Metamaterial tuning by manipulation of near-field interaction

Me ama e ial uning by manipula ion o nea - ield in e ac ion Da id A. Powell,1,*Mikhail Lapine,2,1 Maxim V. Go kuno ,3Ilya V. Shad i o ,1and Yu i S. Ki sha 1 1Nonlinea Physics Cen e, Resea ch School o Physics and Enginee ing, Aus alian Na ional Uni e si y, Canbe a, Aus alian Capi al Te i o y 0200, Aus alia 2Depa men Elec onics and Elec omagne ics, Facul y o Physics, Uni e si y o Se ille, A da. Reina Me cedes s/n, 41015 Se ille, Spain 3A. V. Shubniko Ins i u e o C ys allog aphy, Russian Academy o Sciences, Lenin A e. 59, 119333 Moscow, Russia 共Recei ed 6 Decembe 2009; e ised manusc ip ecei ed 14 June 2010; published 19 Oc obe 2010兲 We analyze he nea - ield in e ac ion be ween he esonan subwa eleng h elemen s o a me ama e ial and p esen a me hod o calcula e he elec ic and magne ic in e ac ion coe icien s. We show ha by adjus ing he ela i e con igu a ion o he neighbo ing spli ing esona o s i becomes possible o manipula e his nea - ield in e ac ion, and hus une he esponse o me ama e ials. We use he esul s o his analysis o explain he expe imen ally obse ed uning o mic owa e me ama e ials. DOI: 10.1103/PhysRe B.82.155128 PACS numbe 共s兲: 41.20.Jb, 78.70.Gq, 42.70.Qs I. INTRODUCTION Me ama e ials, which a e ypically egula a ays o sub- wa eleng h esonan pa icles, o e us a new deg ee o ee- dom in con olling he elec omagne ic esponse o ma e . Thus we a e no longe comple ely cons ained by he p op- e ies o exis ing ma e ials bu can ailo he esponse in an almos a bi a y ashion, o example, achie ing e y high,1 e y low,2and nega i e3 alues o e ac i e index, pe mi - i i y and/o pe meabili y. Because o he inhe en ly s ong dispe sion o esonan me ama e ials, hey mus be modi ied in o de o ope a e in a di e en equency band. The e o e, he e is a signi ican push o ha e a u he deg ee o con ol o e hese ma e ials— unabili y o hei esponse. Fo una ely, he enginee ed na u e o me ama e ials al- lows hei p ope ies o be con olled ex e nally, ei he by dynamically modi ying hei s uc u e o by adding some nonlinea inclusion and con olling wi h ex e nal ields.4Ex- amples o he la e app oach include he in oduc ion o a - ac o diodes,5 e oelec ics6and pho oconduc i e semiconduc o s.7On he o he hand, e en wi hou eso ing o such exo ic 共and o en lossy兲cons i uen s, he e is a g ea deal o eedom o manipula e he s uc u e i sel , and his is he app oach we ake he e. We conside speci ically he spli ing esona o 共SRR兲as one o he mos impo an me ama- e ial elemen s, no ing ha while he ine de ails o nea - ield in e ac ion a e s uc u ally speci ic, ou app oach can be ap- plied o a wide a ie y o s uc u es. An analy ical model o he magne ic esponse o a sub- wa eleng h a ay o iden ically o ien ed wi e loops loaded wi h a capaci ance8 akes in o accoun he mu ual in e ac ion o all he elemen s in he la ice, which is essen ial o de i - ing he e ec i e pe meabili y co ec ly. Al hough ha analy- sis is limi ed o he quasis a ic case accoun ing only o mag- ne ic nea - ield in e ac ions, i is c ucial o e ealing he consequence o la ice changes. These end o be o e looked by o he wise igo ous app oaches which include spa ial dis- pe sion bu de elop Lo en z local ield app oaches, based on nea es -neighbo in e ac ion9o poin -dipole app oxima ion.10 In pa icula , i was poin ed ou in Re . 8 ha he esonan equency o he me ama e ial pe meabili y can be al e ed by a ying he la ice cons an s wi hou changing he s uc u al uni s. This scheme is illus a ed in Fig. 1共a兲and has been e i ied by expe imen s in Re . 11. Howe e , a p ac ical con- sequence o his change in la ice cons an is ha he sample size also changes co espondingly. Mo e ecen ly, an al e na- i e app oach was sugges ed in Re . 