Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions

Retrieval

of

effective

parameters

for

bianisotropic

metamaterials

with

omega

shaped

metallic

inclusions

Zhaofeng

Li

*

,

Koray

Aydin

,

Ekmel

Ozbay

a

NanotechnologyResearchCenter,BilkentUniversity,Bilkent,06800Ankara,Turkey

b

DepartmentofPhysics,BilkentUniversity,Bilkent,06800Ankara,Turkey

c

DepartmentofElectricalandElectronicsEngineering,BilkentUniversity,Bilkent,06800Ankara,Turkey

Received19March2011;receivedinrevisedform17October2011;accepted8November2011

Availableonline18November2011

Abstract

Bianisotropic metamaterials including V-shaped metallic elements are investigated experimentally and theoretically. A passband is observed fora composite metamaterial(CMM) based on an V-medium belowthe plasmonic frequency of the correspondingclosedCMM.Theeffectiveparameters(refractiveindex,impedances,permittivity,permeability,and magnetoelec-triccouplingcoefficient)areretrievedfortheV-mediumandtheCMMbasedonit.Ourretrievalresultsshowthatthepassband observedfortheCMMisabandwithpositiverefractiveindices.Ourretrievalresultsconfirmthedeductionsofourpreviousreports. #_{2011ElsevierB.V.Allrightsreserved.}

Keywords: Metamaterials;Bianisotropy;Omegamedia;Effectiveparameters

1. Introduction

Metamaterials have attracted much attention from the scientific community because of their exotic properties, which usually do not exist in natural materials.Forinstance,metamaterials withanegative refractiveindex(NRI)[1–4]canbeusedtoconstructa superlens[5–9].Moreinterestingly,metamaterialscan evenbeusedtoconstructaninvisiblecloak[10,11].In order to realize such novel properties, metamaterials usuallyincludetwotypesofelementsand,therefore,are calledcompositemetamaterials(CMM).Forinstance, when constructing ametamaterial withNRI, onecan use a periodic thin metallic wiremedium in orderto

obtain negative permittivity [12], and use split ring resonators(SRRs) in orderto obtain negative perme-ability [13]. Although SRRs are widely used in metamaterials operating in the microwave region, studies show that there can be problems when SRRs are used at optical frequencies [14]. Bianisotropy usually should be avoided during the design of metamaterials. However, it was proposed recently

[15]thatitispossibletobenefitfromthebianisotropic properties of an example metamaterial – V medium (omega medium). An omega medium was first introduced by Saadoun and Engheta [16] and was called a pseudochiral medium in 1992. An omega mediumis acompositeelectromagneticmaterial with V-shapedmetallicinclusionsthatare placedinahost dielectric medium. In the omega medium, there is magnetoelectric coupling due to its intrinsic bianiso-tropic characteristics. Following the ideas described above[15,16], westudied andreportedon aseriesof

www.elsevier.com/locate/photonics

Availableonlineatwww.sciencedirect.com

PhotonicsandNanostructures–FundamentalsandApplications10(2012)329–336

*Correspondingauthor.Tel.:+903122901018;

fax:+903122901018.

E-mailaddress:zhaofengli@bilkent.edu.tr(Z.Li).

1569-4410/$–seefrontmatter#_{2011ElsevierB.V.Allrightsreserved.}

metamaterials based on V-shaped metallic inclusions experimentally and numerically [17,18]. The results haveclearlyshownthedifferencesfromthetraditional metamaterialsmadeofSRRs.Atransmissionbandwas observed for a CMM based on an omega medium, whichisbelowthe plasmonicfrequencyof theclosed compositemetamaterial(CCMM),inwhichithasbeen deduced that thistransmission band is not a band of negative refraction, although the effective parameters forthebianisotropicmetamaterialswerenotpresented inprevious reports.

Ithasbeenproposedthatmetamaterialsrespond to electromagneticradiationascontinuousmaterialswhen thewavelengthismuchlargerthanthespacingbetween the composite components and the size of these respectivecomponents. Therefore, it is reasonable to assignvaluesofpermittivityeandpermeabilitymfora metamaterial. Several methods [19–22] have been proposed to retrieve the effective parameters for isotropic metamaterials. Among them, the method basedonthescatteringparameters(Sparameters)ofa slabofmetamaterialiswidelyused[23–26].Compared with isotropic metamaterials, the retrieval of the effective parameters for a bianisotropic metamaterial ismorecomplexduetotheexistenceofthe magneto-electriccouplingeffect.However,itisproposedthatby

using the Sparameters of three orthogonal directions

[27], it is possible to retrieve all of the effective parameters for a bianisotropic metamaterial. Very recently, our study revealedthat it is also possibleto retrieve all of the effective parameters for a bianiso-tropicmetamaterialbyusingtheSparametersofonly onedirection[28].Inthepresentpaper,wewillrecount someofthe briefconclusionsofourretrievalmethod. Then, we will provide the effective parameters includingthe magnetoelectric couplingcoefficient for the bianisotropicomegamedium.Ourretrievalresults validate what was deduced in our previous reports

[17,18].

