Extraordinary transmission through arrays of electrically small holes from a circuit theory perspective

3108 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 12, DECEMBER 2008 Ex ao dina y T ansmission Th ough A ays o Elec ically Small Holes F om a Ci cui Theo y Pe spec i e F ancisco Medina, Senio Membe , IEEE, F ancisco Mesa, Membe , IEEE, and Rica do Ma qués, Membe , IEEE Abs ac —Ex ao dina y op ical ansmission o ligh o elec o- magne ic wa es h ough me al pla es pe iodically pe o a ed wi h subwa eleng h holes has been exhaus i ely analyzed in he las en yea s. The s udy o his phenomenon has a ac ed he a en ion o many scien is s wo king in he ields o op ics and condensed ma e physics. This con luence o scien is s has gi en ise o di - e en heo ies, some o hem con o e sial. The i s heo e ical explana ion was based on he exci a ion o su ace plasmons along he me al–ai in e aces. Howe e , since pe iodically pe o a ed di- elec ic (and pe ec conduc o ) slabs also exhibi ex ao dina y ansmission, di ac ion by a pe iodic a ay o sca e e s was la e conside ed as he unde lying physical phenomenon. F om a mi- c owa e enginee ing poin o iew, pe iodic s uc u es exhibi ing ex ao dina y op ical ansmission a e e y closely ela ed o e- quency-selec i e su aces. In his pape , we use simple concep s om he heo y o equency-selec i e su aces, wa eguides, and ansmission lines o explain ex ao dina y ansmission o bo h hin and hick pe iodically pe o a ed pe ec conduc o sc eens. I will be shown ha a simple ansmission-line equi alen ci cui sa is ac o ily accoun s o ex ao dina y ansmission, explaining all o he de ails o he obse ed ansmission spec a, and easily gi es p edic ions on many ea u es o he phenomenon. Al hough he equi alen ci cui is de eloped o pe ec conduc o sc eens, i s ex ension o dielec ic pe o a ed slabs and/o pene able con- duc o s a op ical equencies is almos s aigh o wa d. Ou ci - cui model also p edic s ex ao dina y ansmission in nonpe iodic sys ems o which his phenomenon has no ye been epo ed. Index Te ms—Ex ao dina y ansmission, equency-selec i e su aces (FSSs), su ace plasmon pola i ons. I. INTRODUCTION PARTIAL anspa ency o opaque slabs (me al slabs) pe- iodically pe o a ed wi h elec ically small holes was e- po ed some yea s ago by Ebbesen e al. [1] (see also he pop- ula a icle [2] in he same issue). This phenomenon, in appa en con adic ion wi h Be he’s heo y o small ape u es [3], was called ex ao dina y op ical ansmission. Since his seminal wo k, hund eds o scien i ic pape s ha e been published gi ing Manusc ip ecei ed Ap il 24, 2008; e ised July 07, 2008. Fi s published No embe 18, 2008; cu en e sion published Decembe 05, 2008. This wo k was suppo ed by he Spanish Minis e io de Educación y Ciencia and Eu opean Union FEDER Funds unde P ojec TEC2007-65376 and P ojec TEC2007- 68013-C02-01), and by Jun a de Andalucía unde P ojec TIC-253. F. Medina and R. Ma qués a e wi h he Mic owa es G oup, Depa men o Elec onics and Elec omagne ism, Facul y o Physics, Uni e si y o Se ille, 41012 Se ille, Spain (e-mail: [email p o ec ed]; [email p o ec ed]). F. Mesa is wi h he Mic owa es G oup, Depa men o Applied Physics 1, ETS de Ingenie ía In o má ica, Uni e si y o Se ille, 41012 Se ille, Spain (e-mail: [email p o ec ed]). Digi al Objec Iden i ie 10.1109/TMTT.2008.2007343 explana ions and epo ing de ails abou his (o ela ed) phe- nomenon. The phenomenon e e s o he appea ance, a ound a ce ain equency, o a na ow and s ong peak o ansmis- sion h ough an opaque sc een pe o a ed wi h small holes. The su p ising ac was ha he diame e s o he holes we e sig- ni ican ly smalle han he co esponding wa eleng h (Be he’s heo y o small holes p edic ed much less ansmi ed powe han obse ed). The cylind ical holes o he o iginal expe i- men al de ice we e a anged in o a wo-dimensional (2-D) pe- iodic squa e la ice whose uni cell had dimensions close o he wa eleng h o he “ex ao dina y” ansmi ed beam. This key ea u e s ongly sugges s ha pe iodici y should play a c ucial ole in he phenomenon. Howe e , he i s heo e ical explana- ions elied basically on he beha io o me als a op ical e- quencies. A hose high equencies, me als a e desc ibed by a complex pe mi i i y wi h a la ge nega i e eal pa (plasma be- ha io ). Me als a e hus pene able ma e ials ha can suppo a special kind o su ace wa es, he so-called su ace plasmons. The exci a ion o such wa es due o he sca e ing o he im- pinging plana ans e se elec omagne ic (TEM) wa e by he pe iodic s uc u e was hen assumed o be he physical ac be- hind ex ao dina y ansmission (see [4]–[6], among o he s). In his in e p e a ion, apa om pe iodici y, he beha io o he me al as an impe ec conduc o (mo e p ecisely, as