12: in oducing a shi be ween laye s in o de o c ea e a monoclinic la ice wi h he shi inc easing linea ly be ween laye s, as shown in Fig. 1共b兲. This con igu a ion keeps he densi y o elemen s wi hin he me ama e ial cons an and can une he coupling be ween neighbo ing pa icles o modi y he esponse o he comple e me ama e ial. Howe e , o ini e-size samples, his shi in- e i ably esul s in a signi ican change in sample shape. Thus, o p ac ical pu poses, we ha e p oposed a supe la ice ype o geome y, whe eby only e e y second laye is shi ed by he same amoun , as shown in Fig. 1共c兲. This uning scheme p o ed o be obus and allows signi ican manipula- ion o he esonan equency wi h only a small change in he sample geome y.12 The e o e, he sample geome y and i s e ec i e p ope ies can be enginee ed almos indepen- den ly o achie e he desi ed manipula ion o elec omag- ne ic wa es. Howe e , as we demons a e below, his s uc u al uning o me ama e ials depends e y s ongly on he na u e o he nea - ield in e ac ions. Since me ama e ial elemen s such as spli ing esona o s a e usually no highly symme ic, he ela i e o ien a ion o pa icles wi hin he la ice is o key impo ance. This e ec is no desc ibed by exis ing ci cui heo y models and can gi e ise o some su p ising expe i- men al esul s, which we p esen he e. ( a ) (b) ( c ) FIG. 1. 共Colo online兲Se e al app oaches o modi y he la ice o me ama e ial unabili y: 共a兲a change in he la ice cons an 共Re . 11兲,共b兲a con inuous shi o he laye s 共Re . 12兲, and 共c兲a supe - la ice o al e na ing shi s o laye s 共Re . 12兲. PHYSICAL REVIEW B 82, 155128 共2010兲 1098-0121/2010/82共15兲/155128共8兲©2010 The Ame ican Physical Socie y155128-1 In o de o unde s and he coupling mechanisms and how hey a e a ec ed by he la ice shi , i is use ul o s a wi h he simples geome y—a pai o spli ing esona o s. Se - e al au ho s ha e conduc ed nume ical and expe imen al in- es iga ions o coupling be ween me ama e ial elemen s o di e en ypes o elemen s and ela i e o ien a ion be ween hem, in mic owa e13,14 and op ical equency anges. 共See he o e iew in Re . 15 and e e ences he ein, as well as Re s. 16–21.兲A de ailed s udy has p e iously been unde - aken on ailo ing he geome ic a angemen o a pai o coupled one-dimensional SRR a ays o enginee he dispe - sion cu es o magne oinduc i e wa es.22 As we ha e shown ecen ly,12 a ough quali a i e unde - s anding o he s uc u al uning can be achie ed by using ci cui heo y wi h pu ely induc i e coupling be ween SRRs. Howe e , his app oxima ion ails o p o ide a quan i a i e unde s anding o he me ama e ial uning in ques ion and, mos impo an ly, does no explain he obse ed s ong in lu- ence o he ela i e SRR o ien a ion. The e o e, below we de elop a dis inc model based on he calcula ion o he un- damen al mode o a single esona o . The knowledge o he cu en and cha ge dis ibu ions wi hin he mode allows cal- cula ing he coupling cons an s o a pai o spli ings. These cons an s a e hen used o explain he expe imen ally ob- se ed uning esponse o ou me ama e ial samples. In Sec. II, we de elop ou app oach o calcula e he me ama e ial coupling, including a discussion o he limi a- ions o o he pu ely analy ical me hods. In Sec. III, we apply ou app oach o he s udy o in e ac ion be ween a pai o spli ing esona o s which a e shi ed la e ally ela i e o each o he and explain quan i a i ely how he shi a ec s he posi ion o he undamen al esonance. In Sec. IV,we apply hese esul s o a bulk me ama e ial and iden i y he mechanisms a wo k in he expe imen al uning o a me ama- e ial slab in a wa eguide. Finally, Sec. Vconcludes he pape wi h u he discussions and ou look. II. NEAR-FIELD INTERACTION IN METAMATERIALS Conside ing a single SRR, i is known13 ha i possesses a disc e e se o eigenmodes 共s anding wa es兲wi h co e- sponding eigen equencies. In an a bi a ily exci ed SRR, he cu en s and cha ges can be ep esen ed as a supe posi ion o he eigenmodes. The undamen al mode wi h he lowes e- quency is ele an o he magne ic esonance in SRRs. On he equency scale his mode is well isola ed om he highe -o de modes, and we can es ic ou sel es o he single-mode app oxima ion neglec ing he exci a ion o highe modes. The ime-dependen cha ge densi y ␳ and cu en densi y Jin a esonan elemen wi h exci ed undamen al mode can be w i en in he mos gene al o m o a s anding wa e ␳ 共x, 兲=Q共 兲q共x兲,共1兲 J共x, 兲=I共 兲j共x兲,共2兲 whe e qand Jdesc ibe he cha ge and cu en dis ibu ions in space. In SRRs, he a ia ion in he cu en dis ibu ion ac oss he wid h o he conduc i e ack could be neglec ed, howe e , o gene ali y ou app oach akes in o accoun he comple e h ee-dimensional su ace-cu en dis ibu ion. To sa is y he conse a ion o cha ge ⵜ·J=− ⳵␳ 共x, 兲 ⳵ 共3兲 we imply ha I共 兲=Q ˙共 兲,共4兲 ⵜ·j共x兲=−q共x兲.