSupplementary information accompanies this paper on www.nature.com/naturephysics.

Reprints and permissions information is available online at http://npg.nature.com/

reprintsandpermissions. Correspondence and requests for materials should be

addressed to X.Z.

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dentcho a. genov, shuang zhang and xiang zhang- mimicking celestial mechanics in metamaterials

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