Tailoring the properties of opticalmetamaterials

Christian Helgert,Thomas Pertsch,Carsten Rockstuhl,Ekaterina Pshenay-Severin,ChristophMenzel,Ernst-Bernhard Kley,Arkadi Chipouline,Christoph Etrich,Uwe Huebner,

Andreas Tuenne rmann4,Falk Lederer2

(1.ZIK“ultra-optics”,Institute of Applied Physics,Friedrich SchillerUniversity Jena,

M ax-W ien-Platz1,07743Jena,Ge rm any;

2.Institute of Condensed M atter Theory and Solid State Optics,Friedrich SchillerUniversity Jena,

M ax-W ien-Platz1,07743Jena,Ge rm any;

3.Institute of Photonic Technology,A lbert-Einstein-Str.9,07745Jena,Ge rm any;

4.Fraunhofer Institute of Applied Optics and Precision Engineering,A lbert-Einstein-Str.7,07745Jena,Ge rm any)

1 Introduction

Metamaterials(MMs)are manmade media in which the propagation properties of electromagnetic radiation are significantly governed by their artificially structured geometry rather than by the natural materials they are composed of.When the electromagnetic waves interacting with these media stem from the optical or near-infrared spectral domain,they are called opticalMMs[1,2].They allow in principal for the control of propagation properties of an optical wave field;like refraction,diffraction,dispersion,phase-and group-velocities. Thus,the knowledge and access to these optical control mechanis msviaopticalMMs enable the realization of optical componentswith comprehensive functionalities[3-7].Moreover,bymeans ofMMs the interaction between light and matter can be extended into domains where nature doesn′t provide any equivalent.Accordingly,although still at the stage of fundamental research,MMs are expected to elicit a boost in the field of modern optics[8,9].

In recent years,a large part of the effort to explore optical MMs was aimed at the derivation of comprehensible design guidelines to realize naturally unattainable optical functionalities.In this paper we consider a class ofMMswhose unit cells have a spatial extent much s maller than the wavelength of the interacting optical radiation.If this condition ismet,the light propagation in theMM will be governed by the normal modes of a homogeneous medium.Such MMs are commonly described as assuming effective properties,which,and this is an important statement,can be deliberately tailored. Usually,an effective electric permittivity and an effective magnetic per meability are introduced that can be utilized to derive formally an effective index of refraction.

The usage of these effective properties greatly facilitates the description of lightpropagation through MMs because detailsof the correct unit cell geometry for ming theMM can be neglected.Despite their unquestionable usefulness forMM designers,itmust be carefully borne in mind that the effective material properties are an intuitive yet simplified approximation in order to model light propagation throughMMs instead of accurately describing the MM itself.The critical issues in deriving meaningful effective propertieswere recently discussed in[10].However,to have such properties at hand,the common approach relies on a retrieval algorithm of effective properties bymeans of the inversion of the scattering problem of a homogenized finite slab at normal incidence[11].This algorithm was recently generalized for oblique incidence including the influence of a substrate[12]as it is necessary for the practical realization of any MM.

In thiscontribution we review our latest achievements to tailor the properties of opticalMMs.We show combined experimental and theoretical studies of three exemplary MM designs.All these samples have been realized by means of a Vistec SB350OS electron-beam writer and lift-off techniques.Their optical responses have been investigated spectroscopically. In each case our experiments are complemented by numerical simulations applying either the FourierModalMethod[13]or Finite-Difference T ime-Domain simulations[14].For the rigorous trea tment both the exact geometry of the respective unit cells and the spectral dependence of the material properties as documented in the literature were appropriately taken into account.

The paper is organized as follows:Section 2 addresses a near-infrared,negative-index MM composed of two distinct unit cell elements. This approach permits an independent tuning of the geometry of the unit cell components. Section 3 deals with a novelMM design that releases the constraint of polarization dependency.The Swiss-crossMM was recently shown to have a polarization independent optical response for nor mal incidence and a negative index of refraction atλ =1.4μm.Moreover,we study the optical properties of the Swiss-crossMM at oblique incidence and reveal its angle-dependent effective properties.