共5兲 Thus i he cu en is known, i is easy o ind he cha ge dis ibu ion and ice e sa. The mode p o ile ob ained nu- me ically o ou SRR geome y is shown in Fig. 2. The cu en jis symme ic and eaches i s maximum a he poin opposi e o he gap. In acco dance wi h Eq. 共5兲, he cha ge dis ibu ion q共x兲is an isymme ic and goes h ough ze o whe e j共x兲is maximal. We see ha q共x兲 eaches i s maxi- mum magni ude nea he gap. In he single-mode app oxima ion, he dynamics o he SRR can be ully desc ibed by he ime-dependen ampli ude Q共 兲, and we may w i e he SRR Lag angian as a sum o e ms quad a ic in Qand Q ˙ L=AQ ˙2−BQ2,共6兲 whe e Aand Ba e cons an s which will be discussed below. Acco dingly, he SRR ene gy eads E=Q ˙ ⳵ L ⳵ Q ˙−L=AI2+BQ2共7兲 and is nicely sepa a ed in o induc i e 共magne ic兲and capaci- i e 共elec ic兲pa s. Clea ly, o a passi e SRR we equi e Aⱖ0, Bⱖ0. The Lag angian equa ion o mo ion d d ⳵ L ⳵ Q ˙= ⳵ L ⳵ Q共8兲 yields ha he dynamics o a single SRR is desc ibed by he oscilla o equa ion o he cha ge ampli ude 2 1 0 1 2 x ( m m ) 3 2 1 0 1 2 3 y ( m m ) ( a ) 2 1 0 1 2 x ( m m ) 3 2 1 0 1 2 3 ( b ) FIG. 2. 共Colo online兲Nume ically calcula ed 共a兲cu en and 共b兲cha ge dis ibu ion ac oss an SRR a esonance. POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲 155128-2 Q ¨共 兲+ ␻ 0 2Q共 兲=0, 共9兲 and he undamen al mode esonance occu s a he equency ␻ 0=冑B/A. No e ha in con as o he known modi ica ions o he Lag angian o malism o me ama e ials 共see, e.g., Re . 23兲, he e we do no ely on an equi alen ci cui model. Al hough one migh iden i y he pa ame e s 2Aand 1/2Bas e ec i e induc ance and capaci ance, espec i ely, below we e alua e hem explici ly om he known undamen al mode shape. In ac , he s ongly inhomogeneous mode p o ile 共see Fig. 2兲 sugges s ha i is unlikely ha he co ec alues o he pa- ame e s would ag ee wi h hose calcula ed om a ci cui analysis. Addi ionally, o ind he esonan equency, we do no need o calcula e Aand Bexplici ly, only hei a io. Fo he pu poses o ou analysis, i is su icien o con- side he case o a pai o SRRs, and i will subsequen ly be shown ha his explains all o he impo an ea u es ob- se ed in ou expe imen s wi h a bulk me ama e ial. In his case, he Lag angian can be w i en as a sum o he single SRR Lag angians and coupling e ms, which we also w i e as quad a ic in cu en s and cha ges L=A共Q ˙1 2+Q ˙2 2+2 ␣ Q ˙1Q ˙2兲−B共Q1 2+Q2 2+2 ␤ Q1Q2兲. 共10兲 The pa ame e s ␣ and ␤ a e he dimensionless cons an s o magne ic and elec ic nea - ield in e ac ion, espec i ely. The co esponding Lag angian equa ions o mo ion yield he sys em o ODEs o he ime-dependen ampli udes Q1,2 Q ¨1+ ␻ 0 2Q1=− ␣ Q ¨2− ␤ ␻ 0 2Q2,共11兲 Q ¨2+ ␻ 0 2Q2=− ␣ Q ¨1− ␤ ␻ 0 2Q1.共12兲 Sol ing hese equa ions one inds ha a pai o esona o s exhibi s wo esonances: symme ic and an isymme ic. Fo he symme ic esonance, Q1=Q2, which yields he esonan equency ␻ S= ␻ 0冑1+ ␤ 1+ ␣ 共13兲 while he an isymme ic mode wi h Q1=−Q2has he equency ␻ AS = ␻ 0冑1− ␤ 1− ␣ .共14兲 The desc ibed esonance spli ing is well known in he heo y o ha monic oscilla o s. Gene ally, b inging oge he wo oscilla o s o he same esonan equency in oduces coupling be ween hem, which esul s in spli ing in o wo modes. Examples ha e been shown o SRR esonan e- quency as a unc ion o some coupling pa ame e , e.g., mu- ual o ien a ion24 o wis angle,25 and ypically demons a e a spli ing o hyb idiza ion o modes. As we see, he di ec ion and s eng h o he esonance shi a e de e mined by he coupling cons an s ␣ and ␤ .To e alua e hem, we use he exp ession o he elec omagne ic ene gy ollowing om Lag angian 共10兲: E=A共I1 2+I2 2+2 ␣ I1I2兲+B共Q1 2+Q2 2+2 ␤ Q1Q2兲.