a lossy solid plasma) seems o be essen ial o he phenomenon. Ne e heless, ex ao dina y ansmission has also been ound in me al s uc- u es a millime e -wa e equencies (see, o ins ance, he pa- pe s by Be ue e e al. [7], [8]). A hese equencies, me als a e desc ibed by a eal conduc i i y and pene a ion o elec omag- ne ic ields (skin e ec ) is ma ginal. In his si ua ion, su ace plasmons a e no suppo ed by he me al–ai in e aces. Mo e- o e , enhanced ansmission o elec omagne ic wa es has also been epo ed in pe iodically pe o a ed pe ec dielec ic slabs [9], [10]. Genuine su ace plasmons (i.e., su ace wa es sup- po ed by a uni o m me al–ai in e ace a op ical equencies) do no appea o be always equi ed o explain ex ao dina y ansmission phenomena, al hough hey can s ill play some ole in modi ying he equency alue a which he ansmission peak is expec ed o occu . Fo una ely, all o he abo e ac s can be explained by means o ull-wa e di ac ion models, which accoun o bo h p op- aga ing and e anescen ields a ound he pe iodic s uc u e. The di ac ion model was i s used o one-dimensional (1-D) pe iodic a ays o in ini ely long sli s [11], [12] (di ac ion g a ings). Howe e , he sli s p oblem is sligh ly di e en om he 2-D a ay o holes. TEM modes wi hou cu o equency 0018-9480/$25.00 © 2008 IEEE Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. MEDINA e al.: EXTRAORDINARY TRANSMISSION THROUGH ARRAYS OF SMALL HOLES FROM A CIRCUIT THEORY PERSPECTIVE 3109 a e possible in he sli s uc u e bu no in he 2-D a ay o holes. Thus, he expe imen al si ua ion ea ed in [1] is be e accoun ed o by he di ac ion models speci ically de eloped o 2-D a ays o subwa eleng h holes [6], [9], [10], [13], [14]. In hese la e models, pe iodici y, di ac ion, and in e e ence a e he ele an concep s while su ace plasmons would play a seconda y ole modula ing he main physical eason behind ex ao dina y op ical ansmission. Ne e heless, he concep o su ace plasmon was escued in his con ex by Pend y e al. in [15]. Al hough he model in [15] is no accu a e (see, o ins ance, [16]), i s unde lying quali a i e idea is wo hy: pe iodically s uc u ed pe ec conduc o su aces can suppo su ace wa es ha mimic su ace plasmons (some esea che s call hese wa es spoo plasmons). A comp ehensi e e iew o his ype o wa e o he 1-D pe iodic case (sli s on a me al su ace) was gi en p e iously in [17]. Ne e heless, o he bes o ou knowledge, hese a e he same ype o su ace wa es suppo ed by pe ec ly conduc ing pe iodic s uc u es ha a e well known by he mic owa e communi y since he 1950s o e en be o e. In o de o gi e p ope c edi o pionee ing wo ks, i should be men ioned ha su ace modes guided by open co uga ed su aces we e epo ed in a classi ied memo andum by Cu le in he 1940s [18]; see also he his o ical e iew pape by he same au ho in [19]. (A co uga ed su ace is basically he same ype o elec omagne ic sys em as ha in ol ed in ex ao dina y op ical ansmission.) A ho ough analysis o co uga ed su aces based on he Floque –Lucke me hod was epo ed as ea ly as 1954 [20], and an in-dep h expe imen al s udy o a ious pe iodic open wa eguides was ca ied ou in [21]. The simila i y be ween he model p oposed in he ecen pape by Pend y e al. [15] o 2-D s uc u ed su aces and he model sugges ed in a no e published almos 50 yea s ago by Golds one and Oline [22] o 1-D s uc u ed su aces is no o ious. Many mo e an eceden s could be gi en, al hough i is enough o men ion ha he well-known classical ex book by Collin [23] includes he opic in he chap e de o ed o su ace wa es. Despi e being a case o “ edisco e ing,” he analysis o ex- ao dina y op ical ansmission in e ms o he coupling o he impinging TEM wa e o spoo plasmons is ac ually appealing (see he excellen e iew pape s by Gene e al. [24] and Ga cía de Abajo [25]). Ne e heless, in ou opinion, he heo y based on su ace plasmons is no easy o use since i s p edic ions a e basically a ained in he o m o nume ical solu ions o e y in ensi e compu a ional p oblems. Mo eo e , he e a e s ill some unclea poin s as well as some si ua ions ha his model canno explain ( o ins ance, si ua ions whe e ex ao di- na y ansmission is possible and plasmons—including spoo plasmons—a e no p esen ). These d awbacks ha e been he mo i a ion o he p esen wo k. Ou p oposal in his pape is o p o ide a much simple pe spec i e and heo y, a leas o hose amilia wi h mic owa e ield and ci cui heo ies, ounded on wa eguide and impedance ma ching concep s. A p elimina y wo k based on hese ideas was epo ed by he au ho s in [26]. Ou p esen pape will ex end conside ably he abo e wo k and will p esen de ailed explana ions o mo e gene al p oblems. In pa icula , we will show how ela i ely simple “ ex book” wa eguide