It is shown that the spectral and angular domains of the negative refractive index as well as its magnitude are closely connected to the propagation direction and the polarization state of the illumination. Generally speaking,the optical response is dominated by spatial dispersion,as it is expected for any thin film MM that has been published to date.The implications for the notion of effective properties of common MMs are discussed. In Section 4 we attempt to evaluate exper imentally the requirement of a periodic arrangement of the unit cells in optical MMs.The ans wer to this question is urgently needed since the serial fabrication methods of today′sMMs are expected to be replaced by faster and less costly self-assembling or chemically randomized fabrication schemes.We investigate a model MM system by gradually increasing the degree of positional disorder with respect to its unit cells.The observable spectral features occurring upon this transition and the impact of the effective properties of the MM are revealed.Most importantly,we confir m that the magnetic properties of commonMMs are hardly affected by an arbitrarily high degree of positional disorder of the unit cells.We elucidate the encouraging conclusions to be drawn with respect to negative index materials and potential devices composed of them.

2 Double-element negative-index structure

In the framework of an effective medium approach,the issue of providing an effective magnetic permeability different from unity,i.e. a magneto-optical activity,was commonly employed by the excitation of localized plas mon polariton eigenmodes in metallic nanostructures[15]. In the optical domain,the double cut-wire structure has attracted particular attention[16].A magnetic moment arises from an antisymmetric localized plasmon polariton that can be excited if the illuminating electric field is polarized parallel to the wires.Combining this structure with continuousmetallic wires decreases its effective plasma frequency and provides control of the effective electric permittivity of the medium. In the wellknown fishnetstructure,these two components merge into a single unit[17].Accordingly,the double cut-wire structure is of interest due to the possibility of tailoring the geometry of both structural elements[18].

Fig.1 (a)Schematic of the double cut-wireMM with structural design parameters,(b)tilted electronmicrograph view of a fabricated double-cutwire sample slicedwith a focused ion beam to visualize the vertical structure,(c)transmission at normal incidence(0,grey lines)and reflection(8°,black lines)spectra for the resonant polarization.Solid and dotted lines represent measured and calculated spectra,respectively,(d)real(grey solid lines)and imaginary(black dotted)parts of the electric permittivityε,magnetic permeabilityμand refractive indexnof the sample derived from the calculated spectra shown in c).

The geometry of the unit cell is shown in Fig.1(a).The periods of the structure arePx=500 nm andPy=600 nm.The width of the continuouswires isW1=130 nm and of the cut-wiresW2=100 nm,the length of the cut-wires isL=430 nm,and the thicknesses of the metal layerdMe=40 nm and the dielectric spacerds=40 nm. The SEM micrograph in Fig.1(b)shows a fabricated sample revealing the vertical structure by an F IB-slice. In the experiment we consider nor mal light incidence and an electric field polarization parallel to the wires.In this configuration the cut-wire structure supports two plas monic eigenmodes with different eigenfrequencies.The excitation of an anti-symmetric mode corresponds to anti-phase current oscillations in the cut-wires and evokes the appearance of a permeability resonance. In Fig.1(c)the measured transmission and reflection spectra are compared to numerical simulations.Regarding the multiple spectral features,we concentrate on the transmission minimum nearλ =2.1μm.Here the anti-symmetric eigenmode ismost excited.This is confirmed by the calculation ofthe effective permeability[12],where a resonance with a Lorentzian line shape is observed atλ =2.1μm.W ith respect to the effective refractive index,we conclude thatn=-0.5+1.9i atλ =2.1μm can be formally attributed to our structure,as shown in Fig.1(d).The options to further tailor the efficiency of the structure are manyfold due to free design parameters.For instance,increasing the strength of the anti-symmetric resonance goes alongwith a decrease of the period in thex-direction or alternatively with an increase of the width of the cut-wires.Another possibility is to break the vertical symmetry of the cut-wire[19].

3 Polarization-independent Swisscross structure and its angular response

The optical MM presented in the former section exhibits its particular optical property,a negative refractive index,for normal incidence and for one polarization state of the electric field component only.This dependency applies to most prototypical unit cells of currently published optical MMs and can be reduced only by a novel design approach.It can be anticipated that for future applications of MMs,a polarization-insensitive response is highly desirable.In addition to that,the angular response of any thin film MM must be known explicitly if it is to be employed in imaging concepts[20].Here we present a first practical approach to address these issues. The Swiss-cross structure[21]was recently shown to have a polarization-independent optical response for normal incidence.The principle design of the unit cell is illustrated in Fig.2(a)and the fabricated sample is shown in Fig.2(b).The structure has a lattice constant of 410 nm in both lateral dimensions.The width and length of the ar ms of the Swiss crosswere designed to bexs=80 nm andxl=310 nm,respectively.The thicknesses of the gold films and the inter mediate dielectric magnesia film were set to bedAu=30 nm anddMgO=37.5 nm,respectively.Remarkably,the metamaterial extends over an area of 9 mm2.