2. Experimental andnumericalresults

Fig.1(a)and(b) showstwounitcellsoftheomega mediumandtheCMMwithomegastructureinclusions under study. Fig.1(c)shows the detailsof the omega structure.Theparametersinthefigurearer=1.19mm, W=0.45mm, andL=1.8mm.The omega structures are made of copper on a FR4 printed circuit board (PCB).Thedielectricconstantandtheconductanceof the FR-4 board are 4.4and 0.0068S/m, respectively. The thicknessof the copper andFR4 are 30mmand 1.6mm, respectively. By arranging these omega

Fig.1. (aandb)TheschematicsofunitcellsfortheomegamediumandtheCMMwithomegastructureinclusions.(c)Theomegastructure.The

dimensionsoftheomegastructurearer=1.19mm,W=0.45mm,andL=1.8mm,respectively.(d)Thecontinuouswirestructurewithits

dimensionst=1.44mmandh=5mm.(e)Theclosedomegastructure,inwhichitsdimensionsareequaltotheomegastructurein(c).Thelattice

structuresperiodicallyinthreeorthogonaldirections,an omegamediumcanbeobtained.Intheexperiments,we arrangeV-resonatorunitsperiodicallywith5,40,and30 unitcellsinthex-,y-,andz-directions,respectively.The lattice constantsare ax=ay=az=5mm.While inthe

simulations,weuseperiodicboundaryconditionsinthey andz directions.Weperformed thenumerical simula-tions by using a commercial software package (CST STUDIO microwave) that is based on the finite integration technique. In order to investigate the propertiesofCMMbasedonomegastructures,aperiodic arrangement of continuous thin copper wires were adopted toachievenegativepermittivityat microwave frequencies.Aunitcellofthecontinuouswireisshownin

Fig.1(d).ThewireisontheoppositesideofthePCB.The thicknessofthe metalis30mm.Thewidthofthethin wireist=1.44mm,andtheheightish=5mm,whichis equaltotheperiodicconstantintheydirection.Inour experiments,thelatticeconstantsandnumberoflayersof continuousmetallicwiresareequaltothatoftheperiodic omegamediainthex-andz-directions.Whileinthe z-direction,thewiresarecontinuousandthetotallengthof thewiresis150mm.

Itwasproposed[29]thatbyusingclosed(orshorted) resonators combined with periodic metallicwires one can obtain the rectified plasmonic frequency. If a transmissionbandofthecorrespondingCMMisbelow therectifiedplasmonicfrequency,thistransmissionband can be thought of as a band of negative refraction. Accordingly, the closed V structure shown in

Fig. 1(e) is adopted in our study as a reference. During the experiments, transmission measurements

wereperformedinfreespacebyusinganHP8510-C network analyzer. Microwave horn antennas were used as transmitters and receivers, in which the transmission through thesamples wasmeasured.

Fig.2(a)and(b)showstheresultsoftheexperiment andsimulation,respectively.Oursimulationresultsare ingoodagreementwiththatoftheexperiment.From

Fig.2,itcanbeclearlyseenthatthereisatransmission band for the CMM medium, which is below the plasmonic frequency of the corresponding CCMM medium.AccordingtoAydinetal.[29],this transmis-sionbandismuchlikelyabandofnegativerefraction. However,unlikethesituationsincomposite metama-terialsbasedonSRRs,thistransmissionbandofCMM does not coincide with the transmission gap of the correspondingomegamedium.Instead,this transmis-sionbandshiftstothelowerfrequencyend.Theabove phenomenon is quite similar to the transmission behaviorofap-structure[30],whichisalsodeduced to not be a left-handed transmission. Therefore, the transmission band ofthe CMM basedon the omega structure is considered to be right-handed in our previously conducted reports [17,18]. In the next section, wewillretrieve theeffective parameters for omega-based metamaterials and see whether our previousconclusionwascorrectornot.

