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2009-nature Stereometamaterials

Stereometamaterials

Na Liu 1,Hui Liu 2,Shining Zhu 2and Harald Giessen 1*

The subdiscipline of chemistry that studies molecular structures in three dimensions is called stereochemistry.One important aspect of stereochemistry is stereoisomers:materials with the same chemical formula but different spatial arrangements of atoms within molecules.The relative positions of atoms have great in?uence on the properties of chemical substances.Here,in analogy to stereoisomers in chemistry,we propose a new concept in nanophotonics,namely stereometamaterials,which refer to metamaterials with the same constituents but different spatial arrangements.As a model system of stereometamaterials,we theoretically and experimentally study meta-dimers,which consist of a stack of two identical split-ring resonators in each unit cell with various twist angles.The interplay of electric and magnetic interactions plays a crucial role for the optical properties.Speci?cally,the in?uence of higher-order electric multipoles becomes clearly evident.The twisting of stereometamaterials offers a way to engineer complex plasmonic nanostructures with a tailored electromagnetic response.

he word ‘stereo’in Greek means ‘relating to space’or ‘three-dimensional’.In stereochemistry,the characteristics of organic compounds depend not only on the nature of the atoms comprising the molecules (constitution)but also on the three-dimensional arrangement of these atoms in space (con?gur-ation)1.For example,the spatial structure of a protein molecule determines its biological activities.In photonics,metamaterials are structured media,consisting of arti?cial ‘atoms’with unit cells much smaller than the wavelength of light 2–4.Such metamaterial atoms can be designed to yield electric as well as magnetic dipole moments,leading to effective negative permittivity and negative permeability.A medium with simultaneous negative permittivity and negative permeability can exhibit negative refraction and unique reversed electromagnetic properties 5,6.As well as their appli-cability in constructing effective media,metamaterials have also been used as prototypes for studying coupling effects between arti-?cial atoms in three-dimensional structures 7–9.However,the mech-anism of interactions arising from different spatial arrangements of such atoms has thus far not been examined.Inspired by the concept of stereochemistry,we now investigate the coupling effects of arti?-cial atoms in three-dimensional metamaterials from a novel per-spective.We study a set of stereometamaterials,each having unit cells consisting of two stacked split-ring resonators (SRRs)with identical geometry (same constitution);however,the two SRR atoms are arranged in space using different twist angles to achieve various structures (different con?gurations).We term these struc-tures stereo-SRR dimers.We theoretically and experimentally demonstrate that the optical properties of these stereo-SRR dimers can be substantially modi?ed by altering the twist angles between the two SRR atoms,arising from the variation of electric and mag-netic interactions between them.Speci?cally,we investigate how the electric and magnetic interactions depend on the spatial arrangement of these SRR constituents.The nontrivial magnetic interaction makes metamaterials more versatile in nanophotonics than stereoisomers in chemistry,where generally only electric interactions are taken into account.Furthermore,we show that the inclusion of the higher-order electric multipolar interactions is essential to understanding the physical implications of the twisting dispersion.A theoretical model based on a Langrangian formalism is used to interpret the evolution of the coupling effects as a function of twist angle.

Stereometamaterial design

Figure 1a illustrates the geometry of the stereo-SRR dimer metamaterials together with their design parameters.Each unit cell consists of two spatially separated SRRs,which are twisted at an angle w with respect to one another.The SRRs are embedded in a homogeneous dielectric with 1?1(that is,air).For excitation of these SRR dimer metamaterials,we use normally incident light with its polarization along the x -direction as shown in Fig.1a.In order to gain insight into the resonant behaviour as well as coupling mechanisms for various stereo-SRR dimer meta-materials,we ?rst study three speci?c dimer systems with twist angles w ?0,90and 1808.The insets of Fig.1b–d present the schematics of the three structures,in which the vertical distance between two SRRs is set at s ?100nm.Numerical simulations were performed based on a commercial ?nite-integration time-domain algorithm,and the simulated transmittance spectra are shown in Fig.1b–d.For each system there are apparently two observable resonances (v 2and v t).To understand these spectral characteristics,current and magnetic ?eld distributions at the rel-evant resonances are calculated.For the 08twisted SRR dimer metamaterial,the electric component of the incident light can excite circulating currents along the two SRRs,giving rise to induced magnetic dipole moments in the individual SRRs.As shown in Fig.2a,the electric dipoles excited in the two SRRs

oscillate anti-phase and in-phase at resonances v 02and v 0t

,respectively.The resulting magnetic dipoles are aligned antiparal-lel at resonance v 02,whereas they are parallel at resonances v 0t

