Accurate circuit modeling of fishnet structures for negative-index-medium applications

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 1, JANUARY 2016 15 Accu a e Ci cui Modeling o Fishne S uc u es o Nega i e-Index-Medium Applica ions Víc o To es, F ancisco Mesa, Fellow, IEEE, Miguel Na a o-Cía, Senio Membe , IEEE, Raúl Rod íguez-Be al, Miguel Be ue e, and F ancisco Medina, Fellow, IEEE Abs ac —Me allic pla es wi h a wo-dimensional (2D) pe iodic dis ibu ion o sub-wa eleng h ape u es a e known o exhibi ex- ao dina y ansmission o elec omagne ic wa es. S acking wo o mo e o such pla es gi es place o he so-called fishne s uc- u es, which cons i u e a popula way o achie ing an e ec i e neg- a i e index medium a equencies anging om mic owa es o op- ics. Un o una ely, a gene al wideband equi alen ci cui has no ye been p oposed o acili a e i s unde s anding and design. This wo k p esen s his ci cui model wi h closed- o m exp essions o he ci cui elemen s, hus making i possible o ob ain he elec ical esponse o his class o s uc u es in a e y e ficien way. This p ocedu e is much as e han al e na i e nume ical me hods a he same ime ha i e ains a high le el o accu acy when com- pa ed wi h some o he o e simplified models. The ci cui model also p o ides a simple a ionale as well as a good physical insigh in o de o explain he quali a i e beha io o such s uc u es, in- dependen ly o he numbe o s acked laye s. Index Te ms—Equi alen ci cui models, ex ao dina y ans- mission (ET), fishne s uc u es, me ama e ials, nega i e index ma- e ials (NIMs). I. INTRODUCTION WITHIN THE ealm o me ama e ials, majo e o s ha e been ocused on ealizing s uc u es wi h e - ec i e nega i e index o e ac ion, he so-called nega i e index me ama e ials (NIMs). Apa om he pu e scien ific in e es on hei peculia elec omagne ic esponse, he s udy o hese media has been igge ed by many po en ial p ac ical applica ions including supe lensing, sensing and o he no el Manusc ip ecei ed Ma ch 12, 2015; e ised Augus 13, 2015; accep ed No embe 17, 2015. Da e o publica ion Decembe 08, 2015; da e o cu en e sion Janua y 01, 2016. This wo k was suppo ed by he Spanish Minis y o Economy and Compe i i eness wi h Eu opean Union Fede Funds (unde g an s TEC2011-28664-C02-01, TEC2014-51902-C2-2-R, TEC2013-41913-P, and Consolide EMET CSD2008-00066, and by he Spanish Jun a de Andalucía unde P ojec P12-TIC-1435). The wo k o M. Na a o-Cía was suppo ed by he Impe ial College Junio Resea ch Fellowship and he Bi mingham Fellow- ship. The wo k o M. Be ue e was suppo ed by he Spanish Go e nmen ia RYC-2011-08221. V. To es and M. Be ue e a e wi h he An enna G oup—TERALAB, Uni- e sidad Pública de Na a a, Pamplona, 31006, Spain (e-mail: ic o . o es@ una a a.es; miguel.be ue e@una a a.es). M. Na a o-Cía was wi h he Impe ial College London, London SW7 2AZ, U.K. He is now wi h he School o Physics and As onomy, Uni e si y o Bi m- ingham, Bi mingham, B15 2TT, U.K. (e-mail: M.Na a o-Ci[email p o ec ed]). R. Rod íguez-Be al and F. Mesa a e wi h he Depa men o Applied Physics 1, ETSII, 41012, Se ille, Spain (e-mail: be[email p o ec ed]s; [email p o ec ed]s). F. Medina is wi h he Depa men o Elec onics and Elec omagne ism, Facul y o Physics, Uni e si y o Se illa, 41012, Se ille, Spain (e-mail: medin[email p o ec ed]). Colo e sions o one o mo e o he figu es in his pape a e a ailable online a h p://ieeexplo e.ieee.o g. Digi al Objec Iden ifie 10.1109/TMTT.2015.2504441 elec omagne ic de ices [1], [2]. The fi s expe imen al imple- men a ion o a NIM was done a mic owa e equencies by combining hin me allic wi es and spli - ing esona o s [3] bu he achie emen o NIMs a highe equencies (in a ed o op- ics) is no i ial because no all opologies e ain he equi ed s ong magne ic esponse in such spec al windows [4]. In his ega d, he fishne s uc u e has ou pe o med o he opologies [5] and can be applied o e a e y wide ange o equencies. The fishne is a mul i-laye ed s uc u e ha comp ises wo (o mo e) me allic pe iodic hole a ays sepa a ed by dielec ic slabs. Roughly speaking, he nega i e index o e ac ion in he fishne s ems om he i ual cu en loop o med by he coupling be ween he laye s and he su ace cu en s ounding he holes p o ided he pe o a ed me al sc eens a e ope a ing in he ex ao dina y ansmission (ET) [6], [7] egime. I s simple geome y acili a es he ab ica ion a he nanoscale, making i one o he mos a ac i e me ama e ials so a . Hence, he fishne is cu en ly he p e e ed op ion o syn hesizing NIMs om mic owa es [8]–[10] o he op ical egime [5], [11], [12], and has been success ully employed o designing quasi-op- ical de ices such as pola ize s, lenses, and demul iplexe s [13]–[17]. In he con ex o elec omagne ic pe iodic s uc u es, analyses based on equi alen ci cui s (ECs) ha e a long a- di ion [18]–[26]. The main ad an age o he ci cui heo y app oach is ha quali a i e as well as quan i a i e p edic ions a e easible wi hou pe o ming in ensi e ull-wa e nume ical simula ions. The e o e, an EC o he fishne me ama e ial is also highly desi ed. In he s udy o he ET phenomenon, ansmission line concep s and ECs ha e been able o p o ide success ul explana ions o ET h ough sub-wa eleng h hole a ays [27]. The elec ically small ape u es a e seen as eac i e discon inui ies modeled wi h lumped elemen s (o , some imes, dis ibu ed elemen s) in he pa h o an