3. Retrievalresults

For the omega medium shown in Fig. 1(a), by assumingthatthemediumisreciprocal[31,32]andthat theharmonictimedependenceiseivt,wherevisan

Fig.2.(a)Thetransmissionspectraofexperimentaldatafortheomegamedium,periodicwiremedium,compositemetamaterial,andclosed

angular frequency, we can write the constitutive relationshipsas follows: D*¼eE*þjH* B * ¼mH*þzE* (1) where e¼e0 ex 0 0 0 ey 0 0 0 ez 0 B @ 1 C A; m¼m0 mx 0 0 0 my 0 0 0 mz 0 B @ 1 C A; j¼1 c 0 0 0 0 0 0 0 ij0 0 0 B @ 1 C A; z¼1 c 0 0 0 0 0 ij0 0 0 0 0 B @ 1 C A (2)

wheree0andm0arethepermittivityandpermeabilityof

thevacuum,respectively,inwhichcisthespeedoflight invacuum.Thesevenunknowns,ex,ey,ez,mx,my,mz,and

j0arequantitieswithoutadimension.Whenaplanewave

that ispolarized inthe zdirection isincident inthe x direction,threeparameters(ez,my,andj0)willbeactive,

whiletheotherfourparameters(ex,ey,mx,andmz)willnot

beinvolvedinthebianisotropicprocessand,therefore, areoutofthescopeofthepresentstudy.Accordingtothe formulaeinRef.[31],onecaneasilyobtainthe expres-sionsfortheeffectiveconstitutiveparametersbasedonez,

my,andj0.However,there isonemainthingthat one

shouldnote.Comparedtoanisotropicmaterial,themost

interestingandimportantfeatureofabianisotropic ma-terialisthatthecharacteristicimpedanceshavedifferent values for the wavespropagating in the two opposite directions of the xaxis. For an electromagnetic (EM) wavetravelinginthexdirection,theimpedanceswill be zþ¼ my nþij0 ; z¼ my nij0 (3) respectively.Wheren istheeffectiverefractiveindex, whichhasthesamevaluefortheEMwavetravelingin twooppositedirectionsofthe xaxis:

n¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiezmyj02

q

(4) Ourrecentstudy[28]hasfoundasimplermethodto retrieve the effective parameters for a bianisotropic metamaterialcomparedtothemethodproposedinRef.

[27]. According to our study, in orderto retrieve the effective parameters of a bianisotropic medium, one only needs to know the S parameters in the two directions(sayxdirectionsforthepresentcase)with the incident waves polarized in the z direction. Consequently, one obtains the analytical expression for therefractiveindex n,whichis

cosðnk0lÞ¼

1S11S22þS212

2S21

(5)

Fig.3.Theintensityspectraoftransmission,reflection,andabsorptionfortheomega(a)andtheCMM(b)media.R1andR2(A1andA2,T1and

wherek0isthewavenumberoftheEMwaveinvacuum,

and l is the thickness of the bianisotropic medium. When solving n from Eq. (5), one must determine onebranchfrommanybranchesofsolutions. Fortunate-ly,therehavebeenseveralreports[22,26]dealingwith thisproblem.Therefore,wewillnotdetailithere.Fora passivemedium,thesolvednmustfollowthefollowing condition:

n000 (6)

where()00_{denotestheimaginarypartoperator.Aftern}

issolved,otherconstitutiveparameterscanbeobtained bythe followingrelations:

j0¼ n 2sinðnk0lÞ   _S 11S22 S21   (7) my¼ in sinðnk0lÞ   2þS11þS22 2S21 cosðnk0lÞ   (8) ez¼ n2_þj 0 2 my (9) Afterobtainingtheabovethreeparameters(ez,my,and

j0), the impedances (z+ and z) can be solved from

Eq. (3). Again, for a passive medium, the following conditionsshouldbe satisfied,

zþ00; z00; (10)

where()0_{denotestherealpartoperator.Sofar,allofthe}

constitutiveparametersthatarerelatedtobianisotropy were retrieved. In the following subsections, we will provide the retrievalresult for the omega medium as wellastheCMMbasedonit.Intheretrievalprocedure, we employed a single layer of omega medium (or CMM) along the x-axis.Hence, the simulation setup coincideswithaslabofomegamedium(orCMM)that consistsofsingleperiodlayer.Theeffectiveparameters werethenderivedfromthetransmissionandreflection

Fig.4. Theretrievalresultsoftheeffectiveparameterforthebianisotropicomegamedium.(a)Effectiverefractiveindexn.(b)Impedancesz+and

coefficientsof thissingle layerof omegamedium (or CMM).

3.1. Sparametersandabsorptionspectraofthe omegaand theCMMmedia

Usingthestructuraldimensions for the omegaand the CMM media that was presented in the previous sections,wecalculatedtheSparametersofoneperiod ofthetwomediaforthetwooppositedirectionsalong thex-axis,andtheintensityspectraareshowninFig.3. InFig.3,wealsoshowtheabsorptionspectraofthetwo media. It is seen that the reflection and absorption spectraaredifferentwhentheEMwavesareincidentin the +x and x directions. This phenomenon happens duetothelossysubstrateandthebianisotropyofthetwo media,ascanbewellexplainedinEqs.(5)–(8) ofthe Ref.[28].