The functionality of the structure can be understood in ter ms of a generalized isotropic cut-wire plate combined with orthogonally oriented wires.Like the double-element MM,an anti-symmetric plasmonic eigenmode is excited at a given resonance wavelength in the cut-wires that are now merged in the Swiss-cross structure.Because of the structure′s fourfold rotational symmetry,a polarization-independent optical response at normal incidence is anticipated.Based on spectroscopic measurements in transmission and reflection we provide experimental evidence of this property(Fig.2(c)and(d)).The measured exper imental data is confir med by comparison to rigorous calculations(Fig.2(e)).Again,we assign effective permeability,permittivity and consequently a refractive indexnto our Swiss-cross structure. For the fabricated sample a value ofn=-1.9+2.7i at the resonance wavelength around 1.4μm is deduced(Fig.2(f)).The design parameters are chosen primarily because of the experimental constraints imposed by our setup and are not meant to be optimized.The Swiss cross improves a particular aspect of the fishnet design as it el iminates the drawback of a polarization-dependent optical response.The free design parameters are the widthxsand the lengthxlof the cross arms and the thicknessesdAuanddMgOof the thin film layers.By changing these values the negative-index domain could be tuned to otherwavelengths aswell.

Fig.2 (a)Schematic of the Swiss-crossMM unit cellwith structural parameters,(b)nor mal view electron micrograph of a fabricated sample(inset:magnification of a unit cell),(c)measured trans mission,(d)reflection spectra of the sampl for a complete set of linear polarization states from 0°to 180°in steps of 5°at normal incidence,(e)measured(solid lines)and calculated(dotted lines)trans mission(grey)and reflection(black)for 0°polarization,(f)real(solid line)and imaginary(dotted line)partof the effective refractive indexnderived from the spectra shown in(e).

Furthermore,we provide insightinto the dependence of the effective MM properties of the Swiss cross on the angle of incident light both experimentally and theoretically[22]. The angular and spectral dependent response was measured using a self-built spectroscopic setup for specular transmission and reflectance.We take the az imuth angleφ,the elevation angleθand the state of polarization to describe the plane wave normal.TE polarization implies that the incidentE-field is tangential to the surface.To exclude the undesired effect of depolarization we consider the four symmetry directions of all possible combinations ofφ =0, φ =45°,TE-and T M-polarization. In these cases no coupling can occur between the TE-like and T M-like polarized eigenstates of the effective MM,hence the polarization states of in-and out-coming waves are maintained.The resulting spectra were measured for their dependence onθandλ and have been compared to the numerically s imulated data. As an excellent agreement is observed,we can rely in future on the simulated data to retrieve the angular-dependent effective properties of the structure[12].

We note that the effective properties of the Swiss-cross structure suffer from strong spatial dispersion and consequently have to be understood as wave parameters rather than genuine material parameters.Any effective property looses its meaning if it has to be deter mined for every incidence angle and polarization state separately.For instance,it can be shown that the spectral and angular domains of the negative refractive index aswell as itsmagnitude are closely connected to the propagation direction and the polarization state of the illumination.We can conclude for the given example of the Swiss-cross MM that its description as effectively homogenous and anisotropic is physically inappropriate[22].However,we dissuade from abandoning the general description ofMMs by effective properties at the present stage.Provided that the limits of their applicability are carefully borne in mind,angular resolved effective properties give preliminary insight into the underlying physics and can serve to simplify the description of light propagation inside a Swiss-cross MM.

Fig.3 (a)Schematic of the cut-wireMM unit cellwith structural parameters,(b)SEM micrograph of a sample with disorderD=1.6,(c)measured transmission and(d)reflection spectra as a function of the disorder parameterD.Both spectra are recorded for discrete values ofDand interpolated to guide the eye,(e)Simulated trans mission spectrum for three discrete valuesofD,(f)FDTD s imulation of the absoluteE-field amplitude in reflection forD=3.0 atλ=1 050 nm.

4 Transition from periodic to amorphousMMs

Almost all fabricated opticalMMs to date are composed ofmeta-atoms or unit cells arranged in periodic lattices.This has been shown to be convenient for numericaltreatmentsince periodic arrangements greatly facilitate rigorous simulations by considering one single unit cell equipped with periodic boundary conditions.Here we provide an intuitive approach to lift the constraintofperiodicity in opticalMMs by investigating the transition form periodic to truly amorphousMMs[23].Besides unraveling the significance of periodic arrangements,the quantitative investigation of disordered and amorphousMMs is usually regarded to be essential for the realization of isotropic MMs.