elec omagne ic mode p opaga ing along an a ificial wa eguide. This app oach educes he o iginal p oblem o a classical wa eguide discon i- nui y p oblem [28]. Fu he wo ks ha e ex ended his echnique o simila ET me ama e ials in o de o analyze he impo ance o he ape u e ype [29]–[31], pola iza ion [32], [33], me al losses [34], he inclusion o dielec ic slabs [24], [35] o he design o pola izing de ices [36]. Likewise, he fishne has also been analyzed om an EC poin o iew in o de o ob ain a mo e in ui i e explana ion han al- e na i e in e p e a ions based on di ac ion o de s o complex wa es, Floque -Bloch modes o su ace plasmons [37]–[39], while e aining he unde lying physics o he p oblem and he accu acy o he esul s. The esponse o he basic fishne s uc- u e can be unde s ood in e ms o an elemen a y backwa d ansmission line ha ing se ies capaci ance and shun induc- ance. The se ies capaci ance comes om he elec ic coupling be ween consecu i e laye s and he shun induc ance ep esen s 0018-9480 © 2015 IEEE. Pe sonal use is pe mi ed, bu epublica ion/ edis ibu ion equi es IEEE pe mission. See h p://www.ieee.o g/publica ions_s anda ds/publica ions/ igh s/index.h ml o mo e in o ma ion. Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. 16 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 1, JANUARY 2016 he induc i e na u e o he sub-wa eleng h ape u es [40]. Se - e al o he EC models can be ound in he li e a u e dealing wi h p oblems in ol ing s acked pe o a ed me al laye s [41]–[46]. Howe e , mos o hese wo ks a e o heu is ic na u e and do no p o ide closed- o m exp essions o he ci cui elemen s [43], [44] o hei scope is limi ed o ce ain configu a ions and s uc- u al pa ame e s (small uni cells, la ge longi udinal pe iod and so on). Thus, a gene al o mula ion alid o s uc u es ha ing uni cells o a bi a y elec ical size sepa a ed by a bi a y dis- ances would be o g ea in e es om a p ac ical poin o iew. Compu a ionally in ensi e b u e o ce calcula ions would hen be a oided a he same ime ha a be e physical insigh could be gained. In his wo k, a simple analy ical ci cui model ha uses lumped elemen s and ansmission lines is de i ed o s udy he ansmission o elec omagne ic wa es h ough a fishne s uc- u e wi h a fini e numbe o laye s (see Fig. 1). The s uc u e is assumed pe iodic and infini e along he and ans e se di- ec ions. The inciden plane wa e is impinging obliquely o he s uc u e and i s elec ic field is assumed o be di ec ed along he di ec ion ( he wa e will be conside ed o TM na u e a no mal incidence). The a en ion is mainly ocused on he sep- a a ion be ween laye s because i unes he e ec i e e ac i e index o he undamen al band [40]. The e o e, wo di e en ope a ion egimes a e s udied: la ge and sho sepa a ion, which espec i ely p oduce weak and s ong coupling be ween laye s h ough highe -o de modes. No e ha in e ac ion h ough he undamen al mode is always p esen o any sepa a ion as his mode has no cu o equency. Fo each egime, we in es iga e he configu a ions wi h ai and a dielec ic ma e ial wi h pe mi i i y as in e -spacing laye s. Di e en closed- o m exp essions o he ci cui elemen s a e p esen ed o all si ua ions. The physical insigh p o ided by he EC is co obo a ed wi h he s udy o he elec omagne ic fields a he mos ele an equencies. Al hough ou s udy is ocused on mic owa es o simpli y he model o me als o pe ec elec ic conduc o , he me hod could be ex ended o conside o he models such as D ude o fini e conduc i i y models [47], [48]. This pape will be o ganized as ollows. Sec ion II will show ou p oposed model o fishne s uc u es wi h a fini e numbe o laye s. Sec ion III will p esen compa ison esul s o he ansmission coe ficien ob ained wi h he ci cui model and wi h ull-wa e simula ions unde no mal and oblique inci- dence o di e en longi udinal pe iods. Also, he s udy o he elec omagne ic fields a he mos ele an equencies. Finally, some concluding ema ks will be summa ized in Sec ion IV. II. CIRCUIT MODELING The use o ECs o model many p opaga ion/sca e ing p ob- lems is a common and e y ui ul p ac ice. Howe e , mos o he analyses jus p opose (in a heu is ic way) he opology o an equi alen ne wo k ha has a beha io e y simila o he o ig- inal p oblem o e a ce ain equency band. The alues o he pa ame e s o he equi alen ci cui a e hen ob ained om some ull-wa e simula ions o he comple e elec omagne ic p oblem ( hese simula ions o en need o be ca ied ou a pa icula e- quency poin s ha a e only known apos e io i). Al hough his s a egy is use ul in many si ua ions, ou aim he e is o de i e he app op ia e ci cui opology in a igo ous way om fi s p inci- ples and o gi e he alues o he ci cui pa ame e s in closed o m. This s andpoin has been used in he pas by some o he au ho s o he p esen wo k o model o he pe iodic s uc u es Fig. 1. Schema ic ep esen a ion o a fishne s uc u e wi h a fini e numbe o laye s. The uppe panel shows a longi udinal c oss sec ion and he bo om panel a on iew o he s uc u e. [35], [49]–[51], and i is now ex ended o s acked 2D geome- ies so ha he p oblem unde s udy (mul ilaye fishne s) can be co e ed. In [51] i was shown ha he key aspec ha makes i pos- sible o easily ob ain an equi alen ne wo k o he p oblem o s acked me allic sli g a ings (1D e sion o he p oblem o in e es in ha pape ) comes om he decomposi ion o he wo-coupled me allic sc eens p