Sub-Wavelength Resonances in Metamaterial-Based Multi-Cylinder Configurations

Transcription

1 Matrial 211, 4, ; di:1.339/ma41117 Articl OPEN ACCESS matrial ISSN Sub-Wavlgth Rac i Mtamatrial-Bad Multi-Cylidr Cfigurati Saml Arlaagić * ad Olav Bribjrg Dpartmt f Elctrical Egirig, Elctrmagtic Sytm, Tchical Uivrity f Dmark, Ørtd Plad, Bld. 348, DK-28, Kg. Lygby, Dmark; b@lktr.dtu.dk * Authr t whm crrpdc huld b addrd; ar@lktr.dtu.dk; Tl.: ; Fax: Rcivd: 31 Octbr 21; i rvid frm: 23 Dcmbr 21 / Accptd: 3 Dcmbr 21 / Publihd: 31 Dcmbr 21 Abtract: Sub-wavlgth rac kw t xit i ilatd mtamatrial-bad tructur f circular cylidrical hap ar ivtigatd with th purp f dtrmiig whthr th idividual rac ar rtaid wh vral f uch rat tructur ar grupd t frm a w tructur. T thi d, tructur citig f 1, 2 ad 4 t f mtamatrial-bad cctric cylidr xcitd by a lctric li currt ar aalyzd umrically. It i dmtratd that th tructur rcvr th rac f th idividual tructur v wh th cylidr ar clly pacd ad th w tructur i thu lctrically mall. Th ivtigati i cductd thrugh a dtaild aalyi f th lctric ar-fild ditributi a wll a th radiati ritac i th ca whr th idividual tructur ar mad f impl dilctric matrial i cjucti with impl, but ly ad dipriv, mtamatrial. Kywrd: mtamatrial; ub-wavlgth rac; cattrig; li urc 1. Itrducti Th fild f mtamatrial (MTM) ha xpricd igificat citific advac i rct yar, ad umru applicati withi th micrwav [1-3] ad th ptical [4] frqucy rgi hav b dvid. Imprtat xampl f MTM iclud dubl-gativ (DNG) matrial, which p a gativ ral part f th prmittivity ad prmability, a wll a pil-gativ (ENG) ad

2 Matrial 211, mu-gativ (MNG) matrial, which p a gativ ral part f th prmittivity ad prmability, rpctivly. Amg th umru rprtd applicati f th MTM, pcific attti ha b dvtd t thir pttial f prvidig ub-wavlgth rat tructur f variu caical hap [5-14] ithr wh ud al r i cmbiati with dubl-pitiv (DPS) matrial, which p a pitiv ral part f prmittivity ad prmability. I particular, it wa hw i [9] that a ilatd t f cctric circular MTM-bad cylidr xcitd by a arby lctric li currt (ELC) p ub-wavlgth rac whr th xcitati f pcific md i fud t lad t larg radiatd pwr fr ctat ELC. Th purp f th prt wrk i t ivtigat hw th ub-wavlgth rac f th ilatd MTM-bad cctric cylidr tructur tudid i [9] ar affctd wh vral f uch tructur ar grupd t frm a w tructur. T thi d, cfigurati citig f 1, 2 ad 4 t f MTM-bad cctric cylidr, hcfrth rfrrd t a 1-, 2-, ad 4-cylidr tructur, ar aalyzd. It i hw that th tructur rcvr th rac f th idividual tructur v wh th cylidr ar clly pacd ad th cfigurati i thu lctrically mall. Th aalyi i cductd with th ANSOFT High Frqucy Structural Simulatr (HFSS) [15] ad iclud dtaild ivtigati f th lctric ar-fild ditributi ad th radiati ritac i ca f impl, but ly ad dipriv, MTM. A cllcti f MTM-bad bjct wr tudid i [6] with th purp f dviig a ffctiv hybrid MTM, i [16] fr clakig purp, ad i [17] fr it cattrig prprti. Th prt wrk i a xti f [18], ad i cmpari, iclud bth a full accut f th HFSS mdl a wll a additial ar-fild ivtigati. Th prt maucript i rgaizd a fllw. I Scti 2, th ivtigatd tructur ar dfid ad th aalyi tchiqu, icludig th xact mthd ud fr th 1-cylidr tructur, a wll a th umrical mthd, ar dcribd. Thi cti al iclud a brif dicui th cditi fr ub-wavlgth rac i th ilatd 1-cylidr tructur; thi i ud i cjucti with th xact aalytical rult t dfi th lctrical ad gmtrical paramtr f a giv 1-cylidr tructur. I Scti 3, th umrical rult ar prtd; i particular, th rac f th idividual tructur ar tudid a th ditac btw th cylidr i chagd. I all ca, th rat tructur ar mad f impl dilctric matrial i cjucti with impl, but ly ad dipriv, MTM, ad th rat prprti f all cfigurati ar aalyzd thrugh dtaild ivtigati f thir lctric fild ditributi ad th radiati ritac. Scti 4 iclud a ummary ad cclui f th prt wrk. Th tim factr xp( j t), with big agular frqucy ad t tim, i aumd thrughut th maucript. 2. Cfigurati ad Thry 2.1. Cfigurati Figur 1 hw th k, th cctric cylidr t () f th 1-, 2- r 4-cylidr tructur ivtigatd i th prt wrk. A circular cylidr (rgi i,) with ctr at O ad radiu i, i cvrd by a cctric circular hll (rgi ) f utr radiu, ad lcatd i fr pac with th prmittivity, prmability, wav umbr k ad itriic impdac /. Rgi i, () cit f impl, ad grally ly ad dipriv, DPS,