共15兲 The i s g oup o e ms gi es he magne ic ene gy and he second g oup desc ibes he elec ic ene gy. A possible ou e o calcula e ␣ and ␤ is o app oxima e he elec ic and magne ic esponse o each ing by a ew e ms o he mul ipole expansion. The p oblem wi h his ap- p oach is ha i is based on he assump ion ha he obse e 共i.e., he second SRR兲is a a la ge dis ance compa ed o he dimensions o he sou ce. This equi emen is s ongly io- la ed in ou me ama e ial samples, whe e he sepa a ion be- ween ings is ac ually much smalle han he ou e ing di- ame e . This is essen ial o achie ing s ong uning by la ice manipula ion. The e o e we ha e chosen o calcula e ␣ and ␤ nume i- cally om he known cha ge and cu en dis ibu ions, q共x兲 and j共x兲, o he undamen al mode in a single SRR. Indeed, in he single-mode app oxima ion, he ene gy o a pai o SRRs eads E=Q1 2WE,11 +Q2 2WE,22 +2Q1Q2WE,12 +I1 2WH,11 +I2 2WH,22 +2I1I2WH,12,共16兲 whe e he pa ame e s WE,mn = 冕 Vm d3x 冕 Vn d3x⬘q共x兲q共x⬘兲 4 ␲ ⑀ 0兩x−x⬘兩,共17兲 WH,mn = 冕 Vm d3x 冕 Vn d3x⬘ ␮ 0j共x兲·j共x⬘兲 4 ␲ 兩x−x⬘兩.共18兲 The in eg als can be easily e alua ed once he cha ge and cu en dis ibu ions a e known. The in eg a ions o e xand x⬘a e o e he same ing i m=no o e di e en ings o he wise. Acco dingly, V1is a olume con aining only he i s ing and V2is a olume con aining only he second. The singula e ms a x=x⬘a e handled using he analy ical o - mulas gi en in Re . 26. Compa ing Eqs. 共15兲and 共16兲shows ha he coupling pa ame e s can be e alua ed as ␣ =WH,12 WH,11 , ␤ =WE,12 WE,11 .共19兲 Fo compa ison pu poses, when induc i e coupling is he dominan in e ac ion mechanism be ween he SRRs in a me ama e ial, we a e able o conside an a ay o spli ings as an a ay o cu en loops wi h some mu ual induc ance be ween hem. This app oach can hen be used o de ine he e ec i e pe meabili y o a me ama e ial sample.8In pa icu- la , o hin wi e loops wi h hei axes o ien ed in he same di ec ion, he mu ual induc ance can be ound27 by nume ical in eg a ion Lnn⬘共 兲= ␮ 0 0 2 4 ␲ 冕 0 2 ␲ 冕 0 2 ␲ d ␸ 1d ␸ 2cos共 ␸ 1− ␸ 2兲 ⫻兵 ␳ 2+z2+2 0 2关1 − cos共 ␸ 1− ␸ 2兲兴 +2 ␳ 0共cos ␸ 2− cos ␸ 1兲其−1/2, whe e he dis ance ec o be ween he ing cen e s has METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲 155128-3 been decomposed in o a adial componen ␳ and axial com- ponen z, 0is he ing adius, and ␸ 1and ␸ 2 ep esen he angle abou each ing. We can use he calcula ed mu ual induc ance o de ine an equi alen magne ic in e ac ion pa- ame e ␣ L=Lnn⬘/L共20兲 共whe e Ldeno es he sel -induc ance o one elemen 兲which should app oxima e he in e ac ion ene gy calcula ed om Eq. 共19兲. Asymp o ically his in e ac ion decays as 1/ so in a la ge a ay he nea es neighbo s p o ide he s onges con- ibu ion bu do no necessa ily domina e o e all o he s. Clea ly, his in e ac ion is highly aniso opic,22 being posi- i e o ings on he same axis, bu nega i e o ings in he same plane. I is also possible o de elop equi alen ci cui models o calcula e he elec ic in e ac ion be ween ings. Fo a pai o coaxial ings, he o al in e ac ion can be eliably modeled as a ci cula pa allel conduc o ansmission line, as o b oadside-coupled 共bc兲spli ing esona o s28 o , al e na- i ely, wi h an ex ended ci cui model accoun ing o he dis ibu ed capaci ance and induc ance.29 Howe e , once we in oduce some o se be ween he ings, hese app oaches a e no applicable and so we do no conside hem he e. III. TUNING INTERACTION BETWEEN A PAIR OF SPLIT RING RESONATORS Ha ing de eloped an app oach o calcula ing nea - ield in e ac ion be ween a pai o ings, we now apply i o a canonical sys em which has he basic p ope ies o ou ex- pe imen al a angemen . We conside a pai o SRRs, ei he iden ically 关gap- o-gap 共g g兲兴 o ien ed o o a ed by 180° wi h espec o each o he 关b oadside-coupled 共bc兲兴 and sub- jec o a la e al o se ␦ a. The geome y and inciden pola - iza ion a e shown in Fig. 3. The ings ha e a e age adius 0=2.25 mm, ack wid h o 0.5 mm, me al hickness o 0.03 mm, gap wid h o 1 mm, and a e sepa a ed in he ans e se di ec ion by 1.5 mm. The esul ing esonan equency is 10.6 GHz. We plo he in e ac ion ene gy calcula ed om Eq. 