oblem in o “in e nal” and “ex- e nal” sub-p oblems as a consequence o he Equi alence The- o em [52]. The ex ension o his idea o a pai o coupled ape - u e- ype equency selec i e su aces (FSS's) ( he mos basic o m o fishne s uc u e) is immedia e as long as he FSS's a e o slo - ype na u e oo ( he p ocedu e is no di ec ly appli- cable o pa ch-based FSS's). Thus, ollowing [51], he p oblem o he wo-coupled ape u e- ype FSS's shown in Fig. 2(a) has he equi alen ne wo k shown in Fig. 2(b), o he in e nal/ex- e nal e sion gi en in Fig. 2(c). I is now s aigh o wa d o ealize ha he equi alen ne wo k o , say, ou cascaded iden- ical FSS's such as ha shown in Fig. 3(a) is he one gi en in Fig. 3(b). Ou a en ion now ocuses on he calcula ion o he ew ad- mi ances in ol ed in he analysis o he coupled ape u e-based FSS. Fo ha pu pose, he s anda d applica ion o he e en/odd exci a ion app oach [52] o he p oblem in Fig. 2(a) makes i possible o educe ou o iginal p oblem o he analysis o p ob- lems wi h jus one single FSS and elec ic/magne ic walls a hal he dis ance be ween he o iginal FSS's, as shown in Fig. 4. The analysis o his much simple s uc u e will ollow some o he guidelines epo ed in [35], [50]. When his analysis is ca ied ou , he final esul s o he equi alen admi ance co esponding o he inciden TM plane wa e (elec ic field di ec ed along he di ec ion) a e he ollowing (see Fig. 4): (1) whe e he p ime in he summa ion means ha i excludes he undamen al ha monic associa ed wi h he impinging, Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. TORRES e al.: ACCURATE CIRCUIT MODELING OF FISHNET STRUCTURES FOR NEGATIVE-INDEX-MEDIUM APPLICATIONS 17 Fig. 2. (a) Two coupled ape u ed-based FSS's. (b) Equi alen ci cui o he sca e ing o he impinging plane wa e in he wo-coupled FSS s uc u e. The coupled sc eens a e ep esen ed by a -ne wo k. (c) Equi alen ci cui al eady shown in (b) bu now showing he decomposi ion o he pa allel elemen in o an ex e nal admi ance and an in e nal admi ance . Fig. 3. (a) S ack o ou coupled ape u ed-based FSS's. (b) Equi alen ci cui o he sca e ing o he impinging plane wa e in he s acked s uc u e. Fig. 4. Pai o coupled FSS's and i s associa ed e en/odd exci a ion (magne ic/ elec ic wall) hal -p oblems wi h hei equi alen ci cui s. eflec ed, and ansmi ed uni o m plane wa es ( he ha - monic is also excluded since i canno exis in he p esen config- u a ion). The supe sc ip s “e/o” s and o “e en/odd” exci a ion and “TM/TE” e e s o he na u e o he conside ed ha monic. The admi ances appea ing in (1) a e gi en by he ollowing analy ical exp essions: e en exci . odd exci . (2) (3) (4) (5) wi h being he acuum wa enumbe , he angula e- quency, he wa e ec o o he impinging plane wa e, and index “(0)” e e s o he ex e nal ee-space egion while index “(1)” e e s o he in e nal egion be ween me al sc eens. The angles and define he di ec ion o he impinging wa e as usual. Howe e , in his pape , i will only be conside ed incidence along p incipal planes; i.e., will be allowed o ake only he alues 0 and 90 . The gene al case o conical incidence could be ea ed using he me hod epo ed in [50]. I is wo h men ioning ha he gi en exp ession o accoun s o all he high-o de ha monics exci ed a he discon inui y. This means ha he possible e ec s associa ed wi h high-o de modes caused by he close p oximi y be ween sc eens a e p ope ly inco po a ed in o he model. Assuming ha he angen ial elec ic field a he ape u e is di ec ed along ( o TM incidence o o TE incidence), he coe ficien s in (1) a e gi en by (6) (7) wi h being he Fou ie ans o m o he spa ial p ofile o he angen ial elec ic field in he ape u e. This spa- ial p ofile is no known ap io ibu can be e y well-app ox- ima ed o many p ac ical cases. In pa icula , o ec angula ape u es and o he ype o illumina ion conside ed in his pape , we ha e employed he spa ial p ofile sugges ed in [53] (see he Appendix). He e, i is impo an o poin ou ha he abo e app oxima ion u ns ou o be qui e accu a e up o e- quencies close o he second sel - esonance o he indi idual ape u e ha is compa ible wi h he impinging wa e [50]. In p ac ice, his means ha he ape u e has no o be sub-wa e- leng h o he ci cui app oach o be applied wi h eliabili y. Ac ually, o no mal incidence, he size o he ape u es can be e en g ea e han a wa eleng h [50]. I is also no ewo hy ha and also ha .Thisisconsis en wi h he ac ha and ha monics canno be exci ed due o he symme ies o he uni cell and he pola iza ion o he inciden plane wa e. Ou nex s ep is o find he ela ionship be ween he defined and he admi ances o he equi alen -ne wo k o a Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. 