3 Matrial 211, DNG, ad/r ENG ad MNG matrial with th prmittivity ad th prmability ' '' i, i, ji, ( ' '' j ) ' '' i, i, j i, ( ' Ek ', Ek j ). Th 1-, 2-, r 4-cylidr tructur i illumiatd by a ifiit lctric li currt (ELC) I that i paralll t th cylidr ad ca b lcatd i ay f th rgi. Th cylidrical (,, z )-crdiat ytm ad th Cartia ( x, y, z )-crdiat ytm ar itrducd with th z-axi cicidig with th cmm axi f th cylidr. Th crdiat f th brvati pit ar (, ), whil th f th ELC ar (, ). Figur 1. Th k' th cctric cylidr t f th 1-, 2-, ad 4-cylidr tructur. Th 1-cylidr tructur cit f a cylidr C1 which ha it ctr at th rigi, i.., O (, C1 ). Th 2-cylidr tructur cit f th prviu cylidr C1 ad a cylidr C2 havig it ctr at O ( d,) 2. Thu, th cylidr C2 i diplacd alg th x-axi with a parati C 2 C1, C ditac d t th cylidr C1. Th 4-cylidr tructur cit f th prviu cylidr C1 ad C2, ad th additial cylidr C3 ad C4 with thir ctr at O d / 2, d / 2 ) ad O d / 2, d / 2 ), rpctivly. C4 ( C1 C Aalyi Mthd C3 ( C1 C3 Fr th 1-cylidr tructur, bth a xact a wll a a umrical luti hav b btaid. Th xact luti i bad th igfucti xpai mthd,.g., [19]. Whra th dtail f th xact luti ca b fud i [9], w mphaiz blw ly th mai pit. Th icidt fild f th ELC, a wll a th ukw fild i th thr rgi, i.., th cattrd fild i th rgi ctaiig th ELC ad ttal fild i th rmaiig rgi, ar all xpadd i trm f cylidrical wav fucti. Th xpai rprt th multipl xpai f th rpctiv fild, ad fr th ukw fild, thy ctai a t f ukw xpai cfficit A, j 1, 2, 3, ad 4 (with j 1 fr rgi i,, j 2 ad 3 fr rgi, ad j 4 fr th ambit fr-pac mdium), whr i th md umbr with rfrrig t th mpl md, 1 t th dipl md ad. Th ukw xpai cfficit A dpd th lctrical ad gmtrical paramtr f i j