共19兲in Fig. 4 o di e en o se s be ween he ings. I can be seen ha he elec ic coupling pa ame e ␤ is nea ly symme ic be ween he wo con igu a ions. This can be unde s ood om Fig. 2, whe e we see ha he cha ge dis ibu ion has a s ong dipole componen o ien ed in he xdi ec ion. Fo he sym- me ic mode o he b oadside-coupled o ien a ion he cha ges accumula ed on he closes sides o he SRRs ha e opposi e signs, he o al-cha ge dis ibu ion has he na u e o a pai o an ipa allel dipoles, hus WE,12⬍0, and ␤ is also nega i e. In con as , in he gap- o-gap o ien a ion, he closes cha ges a e o he same sign, he o al-cha ge dis ibu ion becomes like a pai o pa allel dipoles and he pa ame e ␤ ⬎0. A an o se o abou one ing adius 共 ␦ a⬇ 0兲, he cha ge on one ing is app oxima ely equidis an om bo h posi i e and nega i e cha ges on he opposi e ing, hus he ne cou- pling passes h ough ze o. A la ge o se s, he elec ic cou- pling changes i s sign bu emains smalle , since only he nea es hal es o he SRRs a e e ec i ely in e ac ing wi h his in e ac ion decaying owa d ze o as he o se inc eases. The magne ic in e ac ion ene gy is also qui e di e en o he wo o ien a ions wi h he magne ic in e ac ion ␣ Lcalcu- la ed by Eq. 共20兲lying be ween he gap- o-gap and b oadside cu es. Fo he b oadside-coupled case, he si ua ion is quali- a i ely simila o he analy ical esul ␣ L. A low o se , he magne ic ield o one ing cu s h ough he o he ing in he same di ec ion o he su ace no mal, hus ein o cing he magne ic ield and inc easing he o al ene gy. As he o se is inc eased, he si ua ion g adually shi s o become like a pai o loops in he same plane, whe e he ield om one ing cu s h ough he o he in he opposi e di ec ion wi h espec o he su ace no mal. Hence, ␣ bc unde goes a change in sign. Howe e , in compa ison o he ing wi h uni o m cu en , he coupling is subs an ially mo e nega i e. This is due o he cu en maxima being on opposi e sides o he ings, and hence u he away om each o he . Fo he gap- o-gap o ien a ion he magne ic in e ac ion is much s onge o low ␦ a. This is due o he cu en maxima being loca ed nea each o he which p oduces a s onge con- ibu ion o he in eg al in WH,12 hus inc easing ␣ . As he ings a e u he sepa a ed om each o he , he in e ac ion ene gy educes, bu does no unde go a change in sign.We can in ui i ely unde s and his by neglec ing he small con- ibu ions o he cu en in he egion nea he gaps, hus we e ec i ely ha e wo linea cu en elemen s in he same plane which always in e ac wi h he same sign. Howe e , his balance is no uni e sal and is de e mined by he speci ic geome y and pa ame e s. To check his, we s udied a geom- e y wi h a e y small gap so ha he cu en dis ibu ion was δa (a) (b) E H k δa FIG. 3. Geome y o he pai o ings 共a兲b oadside-coupled and 共b兲gap- o-gap o ien a ion. 0 . 2 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3 N o m a l i z e d I n e a c i o n e n e g y 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 δ a / 0 β b c α b c β g g α g g α L FIG. 4. 共Colo online兲Magne ic 共 ␣ 兲and elec ic 共 ␤ 兲coupling pa ame e s o b oadside coupled 共bc兲and gap- o-gap 共g g兲o ien- a ion o a pai o shi ed SRRs. The magne ic in e ac ion calcula ed om he mu ual induc ance is gi en as ␣ L. POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲 155128-4 much mo e homogeneous wi h lowe esonance equency. In his simula ion 共no shown兲 he magne ic coupling did change sign and bo h alues o ␣ con e ged closely o ␣ L. In o de o e i y ha he calcula ed coupling co ec ly desc ibes he equency spli ing o his sys em, we compa e he equency shi p edic ed by Eqs. 共13兲and 共14兲wi h ha ob ained om he ull nume ic simula ions. Fo consis ency wi h ou in e ac ion-ene gy app oach we assume a homoge- neous ee-space backg ound. We use he equency domain sol e o he comme cial so wa e package CST MICROWAVE STUDIO 共Re . 30兲 o model a pai o ings, in a uni cell wi h pe iodic bounda y condi ions in he di ec ions ans e se o he p opaga ion di ec ion. This pe iodic sys em enables us o de ine a ansmission coe icien , and he bounda ies a e 10 mm om he ings. This alue is chosen o be la ge enough so ha he e is no signi ican in e ac ion wi h pe iodic neigh- bo s, ye small enough o a oid signi ican sca e ing in o highe o de di ac ion modes. Thus we can conside his he limi ing case o a highly dilu e me ama e ial slab. The ansmission spec um as a unc ion o o se is plo - ed in Fig. 5. We see ha all he impo an ea u es o he mode spli ing a e ep esen ed co ec ly by ou single-mode heo y o coupled SRRs. Expansion o Eqs. 