18 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 1, JANUARY 2016 pai o coupled pe o a ed sc eens. Since he e en and odd exci- a ions co espond espec i ely o open and sho -ci cui e mi- na ions a he middle plane o he equi alen ci cui , he pa allel and se ies admi ances o he -ne wo k in Fig. 2 a e ound o be (8) F om hese exp essions, i is s aigh o wa d o iden i y he ad- mi ances appea ing inFig.3(b) oob ain (9) whe e (10) (11) and (12) The o mulas p o ided o all he pa ame e s appea ing in he abo e exp essions a e equency-dependen , which implies ha he abo e infini e se ies summa ions ha e o be ca ied ou , in p inciple, o e e y equency alue. The ac o ha ing ad- mi ances wi h an in ol ed equency-dependen beha io does no ce ainly con ibu e o highligh he possible ad an ages o he equi alen ci cui app oach. Fo una ely, o high-o de ha - monics ), i is possible o w i e (13) This means ha o all he e anescen ha monics ope a ing well below cu o (“localized” modes in he e minology used in [23], whe e a ela ed p oblem is deal wi h), he wa e admi ances can be app oxima ed as (14) (15) whe e he label “ho” indica es “high o de ”. The coe fi- cien s can be in e p e ed as lumped capaci ances associa ed wi h su ficien ly high-o de TM ha monics in medium . Simila ly, a e lumped induc ances associa ed wi h high-o de TE ha - monics (no e ha hey do no depend on pe mi i i ies). In he ligh o he abo e app oxima ion, he admi ances o he -ne wo k can be decomposed in one pa wi h ce ain com- plica ed, al hough known, equency dependence and ano he pa accoun ing o s anda d capaci ance and induc ance ad- mi ances. Thus, assuming ha he e a e and “low o de ” (lo) ha monics ha do no sa is y condi ion (13) (“acces- sible modes in [23]), i is possible o w i e he pa allel admi - ances in (10) and (11) in he ollowing gene al o m: (16) whe e he coe ficien s ha e been pu posely inco po a ed o he “lo” admi ances. Thus, he pa allel admi ances a e com- posed o a egula pa allel ank which is connec ed in pa - allel wi h a ew admi ances ha ing a mo e complex equency dependence. Pa o he admi ance is associa ed wi h he ex e nal p oblem and pa wi h he in e nal one . The se ies admi ance in (12) can equi alen ly be w i en as (17) Ne e heless, he p esence o he a enua ing ex- ponen ial ac o in (12) would allow us o supp ess he e ec o he se ies lumped elemen s ( and ) accoun ing o high-o de mode con ibu ions i he elec ic sepa a ion be ween me al sc eens is la ge. In such case, i is appa en ha he e ec o he high o de ha monics can be neglec ed due o i s esidual e ec . The numbe o modes o be e ained in he desc ip ion o will ob iously depend on he alue o he dielec ic slab hickness and should include, a leas , he fi s p opaga ing ha - monic. The explici exp essions o all he capaci ances and in- duc ances in ol ed in ou p oposed equi alen ci cui a e gi en in he Appendix. A key poin ha should be no ed he e is ha , o una ely, in many p ac ical cases he numbe o equency-dependen ele- men s (namely, he numbe o “accessible ha monics) equi ed o ob ain good quali a i e esul s is jus one o wo. O en, wi h his low numbe o accesible ha monics, sa is ac o y quan i a- i e esul s can also be ob ained. This makes i possible o ha e a “minimal” equi alen ci cui whose opology is igo ously de- i ed and i s componen s a e gi en in closed o m. Thus, le us assume ha he e is only one ele an “low-o de ” ha monic in he equency egion o in e es ; i.e., a single ha monic has o be e ained wi h i s whole dis ibu ed equency-dependence while all he o he s a e ep esen ed by equency-independen and . This ha monic is necessa ily he inciden TM-po- la ized plane wa e—which has been aken as he ha - monic. In he in e io egion (1), his ha monic would be he only p opaga ing ha monic wi h all he highe -o de ha monics being below cu o . This si ua ion would model app op ia ely hose configu a ions whe e he elec ical dis ance be ween suc- cessi e sc eens is la ge enough, say ,whe e is he wa eleng h inside medium (1). In such case, he se ies admi - ance in (17) can be app oxima ed by jus one componen gi en by (18) whe e i has been assumed ha he con ibu ion o he e- maining e anescen ha monics modeled by and is negligible. The in e nal pa o he pa allel admi ance in (11) can be ew i en as (19) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. TORRES e al.: ACCURATE CIRCUIT MODELING OF FISHNET STRUCTURES FOR NEGATIVE-INDEX-MEDIUM APPLICATIONS 19 Fig. 5. Equi alen ne wo k o a pai o coupled FSS's when (a) only one low-o de ha monic ( he inciden plane-wa e) is p esen in he s uc u e,and (b) he e is an addi ional p opaga ing low-o de ha monic. whe e a new pa allel admi ance, , has been in oduced and defined in e ms o he al eady known admi ance .I is easy o ealize ha he equi alen -ci cui o he pai o cou- pled FSS's in Fig. 2 can hen be eph ased as he ci cui shown in Fig. 5(a). The opology o his equi alen ne wo k can eadily be ecognized as he one usually p oposed in he li e a u e o his kind o p oblems when he coupling be ween successi e sc eens is only accoun ed o by he undamen al ha monic (see, o in- s ance, [41], [46] among many o he s). Al hough his si ua ion is well unde s ood, he model in Fig. 5(a) yields quali a i ely and quan i a i ely w ong p edic ions in ypical fishne s uc u es whe e he s uc u ed me al sc eens a e elec ically close o each o he . I mo e han one p opaga ion mode ope a es in he in e- io egion, his simplified equi alen ne wo k is no longe alid. Ne e heless, he only significan di e ence in he opology e- qui ed o accoun o in e ac ions h ough highe o de modes comes om he appea ance o an addi ional se ies admi ance, ,asshowninFig.5(b).The ela ion be ween his new se- ies admi ance and in (12) is gi en by (20) No e ha he con ibu ion o he se ies admi ance o he p op- aga ing ha monic is explici ly ex ac ed ou in (20). Thus, only con ains in o ma ion o he high-o de modes esponsible o in e ac ions be ween he wo me al sc eens. These a e e y ew modes which a e ope a ing abo e cu o ( his is possible due o he p