The system we consider is based on the cut-wire pairMM[16].Fig.3(a)shows the principle design of the unit cell.Each cut wire pair consists of two gold layerswith a thickness ofdAu=30 nm separated by a magnesia spacerwithdMgO=45 nm.The length of the wires isxl=yl=180 nm and the lattice constant isPx=Py=512 nm in the reference sample.Positional disorder is introduced by summing a random displacement to the centre position of each unit cell,independently in both lateral directions.Normalizing this displacement to the periodPx=Py,we obtain an average dimensionless parameterDto quantize the degree of disorder in the system. Keeping the average density of cut-wire pairs and hence the average surface filling fraction constant,several MM samples withDincreasing from 0 to 1 000 were fabricated.Fig.3(b)shows one representative SEM micrograph of a sample withD=1.6.The results of the spectral characterization for trans mission and reflection are summarized in Fig.3(c)and(d).As for the periodic reference sample we note two dips situated atλ=800 nm and λ=1 050 nm in the trans mission and a peak atλ=800 nm in the reflection spectrum.These two resonances are identified as the symmetric and antisymmetric plasmonic eigenmodes of the cut-wire pairs.They evolve differently if the degree of disorderDis increased.While the anti-symmetric resonance atλ=1 050 nm is almost independent ofD,the symmetric resonance rapidly decays even for a moderate degree of disorder. This behaviour is confir med by finite-difference t ime-domain simulations for a supercell of cut-wire pair MMs with no(D=0),moderate(D=0.3)and high(D=3.0)positional disorder corresponding to a periodic,disordered and amorphous MM,respectively(Fig.3(e)).

Our key finding is that the anti-symmetric resonance is nearly invariant to positional disorder.It is important to note that this resonance is the key feature in the majority of today′s negative-indexMMs.Based on a detailed investigation of the eigenmodes supported by near-field calculations for different values ofD(Fig.3(f)),we can explain this result.Basically we argue that the electric quadrupole associated with the anti-symmetric resonance does not have any in-plane component.Thus itmakes the interaction among neighbouring particles negligible.Furthermore,with the claim of evaluating the effective properties of amorphousMMs for the first time,similar conclusions can be drawn.The resonance in the effective magnetic permeability that is related to the anti-symmetric eigenmode does not experience any noteworthy changes upon the transition from a periodic to an amorphous MM. Independent of the degree of disorder the line shape,the strength and the width of this resonance remain unchanged. It can be concluded that the magnetic response of any MM based on this particular eigenmode is solely determined by the response of the individual metaatoms regardlessof their arrangement.This is an important finding when it comes to the fabrication of MMs by self-organized approaches.The implications of our finding facilitate the integration of optical negative-indexMMs in sub-wavelength imaging applications and relax the constraintof the necessity of periodical arrangements in modernMM designs.For instance,the effective properties of large-scale optical MMs fabricated by quick and reliable bottom-up approaches[19]can be evaluated by considering their periodic equivalents.Moreover,the influence of the structural parameters on the tunability of the effective properties of such amorphous MMs can be revealed[23].

5 Conclusions

Effective properties constitute an intuitive way to gain insight into light propagation in optical MMs.We have demonstrated how structural parameters can be employed to design and tailor the response of highly dispersive,optical MMs.We addressed this approach on the basis of three opticalMM designs:the double cut-wire pair structure,the Swiss-cross structure and the amorphous cut-wire pair MM.Combining experimental and theoretical studies it was shown how the operational wavelength of an effective index of refraction smaller than zero and its sensitivity to polarization can be modified.Nevertheless,the valuable concept of effective parameters must a lways be used in the limits of its applicability due to the scaling of the characteristic lengths of the constituentmeta-atoms and the wavelength of the interacting electromagnetic radiation.By retrieving angular-dependent effective propertieswe show through the example of the Swiss-crossMM that in the vicinity of the resonance with negative refraction it cannot be described as effectively homogeneous.Another way to look at the homogenization of opticalMMs is to evaluate the necessity of a periodic arrangement of the unit cells.We investigated the transition from periodic to amorphousMMs and confirm that the opto-magnetic properties of common MMs are virtually unaffected by an arbitrarily high degree of positional disorder.This new degree of freedom in the design and fabrication of opticalMMs opens further paths to tailor their effective properties according to the requirements imposed on them.

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