heo y concep s p o ide a comple e accoun o he obse ed ex ao dina y ansmission phenomena in a wide a ie y o si ua ions. Mo eo e , some new sys ems exhibi ing ex ao dina y ansmission (which, o he au ho s’ knowledge, ha e no ye been epo ed) will be b ie ly discussed. This pape will be o ganized as ollows. Sec ion II will show ou p oposed model o s udy ex ao dina y ansmission. Sec ion III will p esen he basics o ou heo y h ough he analysis o he simples 2-D pe iodic s uc u e exhibi ing ex ao dina y ansmission (a pe iodically pe o a ed ze o hickness sc een). Sec ion IV will in oduce a modi ica ion o he model o ac- coun o ini e hickness sc eens, and Sec ion V will desc ibe a mo e gene al model explaining some addi ional de ails o he dependence o he ansmission spec um wi h espec some geome ical dimensions. In Sec ion VI, we will gi e some insigh and quali a i e explana ions abou he de ails o he ansmission spec um a equencies abo e he onse o he i s g a ing lobes. Also, we will discuss o he possible s uc u es exhibi ing some kind o ex ao dina y ansmission. Finally, some concluding ema ks will be summa ized in Sec ion VII. II. MODELING OF EXTRAORDINARY TRANSMISSION The s a ing poin o ou modeling o ex ao dina y ansmis- sion a op ical and lowe equencies will be he key ole played by pe iodici y, a he han any o he conside a ion abou he ma e ial p ope ies. (In his sense, he di ac ion model s and- poin is e y close o ou poin o iew on he phenomenon.) Any mic owa e o an enna p ac i ione can eadily app ecia e he simila i y be ween he pe iodic s uc u es exhibi ing ex- ao dina y op ical ansmission and he equency-selec i e su aces (FSSs). As i is well known, FSSs a e 2-D a ays o plana me allic sca e e s (s opband FSS) o slo s p ac iced in a me al pla e (passband FSS). The shape and size o he plana sca e e s/slo s a e ailo ed o con ol he equency dependence o he ansmission and/o e lec ion coe icien s [27]. The slo s, usually wi h complex shapes, a e designed o esona e a a ce ain equency o gi e a o al ansmission peak a his equency, p o ided ha ma e ial losses a e neglec ed. In he heo e ical and expe imen al wo ks on ex ao dina y op ical ansmission, he geome y o he holes is commonly e y simple: ci cula /cylind ical o ec angula /p ism. Thus, one ques ion ha is immedia ely aised is why ex ao dina y ansmission was no p e iously epo ed by FSS p ac i ione s. Be o e gi ing a possible explana ion o his ac , i should be poin ed ou ha ex ao dina y ansmission always appea s a equencies e y close o he onse equency o he i s g a ing lobe and ha , a his onse equency, slo -like FSS always exhibi a ze o ansmission poin known as Wood–Rayleigh anomaly [28]–[30]. Fo FSS p ac i ione s, he ange o equen- cies close o he Wood–Rayleigh anomaly is no o p ac ical in e es because o he p esence o undesi ed g a ing lobes, and, ac ually, i has only been explo ed as a limi a ion ac o o he ope a ion o FSS. Mo eo e , o he e y hin me al sc eens commonly employed in FSS, he ansmission peak is ex emely na ow—as i will become appa en la e —and ypical ohmic losses migh se iously mask he phenomenon. On he con a y, ex ao dina y ansmission expe imen s a op ical equencies we e ca ied ou wi h elec ically hick sc eens, Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. 3110 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 12, DECEMBER 2008 Fig. 1. Pe ec conduc o sc een pe o a ed wi h ec angula holes: (a) on iew and (b) wo la e al cu s and (c) on and (d) la e al iews o he s uc- u e uni cell (pa allel-pla e ansmission line wi h diaph agm discon inui y wi h hickness ). gi ing place o wide ansmission peaks ha could be de ec ed by he human eye ( he always-p esen app op ia e “de ec o ”). Thus, assuming ha FSSs and ex ao dina y ansmission s uc u es a e he same hing, we will apply he FSS analysis me hodology o he s udy o s uc u es exhibi ing ex ao dina y ansmission. As is well known, he analysis o an in ini e FSS can be educed o he analysis o a single uni cell. This concep is illus a ed in Fig. 1, whe e he o iginal pe iodic s uc u e and he equi alen uni cell a e depic ed. Fo no mal incidence and linea pola iza ion (along he -di ec ion, in ou case), he uni cell is a pa allel-pla e ansmission line wi h a hin o hick (depending on he alue o ) ec angula i is diaph agm placed ans e sely o he axis o he ansmission line [see Fig. 1(c) and (d)]. This way o hinking is no new o FSS p ac i ione s [27] and mic owa e ield heo y esea che s. Fo ins ance, a ho ough analysis o 1-D ze o- hickness pe iodic s uc u es was ca ied ou mo e han 40 yea s ago using equi - alen ne wo k analysis [31]. Two comp ehensi e pape s abou he di ac ion by an a ay o s ips p in ed on a dielec ic sub- s a e