esence o dielec ic slabs) o below bu close o hei cu o equencies. I all he in ol ed ha monics a e a and below cu o , using again he app oxima ions in (13) and (14), can be in e p e ed as a shun ci cui . Fo elec ically hick dielec ics, he alues o he co esponding pa ame e ends o be ex emely small while is e y la ge, hus gi ing place o a e y low admi ance, as expec ed. This means ha can be neglec ed i he elec ical hickness o he dielec ic is la ge enough, hus explaining why he model employed in some pape s dealing wi h s acked s uc u es [ he model in Fig. 5(a)] wo ks p ope ly. Howe e , i he elec ic leng h be ween sc eens is no la ge enough, he mo e accu a e model in Fig. 5(b) has o be used. In nex sec ion, i will become appa en ha neglec ing he oleo can gi e place o comple ely w ong esul s. III. RESULTS In his sec ion, he beha io o he s acked fishne s uc u e o in e es in his wo k is s udied unde wo di e en ope - a ion egimes: 1) he “sho pe iod” egime co esponding o a longi udinal pe iod less han hal he ans e se pe iod ,; and 2) he “long pe iod” egime co esponding o . Special a en ion is also paid o he impo ance o inco po a ing he se ies admi ance o he model—see Fig. 5(b). In he ollowing, he elec omagne ic field dis ibu ion is analyzed a he mos ele an equencies ob- se ed in he spec a o co ela e hem wi h he equi alen ci - cui elemen s p esen edin he p e ious sec ion. In his analysis, all he equencies a e no malized o he pe iod; i.e., is ex- p essed in uni s o whe e e e s o he wa eleng h o he fi s Rayleigh-Wood anomaly. A. Sho Longi udinal Pe iod The s uc u e analyzed in his and subsequen sec ions is a fini e s acked fishne lossless s uc u e composed o fi e me allic sc eens sepa a ed by ou ai /dielec ic laye s. Fi s , he s uc u e is nume ically analyzed by means o a ull-wa e Floque -mode simula ion o a uni cell using he so wa e CST Mic owa e S udio. The p esen ed nume ical esul s come a e he adap i e mesh efinemen be ween wo consecu i e calcula ions o he sca e ing pa ame e s shows di e ences below 0.001. In Fig. 6(a), he ansmission coe ficien unde no mal inci- dence o he sho pe iod egime wi h ai be ween he pe o a ed me al sc eens is shown. The dip a is asso- cia ed wi h he fi s Rayleigh-Wood anomaly, which is he uppe limi o he spec al window o in e es ( he di ac ion egime is beyond he scope o he p esen wo k). A ansmission band is clea ly obse ed o wi h fi e ecognizable peaks ( ,,,,and ). The ele an componen s and (is negligible) o he elec ic field a he peak e- quencies a e p esen ed in Fig. 7(a). A ,is mainly loca ed on he ou e me al-ai in e aces while is confined a he ape - u e, esembling he so-called ET esonance in a single laye hole a ay [38], [40]. Rega ding he fields a he o he peaks , i can be seen ha is confined nea he ape u e, as well. Howe e , g adually changes: om he highes o he lowes equency, becomes mo e and mo e con- fined be ween he pe o a ed laye s, whe e he fieldampli ude is highe . These peaks co espond wi h in e nal mode esonances [38], [54]. Fu he mo e, in he longi udinal di ec ion, di e en dis ibu ions o maxima and minima a e no iced. In e es ingly, he numbe o a ia ions (maxima and minima) inside he s uc- u e inc eases as he equency dec eases: co - esponds o a , , and mode inside he s uc u e. Hence, he band has a nega i e-index cha ac e , as i is expec ed o sho longi udinal pe iods [40], [54]. The numbe o hese in- e nal esonances is always equal o ,wi h being he numbe o pe o a ed sc eens. Finally, on he igh mos column o Fig. 7, he -componen o he magne ic field, , be ween he 3 d and 4 h laye s is ep esen ed. The dis ibu ion o o all peaks esembles he mode o he uni -cell o he i ual pa allel-pla e wa eguide induced by he ans e se pe- iodici y. This ac oge he wi h he dis ibu ion indica es ha he ele an highe o de mode in ol ed in his ansmis- sion band is he [27], [40]. Simila field dis ibu ion is ob ained be ween each pai o adjacen laye s. This high-o de dominan mode con ols he alues o he se ies admi ance in he ci cui model in Fig. 5(b). The equi alen ci cui (EC) esul s o his “sho -pe iod” case a e shown in Fig. 8(a). Ce ainly, he CPU ime in ol ed in EC calcula ions is comple ely negligible when compa ed wi h Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. 20 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 1, JANUARY 2016 Fig. 6. T ansmission coe ficien o a fini e fishne s uc u e wi h me allic sc eens o se e al configu a ions. , , ;(a) wi h ai be ween me al sc eens; (b) wi h dielec ic ,(c) wi h ai and d) wi h dielec ic be ween laye s. he compu a ional e o equi ed by he nume ical app oach. In his figu e he ull-wa e simula ion esul s [black cu e] a e plo ed along wi h h ee cu es co esponding o esul s o he EC app oach. These h ee cu es a e labeled as ,whe e and s and espec i ely o he numbe o low-o de TE and TM ha monics (in addi ion o he undamen al impinging one) whose exac equency beha io (dis ibu ed na u e) is aken explici ly in o accoun in he EC model. The es o he high-o de modes a e app oxima ed wi h he capaci ance and induc ance limi s discussed in he p e ious sec ion. The cu e EC(0,0) [g een line] co esponds o he opology shown in Fig. 5(b) wi h he coupling be ween he me allic sc eens basi- cally accoun ed o by he ha monic ( he ha monic is no exci ed in he p esen s uc u e) and wi h gi en by , a e applying he