共13兲and 共14兲 o small coupling p edic s equency spli ing o ⌬ ␻ =⫾共 ␤ − ␣ 兲/2, hence he cu es a e app oxima ely symme ic abou ␻ 0. The s ong spli ing obse ed o he b oadside-coupled o ien a ion is due o he opposi e signs o he elec ic and magne ic coupling. A ␦ a/ 0⬇1.1 bo h ␣ bc and ␤ bc change signs, hence he c ossing o he symme ic and an isymme - ic modes is obse ed. In con as , we see ha o he gap- o-gap o ien a ion o small o se ␣ g g and ␤ g g a e o he same sign, and hus hey ha e an opposing e ec , esul ing in small equency spli ing. Since ␤ dec eases much as e and changes sign, he esul in maximum equency spli ing o ␦ a/ 0be ween 1.1 and 1.5. We no e ha in Re . 12 he eso- nan equency based on ␣ Lwas compa ed wi h expe imen al esul s o he b oadside-coupled o ien a ion and s ong dis- ag eemen was ound. I can be seen ha he calcula ed ansmission h ough he cell exhibi s di e en dep hs o he esonance o he sym- me ic and an isymme ic modes. This is due o he di e en e iciency o coupling be ween he modes and he inciden plane wa e. Fo ins ance, i is impossible o exci e he an i- symme ic mode wi h a no mally inciden plane wa e o ␦ a=0, since bo h ings a e exci ed in phase. As he o se is inc eased, some e a da ion be ween he ings occu s and ex- ci a ion o he an isymme ic mode is allowed. The e a e se e al easons o he small quan i a i e dis- ag eemen be ween he exac calcula ions and hose based on he calcula ed in e ac ion ene gy. Fi s , he minimum o ansmission occu s a a equency sligh ly di e en om he esonan equency, due o coupling e ec s 共impedance ma ching兲be ween he inciden wa e and he ing. Second, he e may be some small con ibu ion o highe SRR eigen- modes due o pe u ba ion o he cha ge and cu en dis ibu- ions. Thi d, he e may s ill be some small in luence o he pe iodic bounda ies. Finally, ou de eloped ela ions neglec e a da ion, which is s ic ly alid in he subwa eleng h limi , whe eas he ou e adius o he ings is 0.18␭a ␻ 0. Re a da ion has p e iously been shown o modi y he in e - ac ion be ween SRRs h ough i s in luence on he dispe sion o magne oinduc i e wa es in a ays.31–33 We emphasize ha ou app oach de eloped in Sec. II is ad an ageous o e he di ec nume ical calcula ion. Fi s , once he mode p o ile is known, calcula ion o he equen- cies in Fig. 5 akes app oxima ely 30 s on a single CPU whe eas he di ec calcula ion o he ull spec um akes se - e al hou s on a mul ico e machine. Second, we a e clea ly able o demons a e he na u e o he coupling, which yields insigh in o he uning beha io . IV. TUNING INTERACTION IN A BULK METAMATERIAL We now apply ou app oach o explain expe imen al e- sul s o uning he esponse o a slab o me ama e ial. The me ama e ial is ab ica ed using pho oli hog aphy o e ch coppe acks on o FR4 p in ed ci cui boa d, using he same geome y as in ou nume ical simula ion o a pai o ings. The ab ica ed sample has 30 laye s, each wi h i e ings in he p opaga ion di ec ion and is only one ing in heigh 共i.e., 0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 1 . 0 5 1 . 1 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 δ a / 0 ( a ) B o a d s i d e c o u p l e d o i e n a i o n 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 1 . 0 5 1 . 1 0 ω / ω 0 0 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 δ a / 0 ( b ) G a p o g a p o i e n a i o n 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 1 . 0 FIG. 5. 共Colo online兲Nume ical esul s. T ansmission spec um o a pai o 共a兲b oadside-coupled and 共b兲gap- o-gap o ien ed ings. Solid line: ␻ S om Eq. 共13兲, b oken line: ␻ AS om Eq. 共14兲 METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲 155128-5 a5⫻30⫻1 a ay兲. The longi udinal pe iod o he sample is 7 mm, he ans e se pe iod is dic a ed by he sample hick- ness 共1.5 mm兲as he e is no spacing be ween boa ds. As wi h he pai o ings, we ha e assembled slabs wi h wo ela i e o ien a ions o he spli ings in adjacen planes—gap- o-gap and b oadside-coupled. The sample is placed in he cen e o a WR229 ec angula me allic wa eguide, wi h dimensions 58.17 mm ⫻29.08 mm, exci ed a i s dominan TE10 mode. We emo ed he in luence o he coaxial adap e s and eeding wa eguide sec ions by pe o ming a h ough- e lec - line calib a ion.34 In Fig. 6we show he expe imen ally ob- ained ansmission h ough each slab o closely coupled SRRs wi h he co esponding