4 Matrial 211, 4 12 th tructur i Figur 1 a wll a th lcati f th ELC, ad thy ar radily dtrmid by frcig th budary cditi at th itrfac btw th thr rgi; c th cfficit ar kw, th fild i th diffrt rgi hav b dtrmid. Fr th 2- ad 4-cylidr tructur, a umrical luti i tablihd uig th ANSOFT HFSS ftwar [15] (th umrical luti wa al mplyd t ivtigat th 1-cylidr tructur ad t cmpar it rult with th xact luti i rdr t cfirm th validity f th tablihd HFSS mdl). Figur 2 hw th HFSS mdl, whr th 4-cylidr tructur with th idividual cylidr digatd a C1, C2, C3, ad C4 i dpictd. Th mdl cit f th ELC urc mdld by a fiit lgth currt tub f radiu a, currt I, ad it axi lcatd at (, )largd i th it f th figur, ad th fiit lgth MTM-bad cylidr. Th fiit lgth currt tub ad th MTM-bad cylidr ar pitid btw, ad prpdicular t tw paralll, prfctly lctrically cductig ifiit plat with parati h. Du t imag thry [19], th plat mdl th ifiit MTM-bad cylidr ad th ELC. Btw th prfctly cductig plat, uifrm prfct matchig layr which mdl fr-pac radiati, which hav thick d, circumcrib a quar f id lgth w, ad hav thir crr ad dg jid, ar irtd. Th valu f th pcific paramtr f th HFSS mdl a wll additial dtail ar fud i Scti 3. Figur 2. Th HFSS mdl f th 4-cylidr tructur. 1: Prfctly cductig tp (1a) ad bttm (1b) plat; 2: Elctric currt tub (largd i th it f th figur); 3: Uifrm prfct matchig layr. Th figur i t t cal. S th mai txt fr furthr xplaati Drivd Quatiti ad Rac Cditi I th prt wrk, th attti i dvtd t th radiati ritac, ctat valu f I radiatig i th prc f th matrial tructur, R t, f th ELC fr a giv

5 Matrial 211, rlativ t th radiati ritac, matrial tructur 2P Rt (1) I t 2 R i, f th ELC with th am currt I i th abc f th 2P Ri (2) I whr th quatiti P t ad P i i th abv xpri rprt, rpctivly, th pwr radiatd by th ELC i th prc ad abc f th matrial tructur. Fr th 1-cylidr tructur, th xact xpri fr P t ad P i hav b btaid i [9], ad ar rpatd hr fr th ak f cvic i 2 Nmax 1 2 k 2 2 Pt I (3 ) (3) k Pi I (4) 4 2 I (3), th quatity A 4 wh th ELC i i rgi i, ad, whil J ( k ) A4, whr J ( ) i th Bl fucti f rdr, wh th ELC i utid th 1-cylidr tructur. Th ymbl i th Numa umbr; thu, 1 fr ad 2 thrwi, ad N max i th trucati limit ch uch t ur th cvrgc f th cylidrical wav xpai. Frm (3) ad (1), it i clar that larg valu f th ttal radiati ritac will rult if th amplitud f th xpai cfficit A 4 bcm larg. Wh a giv igl cctric cylidr t i lctrically mall, i.., wh it i ub-wavlgth, th xpai cfficit bcm vry larg, ad thu xhibit a rac wh th cditi i atifid [6,9]. A xplaid i [6,9], at lat f th rgi cmpriig uch a cctric cylidr t mut b mad f DNG ad/r MNG matrial i rdr t atify th cditi i (5), ad mrvr, th xcitati f th ub-wavlgth rac i du t th prc f atural md i th tructur. Th rac cditi i (5) ha b ud i [6,9] t dig rat ub-wavlgth 1-cylidr cfigurati, ad i al ud xt t dig th idividual cctric cylidr t f th 1-, 2-, ad 4-cylidr tructur. i, ' ' ' ( i, )( ) 2 (5) ' ' ' ( i, )( ) 3. Numrical Rult 3.1. Rat Cfigurati ad Furthr Rmark th HFSS Mdl Accrdig t Scti 2.3 ad [6,9], it i pibl t dig a ub-wavlgth 1-cylidr tructur capabl f xcitig a dipl ( 1) md rac, which lad t larg valu f,.g., radiatd pwr ad radiati ritac. I th prt cti, w ivtigat whthr th rac f th idividual 1-cylidr tructur xit, ad udr which cditi wh vral cylidrical tructur