simplifica ions in (13), (14) and (15). I should be no ed ha he esul s o his Fig. 7. Elec ic field ( and )on he -plane and magne ic field on he -plane, a he ele an equencies labeled in Fig. 6(a) and Fig. 6(b) o he fishne wi h and a) wi h ai and b) wi h dielec ic be ween laye s. model and he one in which is aken ze o a e ound quali a- i ely iden ical al hough wi h some quan i a i e disc epancies. Thus, in mos p ac ical cases, he esul s o he cu e EC(0,0) can be associa ed wi h he opology o Fig. 5(a). The EC(0,1) cu e [cyan line] al eady ully co esponds o he ci cui shown in Fig. 5(b), wi h only he fi s high-o de TM mode ( in his case) being co ec ly included in he defini ion o . I is obse ed ha he inclusion in he ci cui model o he co ec equency dependence o jus he ha monic p o- ides a good quali a i e ma ching wi h he ull-wa e esul s. This is in conco dance wi h he field dis ibu ions obse ed in he p e ious EM field inspec ion since high-o de TM modes a e esponsible o such dominan . Hence, in his case o sc eens elec ically close, i is he influence o he fi s e anes- cen ha monic in he coupling be ween me al sc eens wha gi es place o he obse ed passband; i.e., he con en ional connec- ion h ough he undamen al ha monic as well as a w ong e al- ua ion o he in e ac ion due o high-o de modes canno a all accoun o he appea ance o such band. The addi ion o he fi s high-o de TE ha monic is necessa y o a be e quan i- a i e ag eemen , as shown by he ed cu e EC(1,1). Indeed, his ha monic has he same cu o equency as he bu i s influence on he o e all field is smalle (since he eac ion o his eigenfield wi h he assumed ape u e field is smalle oo). In he EM field inspec ion, he TE dis ibu ion is mainly masked. Howe e , by looking a he magne ic field on he c oss-sec- ional -plane in Fig. 9, we can obse e some which ac- coun s o he induc i e coupling be ween holes [45], pa icu- la ly a . Clea ly he ag eemen be ween he ci cui model and he nume ical esul s can be sys ema ically imp o ed by adding highe -o de modes o he defini ion o . Fishne me ama e ials usuallyemploydielec icslabsbe- ween he me allic laye s o he sake o obus ness and ease o ab ica ion. In Fig. 6(b) he nume ically compu ed ansmis- sion coe ficien o he sho pe iod case wi h a dielec ic is p esen ed. The ansmission band now ex ends om o , which indica es ha adding he dielec ic widens he ope a ion bandwid h. In addi ion, six peaks ( , , , , and ) a e p esen . Looking a he field dis ibu ion o and ,showninFig.7(b),aclea Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. TORRES e al.: ACCURATE CIRCUIT MODELING OF FISHNET STRUCTURES FOR NEGATIVE-INDEX-MEDIUM APPLICATIONS 21 Fig. 8. EC esul s o he “sho -pe iod” s uc u es p e iously analyzed in Figs. 6(a) and (b). (a) Ai be ween me al sc eens; (b) dielec ic ma e ial be ween me al laye s. (c) Oblique incidence , wi h he elec ic field di ec ed along he di ec ion, o he s uc u e in (a). This figu e (c) also shows EC esul s when a ela i ely high numbe o ha monics a e explici ly conside ed. Fig. 9. Magne ic field on he -plane in he middle plane o he uni cell a he ele an equencies ,and . conco dance wi h he fields a (case wi hou dielec ic) is obse ed. Ac ually and p esen he same dis ibu ion in bo h and and he e o e is iden ified as he e en ex e nal ET esonance. Fo , he is mainly loca ed on he ou e me al-ai in e aces and is highly confined a he ape u e; al hough a longi udinal esonance is obse ed in his case. The symme ic dis ibu ion o wi h espec o he ans e sal cen al -plane a he ex e nal aces indica es ha can be associa ed wi h he so-called odd ex e nal ET esonance [38]. The magne ic field dis ibu ion also shows he exci a ion o he highe o de mode. The field dis ibu- ions o a e omi ed in Fig. 7(b) since hey a e simila o he cases al eady s udied in . This case can also be con enien ly handled wi h he EC ap- p oach, as i is shown in Fig. 8(b). I he se ies in e ac ion associ- a ed wi h he p esence o a non- anishing is elimina ed o in- co ec ly accoun ed o [case (0,0), g een line], he ci cui model has no any physical meaning again. I is hen essen ial o co - ec ly in oduce he in e ac ion be ween successi e pe o a ed sc eens h ough, a leas , he fi s high-o de mode ( in his s uc u e) o accoun o he exis ence o ansmission peaks in he equency egion o in e es [cyan cu e in Fig. 8(b)]. The p ope inclusion o he con ibu ion o he in ex- plains he appea ance o he ansmission band and yields ea- sonably accu a e esul s o he low- equency pa o he ans- mission band. Howe e , i is clea ha quan i a i e ag eemen is poo , especially as he equency inc eases, and e en one o he ansmission peaks is los . This d awback is alle ia ed by adding o he con ibu ion o he fi s TE mode oo [ ed line in Fig. 8(b)]. Simila o he non-dielec ic case, TE modes a e less exci ed bu s ill no iceable in he field inspec ion, pa icu- la ly a and which, in he EC esul s, appea when he dispe sion is aken accoun . Mo e modes should be added o a be e quan i a i e ag eemen bu he model would become mo e and mo e cumbe some. Ac ually, he cu e co e- sponding o EC(1,2) has been ound o be almos indis inguish- able in he g aphic o ha co esponding o CST Mic owa e S udio. Wi h a compu a ion in ol ing EC(3,3), he nume ical ag eemen wi h CST Mic owa e S udio is e y good. I is wo h men ioning ha he ci cui -model app oach