.The above phenomenon can be interpreted as the plasmon hybridization 10–12between the two SRRs due to their close proxi-mity.In the hybridization scheme,each SRR can be regarded as an arti?cial atom.The two SRR atoms are bonded into an SRR dimer or SRR ‘molecule’due to the strong interaction between them.Such interaction leads to the formation of new plasmonic modes,arising from the hybridization of the original state of an individual SRR.For the con?guration of the 08twisted SRR dimer system,the two excited electric dipoles are transversely coupled,while the two magnetic dipoles are longitudinally coupled.In the case of a transverse dipole–dipole interaction,the antisymmetric and symmetric modes are at the lower and higher resonance frequencies,respectively.In contrast,in the

4.Physikalisches Institut,Universita

¨t Stuttgart,D-70569Stuttgart,Germany,2Department of Physics,Nanjing University,Nanjing 210093,People’s Republic of China;*e-mail:

Giessen@physik.uni-stuttgart.de

case of longitudinal dipole–dipole interaction,the two magnetic dipoles should align parallel at the lower resonance frequency and antiparallel at the higher resonance frequency 12.It is evident that for the 08twisted SRR dimer system (see Fig.2a),the resonance levels are determined according to the picture of transverse electric dipole–dipole interaction,with the antisym-metric (symmetric)mode having the lower (higher)resonance frequency.In essence,the two coupling mechanisms,that is,the electric and magnetic dipolar interactions,counteract one another and the electric interaction dominates in this system.For the 908twisted SRR dimer metamaterial,circular currents in the underlying SRR cannot be directly excited by the incident light due to its orientation with respect to the external electric ?eld.In a sense,the underlying SRR can be regarded as a ‘dark atom’at the resonant frequency of the ring 13.Nevertheless,for the coupled dimer system,on resonance,excitation from the upper SRR can be transferred to the underlying one by the interaction between the two SRRs,which can also lead to the formation of new plasmo-nic modes (v 902and v 90t

).Interestingly,because the electric ?elds in the slit gaps of the two SRRs are perpendicular to one another,the electric dipole–dipole interaction equals zero.In addition,as the higher-order multipolar interaction is negligible in a ?rst approxi-mation,the electric coupling in the 908twisted SRR dimer system

can thus be ignored.As a consequence,the resonance levels are determined in line with the picture of longitudinal magnetic dipole–dipole coupling.As shown in Fig.2b,at resonances v 902and v 90

t,the resulting magnetic dipoles in the two SRRs are aligned parallel and antiparallel,respectively.

For the 1808twisted SRR dimer metamaterial,the interaction between the two SRRs results in new plasmonic modes,v 1802and v 180t

.Notably,from the current and magnetic ?eld distributions

as shown in Fig.2c,resonances v 1802and v 180t

are associated with the excitation of the electric dipoles in the two SRRs oscillating anti-phase and in-phase,respectively.The two resulting magnetic dipoles are aligned parallel and antiparallel,accordingly.In essence,the transverse electric and longitudinal magnetic inter-actions contribute positively in the 1808twisted SRR dimer system.This leads to the largest spectral splitting,which is a direct indication of the coupling strength.Based on the above dis-cussions,we infer that the optical properties of stereometamaterials depend dramatically on the spatial arrangement of metamaterial constituents.Speci?cally,the possibility of tuning the resonant behaviour by simply varying the relative twist angles makes stereo-metamaterials particularly interesting as model systems for explor-ing and comprehending different coupling mechanisms in complex three-dimensional plasmonic structures.