6 Matrial 211, ar grupd t frm a w tructur. T thi d, w ch th idividual 1-cylidr cfigurati f 25, 266, 283, 3 MHz fr which xcit th dipl rac at th dig frquci cylidr C 2, C3, C4, ad C 1, rpctivly, thi ladig t th mallt fr-pac wavlgth 1 m. Th xcitati f th dipl md at th pcifid dig frquci ca,mi b accmplihd if rgi i, i fr pac, i.., ( i,, i, ) (, ), rgi i a MNG matrial with (,, ) (, 4 ), ad i, 6 mm, whil C mm, C mm, C mm, ad C mm. Th valu fr ar th xact valu f th utr radii f th rpctiv hll f th idividual 1-cylidr tructur f which th lctrical ad gmtrical paramtr ar ummarizd i Tabl 1. Tabl 1. Elctrical ad gmtrical paramtr f diplar 1-cylidr cfigurati. Cylidr f [MHz] [mm] C C C C Fr all tructur: i,, i, ) (, ) ; (,, ) (, 4 ) ; 6 mm ( i, I all ca, th currt f th li urc, i th xact a wll a th HFSS-bad xamiati, i t t I 1A. Fr th HFSS mdl, th ELC i mdld by a prfctly cductig tub f radiu a.15 mm, mi / 6666 ; th id lgth f th quar circumcribd by th uifrm prfct matchig layr i w 1 mm ; th parati btw th prfctly cductig plat i, mi h 1mm, mi /1, ad th thick f th prfctly matchig layr i t t d mm, mi / 63 (th lattr big th dfault valu uggtd by HFSS fr th ch valu f w ). Th mdl f th ELC wa ttd thrughly by cmparig th umrically calculatd radiati ritac i fr pac with th kw aalytical rult. With th currt alg th tub big qual t I 1A, th radiati ritac a calculatd by HFSS wa fud t b.57 Ω/mm; a valu which i vry cl t th xact aalytical rult f.59ω/mm [12], thrby vrifyig th tablihd HFSS mdl f th ELC. I rdr t a th frqucy bhavir f th MNG matrial f th 1-, 2-, ad 4-cylidr tructur, th Drud dipri mdl [3] ha b mplyd fr th prmability f rgi fr all cfigurati. I (6), th quatity pm, i th magtic plama frqucy, ad i ch uch that, 4 i btaid at th rpctiv dig frquci. Th paramtr m, i th magtic cllii frqucy rprtig l i th matrial. 2 pm, 1 (6) ( j m, )

7 Matrial 211, Cylidr Structur Th rac f th 1-cylidr tructur ar illutratd i Figur 3(a) whr th quatity 1lg R t [db], whr th radiati ritac R t (1) ha b rmalizd by 1 / mm, i RR 1 hw a a fucti f frqucy wh ach f th cylidr i ctrd at th rigi ad th ELC i lcatd at (, ) ( 2.5 mm, ). Th prmability f th MNG hll f all cfigurati i mdld by th ll Drud mdl fr which i all tructur. Th circl i Figur 3(a) m, rprt th xact aalytical rult whil th full li rprt th crrpdig HFSS rult. Th agrmt btw th xact aalytical rult ad HFSS rult i t b xcllt; a imilar agrmt wa rprtd i [13]. It i clar that th idividual 1-cylidr tructur rat at th dird digd frquci; mrvr, th valu f RR ar cmparabl i th fur ca ad qual t apprximatly 2 db, thi hwig larg hacmt f th radiati ritac f th ELC arby th MTM-bad tructur rlativ t th ca whr th ELC i al i fr pac. Figur 3(b) hw th magitud f th lctric fild i th xy-pla (with th dyamic rag (DR) idicatd blw th figur) f th rat C 2 ( 25 MHz) 1-cylidr tructur at th frqucy f 25 MHz (imilar lctric fild rult ar btaid fr thr 1-cylidr tructur ad ar thrfr t prtly icludd). A clar dipl lctric fild pattr i brvd i Figur 3(b), cfirmig that th rac i Figur 3(a) ar du t th xcitati f th dipl md i th idividual cylidr. It i xt ivtigatd hw th rac f th idividual 1-cylidr tructur rprtd i Figur 3 ar affctd wh vral tructur ar grupd t frm 2- ad 4-cylidr tructur. Figur 3. Th quatity 1lg [db] a a fucti f frqucy fr th rat RR 1 R t 1-cylidr tructur (a), ad th magitud f th lctric fild f th rat C 2 ( 25 MHz) 1-cylidr tructur fr f 25 MHz (b). I all ca, ach f th idividual cylidr ar ctrd at th rigi ad th ELC i lcatd at (, ) ( 2.5 mm, ). Th lctric fild i (b) i dpictd i th xy-pla ad th liar dyamic rag (DR) i V/m i idicatd blw th figur. Curv rprtig th circular urfac f th tructur ad a lft-right arrw, idicatig th iz cal f th figur, ar likwi idicatd i (b). (a) DR: 5.5E1 1.2E5 f 25 MHz (b)