p o ides a simple explana ion o he di e ence be ween he ansmission pa e ns obse ed o he s uc u e wi h and wi hou dielec ics. When he space be ween me al sc eens is occupied by ai , he fi s wo high-o de modes in ol ed in he in e ac ion a e below cu o in he whole equency ange o in e es . Howe e , when he di- elec ics a e p esen , he cu o o he wo lowes o de TM and TEmodesisapp oxima elya ,insuchaway ha he equency dependence o d as ically changes due o he p op- aga ion o he fi s wo high-o de modes, hus allowing o ad- di ional ansmission peaks. Finally, Fig. 8(c) shows he esul s ob ained o oblique TE/TM incidence and o an azhimu al angle .In his figu e, he equi alen -ci cui cu e co - esponding o EC(1,1) means ha all he TM/TE ha - monics in ol ing , 0, 1 a e explici ly conside ed in he dynamic se ies [ o no mal incidence, he e appea s degen- e acy in he ha monics o he ype ]. These EC(1,1) da a show a good quali a i e beha io bu he figu e also makes i appa en ha o a good quan i a i e beha io i is neces- sa y o explici ely conside mo e ha monics. Ac ually, a sa is- ac o y quan i a i e beha io is again achie ed wi h EC(1,2) as in Fig. 8(b) [ ela i e e o s a ound o less han 1%], and nume - ical excellen ag eemen is ob ained o he TE pola iza ion o EC(3,3). Anyway, his figu e clea ly demons a e he abili y o he EC app oach o deal also wi h oblique incidence eliably, wi hou a significan inc ease o he equi ed CPU ime ( he wo housand equency poin s shown in Fig. 8(c) o EC(3,3) we e compu edinabou 1 o2secondsonani7lap op). B. Long Longi udinal Pe iod In he long pe iod case, (a longi udinal pe iodici y o is assumed), conside ing ai be ween he me allic laye s, Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. 22 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 64, NO. 1, JANUARY 2016 a significan change in he ansmission coe ficien is app eci- a ed in Fig. 6(c). Now, wo di e en ansmission bands a e obse ed. The highe band is loca ed be ween and and fi e peaks a e no iced ( ,,,,and ). By examining he fields a hese peaks in Fig. 10(a), he na- u e o his band can be immedia ely ex ac ed. A ,and co esponds wi h he fields al eady obse ed a (i.e., he e en ex e nal ET esonance). Likewise, he fields a also e- semble he fields obse ed a and, he e o e, i is ela ed o in e nal mode esonances as well. Due o he simila i y be ween he field o he sho and long pe iod cases, he fields a o a e omi ed. The lowe band ex ends om o and i con ains ou peaks ( ,,,and ). An- alyzing he fields a he peaks, a comple ely di e en dis ibu- ion is no iced [Fig. 10(a)]. The componen is no longe he dominan con ibu ion since he mu ual coupling be ween adja- cen laye s is pa ially los due o he la ge longi udinal pe iod- ici y. Now, he highes in ensi y o he elec ic field is obse ed in . This kind o field dis ibu ion is well known in s acked s uc u es and co esponds wi h Fab y-Pé o esonances [40], [46], [54]. Mo eo e , peaks a highe equencies show mo e maxima and minima (con a ily o wha happens in he highe band) and, he e o e, his is a band wi h posi i e index o e- ac ion [40], [54]. By examining he magne ic field be ween he laye s [ o ], does no show a ele an a ia ion along he -axis, di e en ly om he p e ious bands. The con inuous dis ibu ion o and along wi h he dom- inan cha ac e o he la e o e esemble he TEM mode, and exci a ion o highe o de modes a e no obse ed. As dis- cussed in [54], he Fab y-Pé o band o sho pe iods akes place abo e he Rayleigh-Wood's anomaly. Hence, i was no obse ed in he p e ious analysis. I is in e es ing now o find ou wha he p oposed EC model p edic s o his si ua ion. The esul s plo ed in Fig. 11(a) show ha he basic ci cui [EC(0,0): g een cu e], whe e in e ac ion be ween me al laye s is ca ied ou h ough he undamen al TEM mode, quali a i ely accoun s o he fi s ansmission band. This model also p edic s he appea ance o a second ansmission band a abou wice he equency ange o he fi s ansmission band, bu his happens beyond he onse o he di ac ion egime. Fo s uc u es wi h an elec ically-small ans e se pe iod, such as hose s udied in [41], he simplified EC(0,0) model has p o en o gi e e y good esul s. Howe e , in he p esen si ua ion, he ans e se pe iod is no so small and he influence o he in e ac ion h ough high-o de modes gi es place o addi ional bands o anspa ency below he Rayleigh-Wood equency poin . Thus, he exis ence o he second ansmission band obse ed in Fig. 11(a) is ound o be in ima ely linked wi h he exis ence o a pa h o in e ac- ion due o high-o de modes, which a e pu ely e anescen in his equency ange o ai filling. As he sepa a ion be ween sc eens is now ela i ely la ge, he accu acy o he model is excellen when only he ull con ibu ion o he fi s high-o de TM and TE modes a e included in . This ag eemen is now be e han in he sho pe iod case because, o he p esen sepa a ion be ween laye s, he in e ac ion h ough modes wi h e en highe o de is comple ely negligible. Finally, he fishne configu a ion co esponding o he long pe iod and dielec ic slabs wi h pe mi i i y be ween he me allic laye s is analyzed. He e i is obse ed he appea - ance o h ee ansmission bands in Fig. 6(d) ins ead o wo. The highes equency band ex ends om o (peaks , , and ), he cen al one om Fig. 10. Elec ic field ( and )on he -plane and magne ic field on he -plane, a he ele an equencies labeled