using ci cui modeling we e published almos 20 yea s ago [32], [33]. Mo e ecen ly, and in close connec ion wi h he opic ea ed in his pape , ci cui models ha e been used in he analysis o s acked pe o a ed sc eens (each o hem exhibi ing ex ao dina y ansmission) o le -handed elec omagne ic Fig. 2. Ci cui model o ze o- hickness diaph agm in pa allel-pla e wa eguide. wa e p opaga ion [34], [35]. Ou con ibu ion in he p esen pape is o show how ex ao dina y ansmission can be ex- plained in all i s de ails by means o equi alen ci cui models. The main ad an age o his app oach is ha quali a i e and semiquan i a i e p edic ions can be done wi hou pe o ming hea y nume ical compu a ions (o limi ing such compu a ions o a ew equency poin s). This will allow us o gi e easy ex- plana ions o mos o he obse ed ea u es o he phenomenon and e en o p edic no el si ua ions exhibi ing ex ao dina y ansmission. De ailed de i a ion o inc easingly complex ci cui models will be gi en in he o hcoming sec ions. III. BASIC THEORY FOR EXTRAORDINARY TRANSMISSION THROUGH ZERO-THICKNESS SCREENS Le us conside he pe iodically pe o a ed pe ec conduc o sc een in Fig. 1 o he pa icula case o in ini esimal hickness . Since he unde lying physics is no a ec ed by he shape o he slo s, we will conside ec angula holes in o de o keep he compu a ions as easy as possible. Using a de ailed ana- ly ical/nume ical app oach [13], i has been es ablished ha his s uc u e exhibi s a single peak o ex ao dina y ansmission (i was ca ied ou o ci cula holes bu he shape is no ele an ). The uni cell unde conside a ion is a pa allel-pla e ansmission line o med by wo e ical magne ic walls sepa a ed by a dis- ance and wo ho izon al pe ec elec ic walls sepa a ed by a dis ance . Mo e p ecisely, due o symme ies o he s uc- u e and he exci a ion, he plane AA’ in Fig. 1(c) is an elec ic wall and he plane BB’ is a magne ic wall. Thus, apa om he TEM mode (TEM o ) ep esen ing he inciden , e lec ed, and ansmi ed wa es in he pe iodic s uc u e, he ansmis- sion line can suppo ( o ) and ( o ) modes (, a e in ege numbe s). I he s uc u e is used as an FSS, TE and TM modes a e always a cu o . In common FSS appli- ca ions, he size o he slo s is chosen in such a way ha hey esona e well below he equency o he i s Wood–Rayleigh anomaly. Fo he s uc u e unde s udy, his equency is gi en by ; no e ha his equency is also he cu o equency o he mode o he wa eguide, ( he i s subindex 0 co esponds o a ia ions along he -di ec ion and he second subindex, 2, co esponds o a ia ions along he -di ec ion). The o iginal p oblem is hen educed o he sca e ing o he inciden TEM mode by a ec angula i is diaph agm p ac iced in a ans e se me al shee o ze o hickness. This is a classical p oblem o discon inui ies in wa eguide heo y and, as is well known [36], a simple equi alen ci cui can accoun o he mos impo an ea u es o such discon inui y (see Fig. 2). The eso- nance o he LC ank ci cui ob iously co esponds o a peak o o al ansmission. F om a physical poin o iew, is ela ed o he elec ical ene gy in excess associa ed wi h below-cu o TM modes exci ed a he discon inui y plane. Equi alen ly, is Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. MEDINA e al.: EXTRAORDINARY TRANSMISSION THROUGH ARRAYS OF SMALL HOLES FROM A CIRCUIT THEORY PERSPECTIVE 3111 associa ed wi h he excess magne ic ene gy o he below-cu o TE modes exci ed a his same plane. The alues o and can be es ima ed om he geome ical dimensions o he s uc u e using some app oxima ions epo ed in [36]. These alues a e conside ed o be weakly dependen on equency in no mal FSS ope a ion. The esonance ( o al ansmission) equency can be ob ained by conside ing he ec angula hole as a sho -ci cui ed sec ion o a slo line. In o de o ob ain e y accu a e esul s, end e ec s should be added, especially o sho sli s (small ). F om his pe spec i e, would be close o a hal wa e- leng h o he slo mode a he desi ed o al ansmission e- quency ( should sa is y such a condi ion o -pola ized elec- ic ield). Highe o de esonances a e also possible, bu hey appea abo e he onse equency o he i s g a ing lobe . Roughly speaking, i he esonance ( ansmission) equency has o be lowe han , should be chosen la ge han . (I is wo h men ioning he e ha and no is he ele an dimension because o he pola iza ion o he im- pinging elec ic ield.) This is he ypical scena io o egula FSS ope a ion. Howe e , ex ao dina y op ical ansmission in ze o- hickness pe ec conduc o sc eens has been epo ed o e y small ape u es [13]. F om a nai e pe spec i e, e y small ape u es should esona e a equencies well abo e and no ansmission peaks would hen be ob ained below . The main aul o he abo e easoning is o o ge ha in Fig. 2 is no a