8 Matrial 211, Cylidr Structur Figur 4 hw th rult fr th 2-cylidr tructur citig f cylidr C1 ad C2 (th tructur i hw i th it tp f th figur). Spcifically, Figur 4(a) ad 4(b) hw th quatity RR [db] a a fucti f frqucy fr th parati ditac d = 5, 4, 3, 2, 1 ad 5 mm with th ELC lcatd at, ) ( / 2, ). Fr all parati ditac d, tw ditict ( i, C1 d rac ar fud. Fr d = 5 ad 4 mm, th rac ccur at f = 25 MHz ad f = 3 MHz, rpctivly, whr th idividual cylidr ar digd t rat ad thir amplitud ar t b lwr tha fr th idividual cylidr i Figur 3(a), ic th ELC i farthr away frm th cylidr i th prt ca. Th rac ar du t th dipl md xcitati i th cylidr, a illutratd i Figur 4(c) ad (d) whr th magitud f th lctric fild i hw fr d = 4 mm ad f = 25 MHz i (c) ad f = 3 MHz i (d). A th parati ditac d dcra furthr, ad/r bth rac hift lightly away frm th rat frquci f th ilatd idividual cylidr. Hwvr, th rac ar till du t th dipl md i th tw cylidr. Thi i, hwvr, t th ca fr th parati ditac f d = 5 mm whr,.g., th firt rac at f = 241 MHz, which attai highr amplitud tha i th ca f idividual cylidr, i du t a md charactrizd by trg cuplig btw th tw cylidr a i illutratd i Figur 5 whr th magitud f th lctric fild i hw. With th diamtr f th idividual cylidr big apprximatly 2 mm, it i thu fud that th ub-wavlgth rac f th idividual cylidr al ccur i 2-cylidr cfigurati f which th vrall iz i a mall a / 2 (fr d 1 mm ), thu prvig uch tructur faibl fr th pttial dig f multi-rat ub-wavlgth ytm. Figur 4. (a) ad (b): Th quatity 1lg [db] a a fucti f frqucy fr RR 1 th rat 2-cylidr tructur fr diffrt parati ditac d. Th magitud f th lctric fild f th rat 2-cylidr tructur fr d 4 mm ad f 25 MHz; (c) ad f 3 MHz (d). I all ca, th ELC i lcatd at (, ) ( / 2, ). R t C1 d Th lctric fild i (c) ad (d) i dpictd i th xy-pla ad th liar dyamic rag (DR) i V/m i idicatd blw th figur. Curv rprtig th circular urfac f th tructur ad a lft-right arrw, idicatig th iz cal f th figur, ar likwi hw i (c) ad (d). It abv th figur hw th 2-cylidr tructur citig f cylidr C1 ad C2.,mi