in Fig. 6(c) and Fig. 6(d) o he fishne wi h and a) wi h ai and b) wi h dielec ic be ween laye s. Bo om figu es in b) also show he pe spec i e iew o he elec ic field on he xy-plane. o (peaks , , and )and he lowe one om o (peaks , , and ). In he lowe panels o Fig. 10(b), he elec ic field in a ans e sal -plane be ween he 3 d and 4 h laye is depic ed o one peak o e e y band: ,,and .The dominan componen s o he fields wi hin all peaks in a band a e e y simila , so hese peaks a e aken as ep esen a i e o he fields in each band. The fields a he lowe and he cen al bands co espond wi h he fields ob ained in he lowe and highe bands, espec i ely, in he ee-s anding long-pe iod fishne . A he elec ic field co esponds o a Fab y-Pé o esonance since i poin s p ima ily o he -axis. On he o he hand, a he elec ic field is mainly longi udinal poin ing o opposi e di ec ions in he uppe and lowe pa o he uni cell. Hence, he peaks a e iden ified as he in e nal mode esonances associa ed wi h he ha monic. Simila ly, he ac ha also poin s like ,is highly confined a he ape u es, and he p esence o p ima ily ex e nal o he whole s uc u e sugges ha his band also co esponds wi h ex e nal esonances asc ibed o he high-o de mode. Be- sides, his band exhibi s posi i e index o e ac ion. The ap- pea ance and bandwid h o his new band is highly dependen o he longi udinal pe iodici y and he pe mi i i y o he dielec- ic and hence i is no always p esen . Despi e his complex beha io , ou ci cui modeling shown in Fig. 11(b) is able o p edic he beha io o his band wi hou including addi ional ci cui elemen s wi h espec o he non-di- elec ic fishne . The ela i ely i ial model deno ed as EC(0,0) p edic s he exis ence o he fi s ansmission band bu com- ple ely loses he in o ma ion o he in e ac ions ha gi es place o he wo addi ional ansmission bands appea ing in he non- Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply. TORRES e al.: ACCURATE CIRCUIT MODELING OF FISHNET STRUCTURES FOR NEGATIVE-INDEX-MEDIUM APPLICATIONS 23 Fig. 11. (a) and (b): Equi alen ci cui (EC) esul s o he “long-pe iod”s uc- u es p e iously analyzed in Fig. 6(c) and (d), espec i ely. (c) Oblique inci- dence , wi h he impinging elec ic field di ec ed along he di- ec ion, o he s uc u e in (a). This figu e (c) also shows EC esul s when a ela i ely high numbe o ha monics a e explici ly conside ed. di ac ion egion. The inclusion in o he in e ac ion h ough he dominan high-o de TM mode is al eady enough o accoun o he exis ence o hose wo high- equency bands, one below he cu o o all he high-o de modes ( hus implying in e ac- ion by means o e anescen fields) and he o he abo e cu o . Howe e , o he co ec p edic ion o he numbe o peaks in he hi d ansmission band, he influence o he TE mode mus be aken in o accoun [ ed cu e]. In Fig. 11(c) i is finally depic ed he case o oblique inci- dence, which simila ly o Fig. 8(c) also shows good ag eemen be ween ou EC esul s and CST Mic owa e S udio. Ac ually, o his longe pe iod case, i is ound a sa is ac o y quan i a i e ag eemen o he case EC(1,1) and excellen ma ching o he case EC(3,3). IV. CONCLUSION This pape discusses he beha io o NIM implemen ed as s acked iden ical me al sc eens wi h 2D pe iodic dis ibu ions o subwa eleng h ape u es ( he so-called fishne s uc u es). A fi s analysis o he s uc u e is ca ied ou by means o a nume - ically in ensi e elec omagne ic sol e . I is demons a ed nex ha he equi alen ci cui app oach, commonly used in he anal- ysis o equency selec i e su aces, yields a as e sol ing p o- cedu e which also sheds ligh on he physics behind he appea - ance o he di e en obse ed ansmission bands. In con as wi h many analy ical models a ailable in he specialized li e a- u e, he p oposed model accoun s o he equency-dependen beha io o he con ibu ions associa ed wi h he lowes -o de exci ed modes and, mos impo an ly, co ec ly accoun s o he high-o de mode in e ac ion be ween successi e sc eens. The la e ac is e y impo an since, o ypical dimensions in ol ed in he implemen a ion o s acked fishne s uc u es, high-o de mode in e ac ion is key o explain he exis ence o some o he ansmission bands. The model au oma ically col- lapses in o simple con en ional ci cui s in ol ing in e ac ion h ough he undamen al TEM wa e i he dis ance be ween sc eens and/o he ans e se pe iod ha e he app op ia e alues. The accu acy o he ci cui model can be sys ema ically en- hanced by adding high-o de con ibu ions a he expenses o gene a ing a mo e complex ci cui opology. In any case, he nume ical e o demanded by hese ci cui app oaches is negli- gible when compa ed wi h usual nume ical codes. APPENDIX In he compu a ion o he nume ical esul s in his wo k, he ollowing app oxima ed exp ession is used o he spa ial p ofile o he ape u e field [53]: (21) whose Fou ie ans o m is gi en by (22) whe e s ands o he ze o h-o de Bessel unc ion o he fi s kind. Using his ape u e field p ofile, analy ical exp essions o he high-o de capaci ances and induc ances in he model can be w i en explici ly in e ms o he s uc u e pa ame e s. Thus, gi en and such ha he high-o de condi ion holds o o , he ollowing exp essions a e ound o he ex e nal pa allel elemen s o he -ne wo k [see (10) and (16)]: (23) (24) wi h (25) Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on May 26,2020 a 15:10:29 UTC om IEEE Xplo e. Res ic ions apply.