e lec ion shown in Fig. 7. The shi o esonan equency shows good quali a i e ag eemen wi h he esul s o a pai o ings p esen ed in Sec. III wi h e y simila changes in he spec um obse ed 共no ing ha ␦ a/a=0.5 co esponds o ␦ a/ 0=1.56兲. How- e e , o he gap- o-gap o ien a ion, nume ical simula ion o a sys em o wo boa ds wi h i e ings each, and pe iodic bounda ies in bo h ans e se di ec ions 共no shown兲,is quali a i ely simila o he expe imen ally obse ed esul s bu quan i a i ely highly inaccu a e. The eason u ns ou o be he loss o symme y when he sys em is placed inside he wa eguide because he uppe and lowe wa eguide walls do no co espond o pe iodic bounda ies bu ins ead ep esen planes o mi o e lec ion. The e o e, his sys em mus be desc ibed as ha ing a supe la ice a angemen in he e ical as well as ho izon al planes wi h each supe cell consis ing o ou SRRs. This cell has al e na ing o ien a ion o he SRRs in he e ical di ec ion co esponding o he planes o mi o symme y, as shown in Fig. 8共b兲. Once his uni cell is aken in o accoun , nume ical simula ions a e in a good ag eemen wi h he expe imen 关Fig. 9共b兲兴. Na u ally, nume ical simula ions o he b oadside- coupled o ien a ion also ag ee well wi h he expe imen 关Fig. 9共a兲兴. In his case a e y simila esul is p o ided wi h simple pe iodic bounda y condi ions 共no shown兲. Fo his o ien a ion he supe la ice e ec i ely o med by he wa e- guide shown in Fig. 8共a兲does no ha e an essen ially di e - en symme y o he o iginal supe la ice. We can conclude ha in bo h cases he dominan mode o he slab co esponds o he dominan symme ic mode o a pai o ings wi h a simila pa e n o esonan equency s o se occu ing. The weake coupling o he modes o he gap- o-gap o ien a ion is due o he shape o he symme ic mode. I s magne ic ield has a la ge componen pa allel o ha o he wa eguide mode, howe e , i s elec ic ield is p ima ily longi udinal, in con as o he ans e se elec ic ield o he wa eguide mode. In Fig. 9共b兲, we see wo highe o de modes, which mos likely co espond o he highe o de esonances obse ed in he expe imen al esul s in Fig. 6共b兲. F om he nume ical 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 T a n s m i s s i o n δ a / a ( a ) B o a d s i d e - c o u p l e d o i e n a i o n 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 ( G H z ) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 T a n s m i s s i o n ( b ) G a p - o - g a p o i e n a i o n FIG. 6. 共Colo online兲Expe imen al ansmission while uning ␦ a o spli ing esona o slab in wa eguide, o 共a兲b oadside- coupled and 共b兲gap- o-gap o ien a ion o adjacen laye s. 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 R e l e c i o n δ a / a ( a ) B o a d s i d e - c o u p l e d o i e n a i o n 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 ( G H z ) 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 R e l e c i o n ( b ) G a p - o - g a p o i e n a i o n FIG. 7. 共Colo online兲Expe imen al e lec ion while uning ␦ a o spli ing esona o slab in wa eguide, o 共a兲b oadside-coupled and 共b兲gap- o-gap o ien a ion o adjacen laye s. ( a) ( b ) FIG. 8. 共Colo online兲Schema ic o he e ec i e supe la ice geome y co esponding o he wa eguide measu emen o 共a兲 b oadside-coupled and 共b兲gap- o-gap o ien a ion. Dashed lines show planes o e lec ion symme y and he shaded egion shows he supe cell wi h ou SRRs. POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲 155128-6 simula ions we obse e ha he cu en dis ibu ions o hese modes a e symme ic, hus hey co espond o highe -o de modes o he me ama e ial slab and no o he an isymme ic mode o a pai o ings. In con as , in Fig. 9共a兲 he e is a weakly coupled an isymme ic mode, which we e i ied by inspec ion o he cu en s. This mode also quali a i ely ag ees wi h he co esponding mode o he pai o ings wi h somewha weake coupling due o he inc eased misma ch o he inciden wa eguide mode. This mode may co espond o some o he smalle ea u es obse able in Fig. 6共a兲, how- e e , due o he size o hese ea u es his canno be eliably de e mined. We do no conside o se s g ea e han 0.5a, since in an in ini e la ice only shi s be ween 0 and 0.5 a e unique, while in a ini e s uc u e, la ge shi s esul in e y i