9 Matrial 211, Figur 4. Ct. (a) (b) DR: 1.9E2 4.2E4 DR: 2.9E 3.8E4 f 25 MHz f 3 MHz (c) (d) Figur 5. Th magitud f th lctric fild f th rat 2-cylidr tructur wh th parati ditac d = 5 mm fr f 241 MHz, which i th frqucy at which th firt rac appar i Figur 4(b). Th ELC i lcatd at (, ) ( d / 2 mm, ), ad th lctric fild i dpictd i th xy-pla with th liar dyamic rag (DR) i V/m idicatd blw th figur. Curv rprtig th circular urfac f th tructur ad a lft-right arrw, idicatig th iz cal f th figur, ar likwi idicatd. C1 1 mm DR: 1.8E2 7.1E4 f 241 MHz

10 Matrial 211, Cylidr Structur Figur 6 hw th rult fr th 4-cylidr tructur (th tructur itlf i hw i th it i th right part f th figur). Figur 6. Th quatity 1lg [db] a a fucti f frqucy fr th rat RR 1 R t 4-cylidr tructur fr diffrt parati ditac d. I all ca, th ELC i lcatd at (, ) ( / 2, ). Th 4-cylidr tructur i hw th right. C1 d Spcifically, Figur 6 hw th quatity 1lg [db] a a fucti f frqucy fr th RR 1 parati ditac d = 5, 4, ad 3 mm wh th ELC i lcatd at (, ) ( d / 2 ). R t i, C1, Fr all parati ditac d, fur ditict rac ar fud, althugh lightly hiftd frm th rat frquci f th idividual cylidr ad with lwr amplitud tha i th ca f th 1-cylidr tructur i Figur 3(a). Fr a giv parati d, thi hift i largr fr th 4-cylidr tha fr th 2-cylidr cfigurati ad i t b largt fr th cylidr C1 ad C2. Th majrity f th rac i Figur 6 ar du t th dipl md xcitati i th idividual cylidr; thi i clar frm Figur 7(a) (d), which hw th magitud f th lctric fild (with th dyamic rag (DR) idicatd blw th rpctiv figur) fr d = 4 mm fr th frquci at which th rac appar i Figur 6 (f = 245.5, 266.5, 282.5, ad 32.5 MHz). Th crrpdig fild at th rat frquci f th ilatd idividual cylidr ar hw i Figur 7() (h); th al hw clar dipl pattr but with diffrt maximum valu f th fild. Whil th fild lvl fr th cylidr C3 ad C4, rpctivly, i Figur 7(b) ad (c) ar cmparabl with th i Figur 7(f) ad (g), thy ar at lat a rdr f magitud lwr fr cylidr C2 ad C1, rpctivly, i Figur 7(a) ad (d), a cmpard t Figur 7() ad (h). Thi xplai why,.g., larg RR valu ar attaid fr cylidr C3 ad C4 t ly at th frquci f = MHz ad MHz, rpctivly, but al at th rigial rac frquci f th idividual cylidr, whra thi i fud t t b th ca fr cylidr C2 ad C1. Mrvr, fr th parati ditac f d = 3 mm, th firt rac ccurrig at f = 24 MHz i t du t a clar dipl md i th cylidr C2, but rathr t a md which i du t cuplig ffct btw th fur cylidr, a i clarly illutratd by th rult i Figur 8, which hw th magitud f th lctric fild i thi particular ca. With th diamtr f th idividual cylidr big apprximatly 2 mm, it i thu fud that th ub-wavlgth rac f th idividual cylidr al ccur i 4-cylidr