Moreover,itisalsoseenfromFig.3(a)thatthepeaks of the absorption spectra of the omega medium are shifted to higher frequencies compared to thatof the reflectionspectra.Thisdeference ofthepeak frequen-cies betweenthe scatteringandabsorptionspectrahas been found for varied scattering particles [33]. This phenomenonwillbefurtherstudiedinourfutureworks. 3.2. Retrievalresults ofthe omegamedium

Fig. 4 shows the retrieval results for the omega mediumbasedonthecalculatedSparameters.Fig.4(a) and(b)illustratestheresultsoftheeffectiverefractive index n andimpedances z+ andz,respectively. In a wide frequency range, z+ and z definitely have different values.Fig. 4(c) and(d) showsthe retrieval results of permittivity and permeability, respectively. The omega medium shows strong features of electric

Fig.5. TheretrievalresultsoftheeffectiveparameterforthebianisotropicCMMbasedontheVmedium.(a)Effectiverefractiveindexn.(b)

resonance rather than magnetic resonance, which is quite different from the results of SRR media [22]. Moreover, one did not see any frequency range with negative permeability. Fig. 4(e) shows the retrieval result for the magnetoelectric couplingcoefficient. It can be seen clearly that near the resonant frequency, there obviously exists a strong magnetoelectric cou-plingintheomegamedium.Theexistenceofthestrong magnetoelectric coupling results in the different impedancesof z+ andzasshowninFig.4(b). 3.3. Retrievalresults ofthe CMMmedium

Fig. 5 shows the retrieval results for the CMM medium, which is a combination of the omega structures andinfinitewirestructures.Fig.5(a)shows theretrievedeffectiverefractiveindexnforthe CMM medium.Oneseesthatthereisafrequencyrange(9.2– 10.1GHz) with positive n0 but very small n00, which exactly corresponds to the transmission band of the CMMshowninFig.2.Fig.5(b)showstheresultsforthe impedances z+ and z, in which again the two impedanceshave differentvalues.Fig.5(c)shows the resultsofthepermittivity.Bycomparingthedatashown in Fig. 4(c), one sees that even though the CMM medium includes periodic thin wire structures that provide negative permittivity, there still exists a frequencyrange (9.2–10.1GHz) withpositive permit-tivity. This frequency range also corresponds to the transmission band of the CMMmedium as shownin

Fig.2.Fig.5(d)and(e)illustratestheretrievedresults for the permeability and magnetoelectric coupling coefficient,respectively.Thesetwosetsofdataarequite similar to that of the omega medium as shown in

Fig.4(d)and(e),whichmeansthattheadditionofthe thin wire structure into the omega medium only has littleinfluenceonthetwo parametersofmyandj0.

Accordingtotheretrievalresultsfortheomegaand CMMmedia,itisconcludedthatthetransmissionband oftheCMMmediumbelowtheplasmonicfrequencyof CCMM isabandwithpositiverefractiveindices.Our retrieval results confirm what we deduced in our previous reports [17,18]. Furthermore, our investiga-tions on the omega medium illustrate that for a bianisotropic metamaterial it is insufficient to decide whetheratransmissionbandofaCMMisofapositive or negative refractive index by comparing it to the transmission spectrum of a corresponding CCMM. Althoughwe did not obtainametamaterialwithNRI basedonanomegamediumhere,ourretrievalmethod and procedurerevealed the underlying physicsof the omegamedium anditscorrespondingCMM medium.

By using the retrievalprocedure as a tool, it will be easier totailorthe properties of omega-related media andbianisotropicmetamaterialsinfuturestudies. 4. Conclusions

In the present paper, we retrieved the effective parametersforanomegamediumanditscorresponding CMM. Since omega-related media are intrinsically bianisotropic,we retrievednot onlythe parametersof permittivity and permeability but also the magneto-electriccouplingcoefficientthatmeasurestheintensity ofthebianisotropy.Furthermore,wealsoobtainedthe effective parameters of the refractive index and impedances. Based on our retrieval results, it was foundthatthetransmissionbandoftheCMMbelowthe plasmonicfrequencyof itscorrespondingCCMMisa band with positive refractive indices. The retrieval results confirm what was deduced in our previous reports.

Acknowledgments

ThisworkissupportedbytheEuropeanUnionunder the projects DPT-HAMIT, EU-PHOME, EU-N4E, NATO-SET-181 and TUBITAK under Project Nos. 107A004, 107A012, 109E301. One of the authors (E.O.) also acknowledges partial support from the TurkishAcademyof Sciences.

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