smoo h unc ion o equency o equencies nea . As a ma e o ac , i is well known ha he inpu impedance co esponding o a TM mode below cu o exci ed in an in in- i ely long wa eguide is gi en by (1) whe e is he cha ac e is ic impedance o acuum, is he cu o equency o he TM mode, and is he ope a ion e- quency. I is hen clea ha he equi alen capaci ance associ- a ed wi h his mode is (2) whe e is a coe icien accoun ing o he ela i e deg ee o exci a ion o his pa icula TM mode (in compa ison wi h he o he highe o de TM modes). Fo simplici y in he o h- coming discussion, he equency dependence o will be igno ed. (Al hough i has been e i ied ha his coe icien shows a signi ican inc ease a ound he ex ao dina y ansmis- sion equency, his ac is ha dly ele an o he ollowing quali a i e conside a ions). The o e all capaci ance in he ci cui model in Fig. 2 is he esul o he pa allel connec ion o an in ini e numbe o elemen a y con ibu ions such as ha in (2). Fo a wo king equency below , he con ibu ion o highe o de TM modes di e en om o is a weakly depending unc ion on equency (because is well below hei co esponding cu o equencies). Hence, o ou pu poses, his con ibu ion o he o al capaci ance can be conside ed o be cons an and will be deno ed as . The equency-dependen con ibu ion is assumed o be gi en by he TM mode wi h he smalles cu o equency ( in ou case). The o e all capaci ance can hen be w i en as ollows: (3) The impo an poin he e is ha , since as , ou model p edic s ha one ansmission peak is always p esen below he i s Wood–Rayleigh anomaly o any alue o . Fo small ape u es, is also small, and hen has o each e y high alues o ul ill he esonance condi ion. This is he eason o ind ex ao dina y ansmission only close (bu below) o he Wood–Rayleigh anomaly o his si ua ion. An in e es ing es o his poin o iew is he compa ison o he ex ao dina y ansmission equencies o wo iden ical ec - angula slo s bu wi h pe pendicula o ien a ions. In his case, nume ical simula ions say ha he o al ansmission peak co - esponding o he ho izon al o ien a ion appea s a a lowe e- quency. The equi alen ci cui heo y gi es an easy explana ion o his ac . The ho izon ally o ien ed ec angula slo will pe - u b mo e s ongly he su ace cu en s (wi h espec o he non- pe o a ed sc een case) han he e ically o ien ed one. The co - esponding highe alue o he induc ance o he ho izon ally o ien ed slo equi es a lowe capaci ance o sa is y he eso- nance condi ion, which will hen occu a a lowe equency. The simplici y o he geome y unde s udy has allowed us o implemen a ela i ely easy compu e code based on he mode ma ching echnique [37] o accu a ely compu e he ansmission and e lec ion coe icien s. (In ou mode-ma ching compu a ions o he ze o- hickness case, we in oduce a e y small sc een hickness and use many modes o each con- e gence. A mo e e icien nume ical p ocedu e o his case would ha e been he solu ion o an in eg al equa ion o he unknown equi alen magne ic cu en in he ape u e [38], bu his nume ical ac is no ele an o he pu poses o he p esen pape ). The unknown pa ame e s o he ci cui model in Fig. 2, , , and , can be easily compu ed om a ew low- e- quency alues o he ansmission coe icien and om he alue o he o al ansmission equency. This in o ma ion is gene a ed using he mode-ma ching code. Fo ins ance, o he equi alen ci cui in Fig. 2, can be ob ained om (4) whe e is he angula equency and is he cha ac e is ic admi ance o he inpu and ou pu ansmis- sion lines. Fo low equencies (in compa ison wi h ), and a e cons an s and (4) is jus a polynomial whose coe - icien s can be p ope ly i ed om a ew low- equency da a. The pa ame e is ob ained by adding he in o ma ion o he esonance equency. Wi h his educed se o pa ame e s, we should be able o ep oduce he whole ansmission spec- um. I is wo h men ioning ha he equi alen ci cui wi h he single-mode equency-dependen capaci ance con ibu ion in (2) pe ec ly accoun s o he Wood–Rayleigh anomaly. Ce - ainly, a , he capaci ance is in ini y and he diaph agm will beha e as a sho ci cui . This has been nu- me ically checked e i ying o al e lec ion and ha he phase o he e lec ed wa e a ha equency is . Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. 