11 Matrial 211, cfigurati f which th vrall iz i a mall a /12. 5 (fr d 4mm ), thu prvig uch tructur faibl fr th pttial dig f multi-rat ub-wavlgth ytm. It i td that if th idividual cylidr ar digd uch that thir rac ar v clr t ach thr, th cuplig bcm mr viibl tha i th ca f th prtly ivtigatd cylidr. Thi i upprtd by th rult i Figur 3(a) which uggt that fr cl ugh rac frquci, th radiati ritac curv (th part with igificat valu f th radiati ritac) fr th idividual cylidr will cidrably vrlap ach thr thu idicatig a trgr cuplig. Figur 7. Th magitud f th lctric fild f th 4-cylidr tructur fr th parati ditac d = 4 mm fr diffrt frquci. Th ELC i lcatd at (, ) ( d / 2 mm, ), ad th lctric fild i dpictd i th xy-pla with th C1 liar dyamic rag (DR) i V/m idicatd blw th figur. Curv rprtig th circular urfac f th tructur ad a lft-right arrw idicatig th iz cal f th figur, ar likwi idicatd. (a) (d): frquci at which th rac ccur i Figur 6; () (h): iitial rac frquci f th idividual tructur.,mi DR: 2.1E1-1.1E5 DR: 2.5E2-5.9E4 DR: 2.8E-3.E4 DR:1.4E1-2.6E4 f MHz f MHz f MHz f 32.5 MHz (a) (b) (c) (d) DR: 1.9E2-4.3E3 DR: 4.2E1-2.3E4 DR: 2.4E1-1.5E4 DR:2.1E1-3.5E3 f 25 MHz f 26 MHz f 283 MHz f 3 MHz () (f) (g) (h)

12 Matrial 211, Figur 8. Th magitud f th lctric fild f th 4-cylidr tructur fr th parati ditac d = 3 mm ad f 24 MHz; thi i th frqucy at which th firt rac appar i Figur 6. Th ELC i lcatd at (, ) ( d / 2 mm, ), ad th lctric fild i dpictd i th xy-pla with th liar dyamic rag (DR) i V/m idicatd blw th figur. Curv rprtig th circular urfac f th tructur ad a lft-right arrw, idicatig th iz cal f th figur, ar likwi idicatd. C1 DR: 2.4E2 1.7E5 f 24 MHz Th aalyi thu far ha cctratd th ca whr th MNG matrial f th 1-, 2-. ad 4-cylidr tructur wa mdld by a ll Drud dipri mdl fr which. m, Prtly, th ifluc f l th rprtd rat prprti f th tructur i ad by icrpratig l i th mdl. Mr pcifically, th paramtr m, wa t t f, whr f i th dig frqucy f th rpctiv cylidr, ad th radiati ritac wa fr th 4-cylidr cfigurati with th parati ditac d = 4 mm. Th btaid rult ar rprtd i Figur 9 i trm f th quatity 1lg [db] a a fucti f frqucy. Thi figur al RR 1 R t iclud th crrpdig ll-ca rult fr cmpari purp. It i brvd that rac ccur at th am frquci a i th ll ca, but that th crrpdig amplitud, a xpctd, ar rducd. Figur 9. Th quatity 1lg [db] a a fucti f frqucy fr th ly RR 1 R t ad ll 4-cylidr tructur fr parati ditac d = 4 mm. I all ca, th ELC i lcatd at (, ) ( / 2, ). S th mai txt fr furthr xplaati. C1 d