(x )

H y )

150

175200a

b T r a n s m i t t a n

c e

Frequency (THz)

150

175200Frequency (THz)

150

175200Frequency (THz)

Figure 1|Structural geometry and numerical simulation.a ,Schematic of the stereo-SRR dimer metamaterials with de?nitions of the geometrical parameters:l ?230nm,h ?230nm,w ?90nm,t ?50nm,and s ?100nm.The periods in both x and y directions are 700nm.b –d ,Simulated transmittance spectra for the 08(b ),908(c )and 1808(d )twisted SRR dimer metamaterials.All the structures are embedded in air.

T wist angle

To provide deeper insight,the dependence of the optical properties of the stereo-SRR dimer metamaterials on twist angle is investigated. Figure3presents the simulated twisting dispersion curves(black squares)of these stereometamaterials,in which the resonance pos-itions are extracted from the transmittance spectra of different struc-tures.It is apparent that by increasing the twist angle w,the two resonance branches?rst tend to converge,with the vtbranch shift-ing to lower frequencies,while the v2branch shifts to higher fre-

quencies.An avoided crossing is observed at w t,which is around608.Subsequently,the two branches shift away from one another.In order to clarify the underlying physics of the twisting dispersion curves,we introduce a Lagrangian formalism14.We start from the analysis of a single SRR and then expand it to coupled stereo-SRR dimer systems.One SRR can be modelled by an equivalent LC circuit with a resonance frequency

v f ?1/(LC)1/2.It consists of a magnetic coil(the metal ring)

with inductance L and a capacitor(the slit of the ring)of capacitance

C.If we de?ne the total charge Q accumulated in the slit as a gen-

eralized coordinate,the Lagrangian of an SRR can be written as

G?(LQ˙2/2)2(Q2/2C).Here,LQ˙/2refers to the kinetic energy

of the oscillations,and Q2/2C is the electrostatic energy stored in

the slit.Consequently,the Lagrangian of the coupled SRR dimer

systems is a combination of two individual SRRs with the additional

electric and magnetic interaction terms

_Q2

àv2f Q21

_Q2

àv2f Q22

tM H_Q1_Q2àM E v2f Q1Q2

ácos wàaácos w

eT2tbácos w

eT3

àá

e1T

Here,Q1and Q2are oscillating charges in the respective SRRs,and

M H and M E are the mutual inductances for the magnetic and elec-

tric interactions,respectively.Apart from the electric dipole–dipole

interaction,the contributions from the higher-order electric multi-

polar15interactions are also included.a and b are the coef?cients of

the quadrupolar and octupolar plasmon interactions16,respectively.

They serve as correction terms to the electric dipolar interaction.It

is straightforward to derive from equation(1)that the major inter-

action items for08and1808cases are M H_Q1_Q2àM E v2

Q1Q2and

M H_Q1_Q2tM E v2

Q1Q2,respectively.It is in accord with the above

simulation results that the magnetic and electric interactions con-

tribute oppositely and positively for08and1808twisted SRR

dimer metamaterials,respectively.For the908twisted SRR dimer

metamaterial,only the magnetic interaction plays a key role,as rep-

resented by the interaction term M H_Q1_Q2.Subsequently,by solving

the Euler–Lagrange equations

d t

@_Q i

@Q i

?0;i?1;2e2T

the eigenfrequencies of these stereo-SRR dimer systems can be

obtained as

v+?v0á

?????????????????????????????????????????????????????????????????????????????????????

1+k Eácos wàaácos w

eT2tbácos w

eT3

àá

1+k H

e3T

at ω90+

at ω

180

at ω

180

I I

Current density Magnetic ?eld Current density Magnetic ?eld

Figure2|Numerical current and magnetic?eld distributions.a–c,Current and magnetic?eld distributions at respective resonances for the08(a), 908(b)and1808(c)twisted SRR dimer metamaterials.Lower left: schematics of currents(I)in two SRRs.Lower right:schematics of the alignments of the magnetic(m)and electric(p)dipoles.At08,transverse electric and longitudinal magnetic interactions work against one another, whereas at1808they add together.At908,only longitudinal magnetic interaction is present.