3112 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 12, DECEMBER 2008 Fig. 3. T ansmission coe icien j S j h ough a ze o- hickness pe ec con- duc o sc een wi h ec angula holes o a ious sizes. The uni cell is squa e: a = a = 5 mm, which co esponds o a Wood–Rayleigh anomaly equency o = 59.9585 GHz. Fo la ge slo size ( w = 3mm, w = 2.5 mm; w = 0.5 mm), we ha e no mal FSS ope a ion, and he passband is wide and well below he Wood–Rayleigh anomaly. Fo small slo s ( w = 2mm, w = 1.5 mm; w = 0.5 mm), wi h in insic esonance equency abo e , he pe iodici y o he s uc u e is esponsible o ex ao dina y ansmis- sion nea bu below wi h na ow bandwid hs. In o de o alida e he abo e p oposals, we ha e compu ed he ansmission coe icien o a ious slo s keeping he same spa ial pe iodici y. The “exac ” mode-ma ching esul s ( he numbe o modes has been inc eased un il each good con- e gence) and he p edic ions o he ci cui model a e gi en in Fig. 3. The i s appa en conclusion is ha he equi alen ci - cui model ma ches e y well he compu ed ull-wa e esul s in he whole equency ange. This is a signi ican hin o he a- lidi y o ou model. In Fig. 3, i can also be obse ed ha la ge slo s yield wideband esonances a away om and below , as i is quali a i ely expec ed since his co esponds o no mal FSS ope a ion. Fo e y small slo s o subwa eleng h size, he esonance ( ansmission) peaks mo e up o he p oximi ies o and hei co esponding bandwid hs become smalle and smalle as he slo size is educed. This is consis en wi h he quali a i ely expec ed beha io o ou equi alen ci cui . When he holes a e e y small and he induc ances a e co espondingly small, esonance mus be e y close o he singula i y o he ca- paci ance a . No e ha he bandwid h o esona o s wi h high and small is small. Mo eo e , due o he as a ia ion o nea , a small a ia ion o equency a ound he es- onance equency makes he ci cui a om esonance condi- ions. Thus, e y na ow bandwid hs a e quali a i ely expec ed i he ansmission peak is nea (which is clea om he esul s epo ed in Fig. 3). P e iously i has been shown ha o al ansmission is p e- dic ed by simple wa eguide heo y a gumen s in he case o pe- iodically pe o a ed pe ec conduc o ze o- hickness sc eens wi h a bi a ily sized holes. Some addi ional in e es ing heo- e ical conclusions can also be deduced om he equi alen ci - cui model. Fo example, i is he beha io o he mode nea cu o ha is mo e ele an o he ex ao dina y ansmis- sion phenomenon. No e ha he mode is also nea cu o ( o he conside ed squa e la ice). This mode also con ibu es wi h a singula induc ance nea i s cu o equency (which could each e y high alues). Howe e , his ac does no a ec ou p e ious conclusions abou he domi- nan ole o he mode in he ex ao dina y ansmission occu ence. I is only he shun -connec ed la ge capaci ance ha yields no iceable a ia ions in he ansmission coe icien , since la ge in pa allel wi h and o he ’s coming om o he TE modes will no a ec he esonance condi ion and equency e- sponse. In summa y, only he o e all capaci ance (and no he o e all induc ance) becomes singula a . Inciden ally, his also explains ha i is he pe iodici y along he di ec ion o he pola ized elec ic ield (-di ec ion) ha ac ually de e mines he alue o he ex ao dina y ansmission equency. The pe i- odici y is no ele an a all. Indeed, pe iodici y along he -di- ec ion is no equi ed o he obse a ion o enhanced ans- mission peaks. Fo ins ance, in [39], i has been demons a ed ha a single ow o holes (1-D pe iodici y) exhibi s ex ao di- na y ansmission peaks. In his case, ex ao dina y ansmis- sion e e s o ansmi ing much mo e powe han he powe im- pinging on he a ea o each indi idual hole ( o al ansmission has no sense in his case, ob iously). This quali a i e p edic ion o ou model is an addi ional alida ion o i s physical sound- ness as well as i s p edic i e po en ial. Ohmic losses we e neglec ed in ou p e ious discussion, bu i is expec ed ha hei p esence leads o a educ ion o ansmi ed powe a he c i ical equencies o o he wise pe ec ansmis- sion sys ems. Following ou equi alen ci cui model, we can quali a i ely ad ance ha losses will be mo e signi ican o he case o small holes (ex ao dina y ansmission) han o he case o la ge holes ( egula FSS ope a ion). In he ci cui model, losses would be modeled as a esis ance connec ed in se ies wi h . Taking in o accoun ha he alues o in ol ed in ex- ao dina y ansmission peaks a e ypically much smalle han hose in ol ed in common FSS ansmission peaks, he e ec o he losses on he quali y ac o o he peaks would be mo e p onounced o ex ao dina y ansmission ope a ion han o usual FSS ope a ion. Thus, p ac ical applica ion o na owband spa ial il e s and pola ize s based on ex ao dina y ansmis- sion could be se iously a ec ed by ohmic losses a mic owa e and millime e -wa e equencies. In op ical applica ions, me als a e no cha ac e ized by a eal conduc i i y bu a he by a e- quency-dependen complex pe mi i i y wi h a la ge and nega- i e eal pa . Maybe, in his case, he phenomenon o ex ao - dina y op ical ansmission could be ui ully exploi ed in he design o new de ices such as hose epo ed in [40]. IV. EXTENDING THE MODEL TO THICK SCREENS Mos o he expe imen s and nume ical simula ions o ex