egula bounda ies. No e ha in he simula ions we ha e neglec ed he e ec o he mode p o ile o he ec angula wa eguide, which would co espond o an e ec i e a ia ion in he angle o incidence o he plane wa e as a unc ion o equency, which can esul in a di e en esponse due o he aniso opy and non-negligible spa ial dispe sion o he medium.10 Clea ly he coupling in he comple e la ice is much mo e complica ed han in he simple wo- ing sys em, as he in e - ac ions be ween a la ge numbe o ings mus be aken in o accoun . In p inciple i is possible o ex end he analysis o Sec. III o an a bi a y numbe o ings. Howe e , he quali- a i e ag eemen be ween he expe imen al esul s and he modeled pai o ings sugges s ha he phenomenology de- eloped o he wo ings is gene ally applicable and leads o co ec p edic ions. Al hough an accu a e gene aliza ion o ou modeling ap- p oach o a bulk sys em lies beyond he scope o his pape , i is clea ha he esul ing homogenized e ec i e me ama- e ial pa ame e s will exhibi simila uning pa e n due o he esonance shi . No e ha he in oduc ion o he e ec i e pa ame e s is jus i ied when he a io o he uni -cell size o he inciden wa eleng h is small. The e o e, when conside - ing modi ica ions o he la ice which c ea e a supe la ice s uc u e, he size o he supe cell should be smalle han he wa eleng h. The e a e homogeniza ion app oaches in he li - e a u e 共e.g., Re . 35兲which include unknown pa ame e s o in e ac ion be ween esonan elemen s and ou semianaly i- cal app oach would make an ideal ool o e alua ing hese cons an s. V. CONCLUSION We ha e analyzed he nea - ield coupling wi hin me ama- e ials, conside ing bo h he ela i e o ien a ion and he o - se be ween he cen e s o wo neighbo ing esona o s. Using a pai o spli ing esona o s as a simple model, we ha e shown he coupling mechanisms a wo k in ou ecen ly p o- posed uning scheme, based on he di ec calcula ion o he in e ac ion ene gy. We ha e con i med ha hese mecha- nisms can p edic quali a i ely he pe o mance o a ealis ic me ama e ial s uc u e. This pa es a oad owa d a eliable design and de elopmen o unable me ama e ials o a ious applica ions. We no e ha he speci ic geome y o he spli ings can ha e a e y signi ican in luence on he quali a i e na u e o he coupling, including cases which un coun e o ou in ui- i e unde s anding o cu en loops in e ac ing magne ically. Finally, we poin ou ha he app oach de eloped he e o modeling nea - ield e ec s is pa icula ly p omising o me ama e ials scaled down o ope a e a op ical equencies. In he isible, he pa adigm o ideally conduc ing me al ails and he a ea o applicabili y o ci cui models is a he lim- i ed. In con as , he conside a ion in e ms o exci a ion and in e ac ion o plasmonic s anding wa es will p o ide a clea physical pic u e. ACKNOWLEDGMENTS The au ho s a e g a e ul o Lukas Jelinek and Rica do Ma qués 共Uni e si y o Se ille兲 o help ul discussions. This wo k was suppo ed by he Aus alian Resea ch Council. M.L. acknowledges hospi ali y o Nonlinea Physics Cen e and suppo o he Spanish Jun a de Andalusia unde P ojec No. P06-TIC-01368. M.G. acknowledges suppo om he Russian Academy o Sciences, BPS P og am “Physics o new ma e ials and s uc u es.” 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 ( G H z ) 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 δ a / a ( a ) B o a d s i d e c o u p l e d o i e n a i o n 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 3 . 0 3 . 5 4 . 0 4 . 5 5 . 0 5 . 5 6 . 0 ( G H z ) 0 . 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 δ a / a ( b ) G a p o g a p o i e n a i o n 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 0 . 6 0 . 7 0 . 8 0 . 9 FIG. 9. 共Colo online兲Nume ically ob ained ansmission spec- um o me ama e ial in wa eguide, 共a兲b oadside-coupled and 共b兲 gap- o-gap o ien a ion. The whi e line indica es expe imen ally ob- ained esonan equencies. METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲 155128-7 *[email p o ec ed] 1M. G. Sil ei inha and C. A. Fe nandes, Phys. Re . B 78, 033108 共2008兲. 2R. W. Ziolkowski, Phys. Re . E 70, 046608 共2004兲. 3J. Pend y, A. Holden, D. Robbins, and W. 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