140

160

180

200

220

240

260

(

)

Twist angle φ(deg)

Figure3|T wisting dispersion of the stereo-SRR dimer metamaterials.

Black squares represent the numerical data.Red lines represent the?tting curves calculated from the Lagrangian model,in which the avoided crossing

is clearly visible at w

.The black arrows represent the alignment of the magnetic dipoles at lower and higher resonance frequencies at twist angles w?08and1808.The grey lines represent the?tting curves calculated from the Lagrangian model without considering the higher-order electric multipolar interactions.No avoided crossing is observable in this case.

where k E ?M E /L and k H ?M H /L are the coef?cients of the overall electric and magnetic interactions,respectively.By ?tting the twist-ing dispersion curves,the corresponding coef?cients are estimated to be k E ?0.14,k H ?0.09,a ?0.8and b ?–0.4.Notably from Fig.3,the ?tting curves (in red lines)reproduce the numerical data quite well and the avoided crossing is clearly observable around 608.This shows that the Lagrangian model can

quantitatively corroborate the results from the numerical simu-lations.It is of crucial importance that the higher-order electric mul-tipolar interactions account for the existence of the avoided crossing.Owing to the ?nite length of the SRR ring,discrete electric plasmon modes characterized by different spatial symmetries can be excited by the incident light.The surface charges in the SRR ring are a superposition of such fundamental plasmon modes of the ring 16

120 130 140 150 160 170

0.0

0.1

0.20.30.40.50.60.70.80.91.0T r a n s m i t t a n c e

Frequency (THz)

120 130 140 150 160 170

0.0

0.1

0.20.30.40.50.60.7

0.80.91.0T r a n s m i t t a n c e

Frequency (THz)

120 130 140 150 160 170

0.0

0.1

0.20.30.40.50.60.7

0.80.91.0T r a n s m i t t a n c e

Frequency (THz)

b c d

Figure 4|Field-emission electron microscopy images and experimental measurement.a –c ,Oblique views of the 08(a ),908(b )and 1808(c )twisted gold

SRR dimer metamaterials.Insets:normal views.The structures were fabricated on a glass substrate.The SRRs were embedded in a photopolymer (PC403),which served as the dielectric spacer.d –f ,Experimental transmittance spectra for the 08(d ),908(e ),and 1808(f )twisted SRR dimer metamaterials.The black and red curves represent the experimental and simulated results,respectively.For the 908twisted SRR dimer structure,an analyser is applied behind the sample,which is rotated by 758with respect to the polarization of the incident light.

To reveal the signi?cant role of the higher-order electric multipolar interactions,the grey lines in Fig.3display the twisting dispersion curves,in which only the dipolar coupling effect is taken into account;that is,a ?0and b ?0.The best ?t is achieved with k E ?0.2and k H ?0.09.Obviously,despite the fact that the grey curves can ?t most parts of the numerical data,no avoided crossing is predicted.Instead,the v tand v 2branches converge at w t .Therefore,it has to be emphasized that although the electric and magnetic dipolar interactions are the essential mechanisms,the higher-order electric multipolar interactions should also be carefully considered for fully understanding the origin of the spectral charac-teristics of the stereometamaterial systems.