a- o dina y ansmission sys ems ha e been ca ied ou wi h ela- i ely hick sc eens. Nume ical simula ions and some simpli ied analy ical models p edic ha wo ins ead o jus one ansmis- sion peaks should be obse ed; see, o ins ance [7, Fig. 2], [6, Figs. 3 and 4], [25, Fig. 8], [41, Fig. 2], and [42, Fig. 2]. Ne - e heless, losses and expe imen al limi a ions can make di icul he obse a ion o wo sepa a e peaks. I is clea ha he equi - alen ci cui in Fig. 2 canno accoun o such a pai o peaks. The physical eason is ha he eac i e ene gy s o ed inside he Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. MEDINA e al.: EXTRAORDINARY TRANSMISSION THROUGH ARRAYS OF SMALL HOLES FROM A CIRCUIT THEORY PERSPECTIVE 3113 Fig. 4. (a) Lumped-elemen s ci cui model o hick diaph agm in pa allel-pla e wa eguide. (b) E en- and odd-mode ci cui s used o ob ain sca e ing pa ame- e s and c i ical ansmission equencies. The equi alence be ween pa ame e s o he model in (a) and he e en-/odd-mode model in (b) is included in he igu e. hole was neglec ed in ha simple equi alen ci cui (only alid o ze o- hickness sc eens). Fo subwa eleng h holes, he dominan e anescen mode in- side he hole is he mode. The hole can hen be iewed as a ec angula wa eguide sec ion ope a ing well below he cu o o he i s p opaga ing mode. F om a mic owa e enginee ing poin o iew, he p oblem co esponds o he sca e ing by a i- ni e hickness diaph agm. Fo diaph agms ha a e no oo hick, he in oduc ion o a se ies induc ance [ in he ci cui model depic ed in Fig. 4(a)] is an app op ia e manne o accoun ing o he modi ica ion in oduced by he nonnegligible hickness o he me al sc een [36]. Fo hin sc eens, i is ela i ely ob- ious ha should be p opo ional o he hickness . The in- clusion o jus his single se ies induc o will al eady p o ide wo esonance peaks. Un o una ely, his simple model (com- monly used o model p ac ical hin diaph agms in closed wa e- guides [36]) would no accoun o sub le de ails o he ans- mission spec um ob ained om nume ical simula ions. Due o his, we p opose he sligh ly mo e sophis ica ed -ne wo k o induc ances shown in Fig. 4(a). The idea unde lying his -ci - cui is ha he equi alen ci cui o a TE mode below cu o is a ladde ne wo k whose elemen s a e in ini esimal induc ances. The dis ibu ed ne wo k is he e eplaced by a single cell wi h i- ni e induc ance alues. Al e na i ely, he abo e -ci cui can be eached by no ing he p esence o a e ical symme y plane a he middle o he hole. Fo e en exci a ion om he wo sides o he sc een, his plane is a magne ic wall, and, o odd exci a ion, i is an elec ic wall. The co esponding equi alen ci cui s o hese wo si ua ions a e depic ed in Fig. 4(b), whe e he induc- ances and a e ela ed o he magne ic ene gy s o ed inside he hole unde e en (e) and odd (o) mode condi ions. The ela- ionship be ween and wi h he induc ances o he -ci - cui is ob ious and has been explici ly shown in Fig. 4(b). Keeping in he ci cui model only (i.e., aking )is equi alen o neglec ing he magne ic ene gy s o ed inside he hole by he below-cu o TE modes in he case o e en exci- a ion. I we wan o accu a ely accoun o nonze o hickness e ec s, his la e con ibu ion mus be aken in o accoun , and a ini e alue o has o be used in he model. Fo e y hin sc eens, i is ound ha (no e ha accoun s o TE modes in he ex e nal wa eguides, no in he hole). In his case, he e ec o is p edominan since and a e shun -con- nec ed. Fo in ini esimally hin sc eens, i is addi ionally ound ha , gi ing place o a sho ci cui in such a way ha he esul s o he ze o- hickness sc eens o p e ious sec ion a e e- co e ed. Howe e , o app eciably hick sc eens, is no small and can be o he same o de o magni ude as . The ci cui model in Fig. 4(a) p edic s wo o al ansmission peaks a e- quencies below and, mo eo e , i can gi e some quali a i e insigh abou he e olu ion o hose peaks as a unc ion o he sc een hickness . A simple e en–odd exci a ion analysis o his ci cui leads o he ollowing ansmission coe icien : (5) whe e he e lec ion coe icien s o e en and odd exci a ions a e gi en by (6) wi h he ollowing alues o he equi alen induc ances o e en and odd exci a ions: (7) These a e he equi alen induc ances o he shun associa ions o induc ances loading he ansmission lines in Fig. 4(b). The induc ances and depend on he sc een hickness bu , p o ided ha he elec ical hickness o he sc een is no oo la ge, hey only sligh ly depend on equency. The alues o and a e ela ed o he ields ou side he hole, and hey a e almos independen o . Equa ion (5)–(7), oge he wi h he equency dependence o in (2), p edic he ollowing wo o al ansmission equencies: (8) p o ided he ollowing condi ion is ul illed: . Fo una ely, his las condi ion is always sa