13 Matrial 211, Summary ad Cclui Thi wrk prtd a dtaild tudy f rat prprti f a umbr f ub-wavlgth MTM-bad tructur f circular cylidrical hap. I particular, attti wa dvtd t ub-wavlgth rac kw t xit i ilatd MTM-bad tructur f circular cylidrical hap with th aim f dtrmiig whthr th idividual rac ar rtaid wh vral f uch rat tructur ar grupd t frm a w tructur. T thi d, tructur cmpd f 1, 2 ad 4 t f MTM-bad cctric cylidr xcitd by a ELC wr aalyzd umrically i ANSOFT HFSS with rgard t thir ar-fild prprti ad radiati ritac. Th MTM f th idividual tructur wr aumd t b impl, but ly ad dipriv, whr th ffct f th lattr wr accutd fr by th Drud dipri md. It wa dmtratd that th ub-wavlgth rac f th ilatd MTM-bad cctric cylidr tructur al ccur fr th tructur cmpd f 2 ad 4 t f MTM-bad cctric cylidr v i th ca whr th cylidr ar clly pacd ad th tir tructur i thu lctrically mall. Spcifically, vrall iz f abut 1/2 ad 1/12.5 f th mallt fr-pac wavlgth wr fud fr 2- ad 4-cylidr tructur, rpctivly, i which th rpctiv rac wr du t th dipl md xcitati i th ctitut cylidr. Th MTM-bad tructur thu ffr th pibility fr multi-rat ub-wavlgth cfigurati. Rfrc ad Nt 1. Elfthriad, G.V.; Balmai, K.G. Ngativ-Rfracti Mtamatrial Fudamtal Pricipl ad Applicati; Jh Wily & S: Nw Yrk, NY, USA, Calz, C.; Ith, T. Elctrmagtic Mtamatrial Tramii Li Thry ad Micrwav Applicati; Jh Wily & S: Nw Yrk, NY, USA, Zilwki, R.W.; Eghta, N. Mtamatrial Phyic ad Egirig Applicati; Jh Wily & S: Nw Yrk, NY, USA, Cai, W.; Shalav, V. Optical Mtamatrial; Sprigr: Brli, Grmay, Zilkwki, R.W.; Kippl, A. Applicati f dubl gativ mtamatrial t icra th pwr radiatd by lctrically mall ata. IEEE Tra. At. Prpagat. 23, 51, Alú, A.; Eghta, N. Plarizabiliti ad ffctiv paramtr fr cllcti f phrical aparticl frmd by pair f cctric dubl-gativ, igl-gativ, ad/r dubl-pitiv mtamatrial layr. J. Appl. Phy. 25, 97, Stuart, H.; Pidwrbtky, A. Elctrically mall ata lmt uig gativ prmittivity ratr. IEEE Tra. At. Prpagat. 26, 54, Arlaagić, S.; Zilkwki, R.W.; Bribjrg, O. Aalytical ad umrical ivtigati f th radiati frm cctric mtamatrial phr xcitd by a lctric Hrtzia dipl. Radi Sci. 27, 42, RS6S16:1-RS6S16:2. 9. Arlaagić, S.; Zilkwki, R.W.; Bribjrg, O. Aalytical ad umrical ivtigati f th radiati ad cattrig frm cctric mtamatrial cylidr xcitd by a lctric li urc. Radi Sci. 27, 42, RS6S15:1-RS6S15: Sh, H.-Y.; Li, L.-W.; Marti, O.J.F.; Mig, J.R. Surfac plarit mall catd cylidr illumiatd by rmal icidt TM ad TE pla wav. Opt. Expr 28, 16,

14 Matrial 211, Wallé, H.; Kttu, H.; Sihvla, A. Elctrtatic rac f gativ-prmittivity itrfac, phr ad wdg. I Prcdig f th 1t Itl. Cgr Advacd Elctrmagtic Matrial fr Micrwav ad Optic, Rm, Italy, Octbr, Arlaagić, S.; Clau, N.C.J.; Bribjrg, O. Prprti f ub-wavlgth rac i mtamatrial cylidr. I Prcdig f NATO Advacd Rarch Wrkhp: Mtamatrial fr Scur Ifrmati ad Cmmuicati Tchlgi, Marrakh, Mrcc May Arlaagić, S.; Bribjrg, O. A umrical ivtigati f ub-wavlgth rac i plygal mtamatrial cylidr. Opt. Expr 29, 17, Arlaagić, S.; Zilkwki, R.W. Activ catd a-particl xcitd by a arbitrarily lcatd lctric Hrtzia dipl rac ad traparcy ffct. J. Optic 21, 12, ANSOFT HFSS, Vri 1.1.3; Aft Crprati. Pittburgh, PA, USA; Cpyright (C), Alú, A.; Eghta, N. Clakig ad traparcy fr cllcti f particl with mtamatrial ad plamic cvr. Opt. Expr 27, 15, Shhtari, A.; Sbal, A.R. Elctrmagtic cattrig by paralll mtamatrial cylidr. PIER 26, 57, Arlaagić, S.; Bribjrg, O. Sub-wavlgth mtamatrial cylidr with multipl dipl rac. I Prcdig f IEEE AP-S USNC/URSI Nat. Radi Sci. Mtig, Charlt, SC, USA, 1 5 Ju Balai, C.A. Advacd Egirig Elctrmagtic; Jh Wily & S: Nw Yrk, NY, USA, by th authr; lic MDPI, Bal, Switzrlad. Thi articl i a p acc articl ditributd udr th trm ad cditi f th Crativ Cmm Attributi lic (