The angle where the avoided crossing occurs in the twisting dispersion spectrum is correlated with the geometry of the SRRs as well as the vertical distance between the two SRRs.For the speci?c stereo-SRR dimer metamaterials we investigated here,the avoided crossing appears at $608.Based on detailed simulated ?eld distri-bution studies,we found that this angle is also a transition angle where the higher and lower frequency modes exchange their magnetic dipole alignments from parallel to antiparallel.At angles smaller than angle w t ,the two magnetic dipoles are aligned parallel (antiparallel)at resonance v t(v 2).The electric coupling effect dominates in this regime.With continuous increase of the twist angle,due to the displa-cement of the two SRRs,the electric coupling contributes less effec-tively.Consequently,the splitting of the two resonance branches starts to decrease.This situation remains until the transition angle is reached,where the electric and magnetic dipole–dipole interactions cancel one another.The higher-order electric multipolar interactions account for the avoided crossing of the two resonance branches.After angle w t ,the electric coupling continues to decrease.As a result,the resonance levels are determined according to the scheme of magnetic dipole–dipole coupling,that is,the parallel and antiparallel align-ments of the magnetic dipoles in the two SRRs correspond to the lower and higher frequency resonances,respectively.When the twist angle reaches 908,the electric coupling quenches and is negli-gible.This represents a purely magnetic dipole–dipole coupling situ-ation.Subsequently,with further increase of the twist angle,the displacement of the two SRRs is reduced and the electric coupling comes into play again.Because of the orientation of the two SRRs,the electric and magnetic coupling can contribute positively,giving rise to a larger splitting of the two resonance branches with increasing twist angle.The splitting ?nally reaches its maximum at w ?1808.The structures of stereometamaterials are compatible with nano-fabrication stacking techniques 7,9.We fabricated three stereo-SRR dimer metamaterials with speci?c twist angles w ?0,90and 1808,as illustrated in the insets of Fig.1b–d.In the experiment,the struc-tures were fabricated on a glass substrate.Gold SRRs were embedded in a photopolymer (PC403),which served as the dielectric spacer.A spacer of s ?120nm was applied in order to achieve surface planar-ization for stacking the second SRR layer.The electron micrographs of the fabricated SRR dimer metamaterials were obtained by ?eld-emis-sion scanning electron microscopy.Figure 4a–c presents oblique views of the 0,90and 1808twisted SRR dimer metamaterials,in which the underlying SRRs are clearly visible.The insets of Fig.4a–c show the normal views,demonstrating the good accuracy of lateral alignment for the different SRR layers.To experimentally investigate the optical properties of these SRR dimer metamaterials,the near-infrared transmittance spectra of the samples at normal inci-dence were measured by a Fourier-transform infrared spectrometer with electric ?eld polarization as illustrated in Fig.1.The measured transmittance spectra are presented by black curves in Fig.4d–f and the simulated spectra as red dashed curves.The resonance pos-itions are redshifted compared to those of the corresponding reson-ances in Fig.1b–d due to the presence of glass substrate and dielectric spacers.For a reasonable comparison with the experiment,in the simulations in Fig.4d–f,gold with a three times higher

damping constant as that used in Fig.1b–d was used to account for the surface scattering and grain boundary effects in the thin ?lm of the real systems 17.The overall qualitative agreement between exper-imental and simulated results is quite good.The discrepancies are most likely due to tolerances in fabrication and assembly,as well as signi?cant broadening in the experiment.For 08and 1808twisted

SRR dimer structures,the lower resonances v 02and v 1802

are less dis-tinctly visible than the higher resonances v 0tand v 180t

,respectively (see spectra in Fig.4d,f).This is due to the fact that for both dimer structures,the electric coupling plays a key role.At the lower

resonance frequencies (v 02and v 1802

),the electric dipoles in the two SRRs oscillate anti-phase.Such resonances are not easily excited by light.On the other hand,at the higher resonance frequencies (v 0tand v 180

t),the electric dipoles in the two SRRs oscillate in-phase.Such resonances can strongly couple to light.For the 908twisted SRR dimer structure,the splitting of the resonances is clearly observa-ble when an analyser is applied behind the sample,which is rotated by 758with respect to the polarization of the incident light.This is due to the polarization rotation effect arising from the chirality 18,19of the 908twisted structure.

The new concept of stereometamaterials adds a signi?cant degree of freedom through the interplay of electric and magnetic interactions,and tremendously enhances the versatility of nanophotonic structures.Stereometamaterials allow us to use higher-order electric multipolar as well as magnetic interactions,which can be nearly as large as the electric dipolar interaction.This is completely different from molecules,where electric dipolar interaction is the essential contribution determining optical properties.It will also be interesting to study the geometry and distance dependence of the different coupling effects.Our concept can be extended to more complex arti?cial molecules,such as stereotrimers,stereoquadrumers and so on.The tuneabil-ity of the resonant behaviour of these new arti?cial materials by altering the spatial arrangement of their constituents offers great ?exibility for exploring useful metamaterial applications,such as chiral structures with negative refraction 20,invisibility cloaks 21and magneto-optically active materials 22.Stereometamaterials open up the potential for optical polarization control,which so far has been dominated by stereoisomers 1and liquid crystals 23.(See Supplementary Information for more details on optical stereoisomers as well as left-and right-handed enantiomers.)Stereometamaterials might also serve as arti?cial nanosystems for emulating the optical properties of complex biomolecules,such as double helix DNA chiral proteins and drug enzymes,which have profound application potentials in biophotonics,pharmacology,as well as diagnostics.

Methods

Structure fabrication.Three (or more)gold alignment marks (size 4?100m m)with a gold thickness of 250nm were ?rst fabricated using a lift-off process on a quartz substrate.The substrate was then covered with a 50-nm gold ?lm using electron-beam evaporation.Next,SRR structures were de?ned in negative resist (AR-N,ALLRESIST GmbH)by electron-beam lithography.Ion beam etching (Ar tions)of the gold layer was then performed to generate the gold SRR structures.Subsequently,a 120-nm-thick spacer layer was applied on the ?rst layer by spin-coating.A solidi?able photopolymer,PC403(JCR),was used as the planarized spacer layer.A pre-baking process in which the baking temperature was

continuously increased from 90to 1308C was ?rst performed to remove the solvent from the polymer.A suf?ciently long bake at a higher temperature (30min in a 1808C oven)further hardened the layer.A 50-nm gold ?lm and a spin-coated AR-N resist layer were subsequently deposited on the sample.Next,the stacking alignment using the gold alignment marks was applied to ensure accurate stacking of the second SRR layer.The procedures of in-plane fabrication were

repeated with the ?nal layer being PC403.All structures had a total area of 200?200m m.

Optical and structure characterization.Transmittance spectra were measured with a Fourier-transform infrared spectrometer (Bruker IFS 66v /S,tungsten lamp)

combined with an infrared microscope (?15Cassegrain objective,NA ?0.4,liquid

N2-cooled MCT77K detector,infrared polarizer).The measured spectra were normalized with respect to a bare glass substrate.

The simulated transmittance spectra and?eld distributions were performed using the software package CST Microwave Studio.For the spectra in Fig.1b–d,the permittivity of bulk gold in the infrared spectral regime was described by the Drude model with plasma frequency v pl?1.37?1016s21and the damping constant

v c ?4.08?1013s21.For the spectra in Fig.4d–f,owing to the surface scattering

and grain boundary effects in the thin?lm of the real systems,the simulation results were obtained using a damping constant that was three times larger than the bulk value.The optical parameters were the refractive index of PC403n PC403?1.55and the quartz substrate refractive index n glass?1.5.

The electron micrographs of the fabricated structures were taken with an FEI-Nova Nanolab600scanning electron microscope.

Received13October2008;accepted21January2009; published online22February2009

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The authors would like to thank M.Stockman,T.Pfau,F.Giesselmann and M.Dressel for useful discussions and comments.We thank S.Linden for stimulating us to study the twisted SRRs with different angles.We acknowledge S.Hein for his metamaterial visualizations.We gratefully thank M.Hirscher and U.Eigenthaler at the Max-Planck-Institut fu¨r Metallforschung for their electron microscopy support.We acknowledge S.Kaiser,H.Graebeldinger and M.Ubl for technical assistance.This work was?nancially supported by Deutsche Forschungsgemeinschaft(SPP1113and FOR557),Landesstiftung BW and BMBF(13N9155and13N10146).The research of H.L.and S.Z.was?nancially supported by the National Natural Science Foundation of China(no.10604029,no. 10704036and no.10874081)and the National Key Projects for Basic Researches of China (no.2009CB930501,no.2006CB921804and no.2004CB619003).

Additional information

Supplementary Information accompanies this paper at https://www.sodocs.net/doc/23191629.html,/naturephotonics. Reprints and permission information is available online at https://www.sodocs.net/doc/23191629.html,/ reprintsandpermissions/.Correspondence and requests for materials should be addressed to H.G.

2009-nature Stereometamaterials

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