is ied. The equa ions in (8) a e a se o implici equa ions ha de e mine he esonance ( o al ansmission) equencies using he capaci ance in (3) and he induc ances in (7). Equi alen ly, i he esonance equencies a e known om a mode-ma ching analysis, (8) would p o ide a me hod o ob ain he induc ances in (7). The equencies in (8) a e always below (onse o he i s g a ing lobe) due o he singula beha io o a . No e ha each o he ansmission peaks can be ela ed o he esonance o he eac i e load associa ed wi h each o he exci a ion modes [e en o odd, see Fig. 4(b)]. Since , his implies ha and, he e o e, he bandwid h a ound will be smalle han ha a ound . The abo e ea u es coincide wi h he epo ed beha io o hick sc eens in many p e ious pape s based on pu ely nume ical app oaches o cumbe some analy ical de elopmen s; see, o ins ance, [6, Fig. 3] o [25, Fig. 8]. A di e en si ua ion is ound i he slo wid h is su i- cien ly la ge ( ypically when ) o allow ansmission o he mode inside he hole a equencies below . While he equi alen ci cui in Fig. 4 is s ill alid below he onse Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. 3114 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 12, DECEMBER 2008 o his mode, , a di e en ci cui model should be used o he e en exci a ion mode in Fig. 4(b) o e- quencies . In his la e case, he con ibu ion o he mode ields inside he hole is a capaci ance ha should be added o , while he ele an induc ance is only he ex- e nal induc ance . This is because a sho sec ion o wa e- guide abo e cu o e mina ed wi h a magne ic wall (open ci - cui ) is equi alen o a capaci ance. The esonance associa ed wi h and he sum o his capaci ance plus in Fig. 4(b) yields he ypical wide ansmission peak obse ed in common FSS ope a ion. In he case o he odd mode, he ci cui model in Fig. 4(b) is s ill alid abo e because a sho sec ion o wa eguide abo e he cu o e mina ed wi h an elec ic wall (sho ci cui ) is equi alen o a lumped induc ance . This lumped induc ance is expec ed o be much lowe han , and i s e ec will p e ail o e he e ec o . This small alue o he o e all induc ance oge he wi h he equency beha io o will lead o a second na ow esonance e y close o . This esonance has been unno iced in he pas because is ex emely small o he elec ically e y hin sc eens used in FSS applica ions. I s associa ed ansmission peak would hen be ex emely na ow and, in p ac ice, possibly masked by ohmic losses. Le us now discuss in de ail he wo di e en si ua ions p e- iously men ioned using some examples. A. Sc eens Wi h La ge Holes Fo ela i ely la ge slo s (say ), he lowe e- quency in (8), associa ed wi h he case o e en exci a ion eso- nance , co esponds o con en ional FSS ope a ion. (No e ha he alue o is no ele an because o he o ien a ion o he impinging elec ic ield). This equency is mainly con olled by and in Fig. 4, al hough should be sligh ly inc eased wi h he alue o he capaci ance coming om he con ibu ion o he mode unde e en exci a ion con- di ions o equencies abo e . As a consequence, he posi ion o his ansmission peak mainly depends on , and no hing “ex ao dina y” happens in such a case ( egula FSS ope a ion). This ansmission peak has been called elsewhe e localized wa eguide esonance; see, o ins ance, [41] among o he s. Wi h his e minology, he au ho s o [41] seem o e e o a si ua ion whe e he ansmission equency oughly ma ches he cu o equency o he i s mode launched abo e cu o in he hole ( in ou case). Howe e , his e minology is misleading because he p opaga ion o a wa eguide mode in- side he hole does no necessa ily imply s ong ansmission. In ac , s ong ansmission is only obse ed a ound a speci ic equency, al hough he mode p opaga es o all o he equencies abo e . The ci cums ance ac ually equi ed o o al ansmission is impedance ma ching, and his condi- ion is only eached when a e age elec ic and magne ic ene - gies s o ed a ound he holes a e iden ical. This condi ion only angen ially migh be ela ed o he onse o a wa eguide mode inside he hole. Su p isingly, his poin seems o be sys ema i- cally igno ed in mos o he physical explana ions epo ed in many (i no all) o he published pape s on he opic. In Fig. 5, we ha e plo ed ansmi ance esul s o se e al slo s wi h di e en alues o . These esul s ha e been ob ained Fig. 5. T ansmission spec um j S j o a pe o a ed pe ec conduc o ini e- hickness pla e o se e al wid hs o he ec angula holes. Da a om [41] (black ci cles) ha e been included o compa ison pu poses. Ve ical a ows ma k he posi ions o he i s maximum de i ed om a simple easoning based on sho - ci cui ed slo esonance. Sligh shi s o hese heo e ical esonances wi h e- spec o nume ical da a is due o he neglec ed end e ec s. Dimensions: a = a , w = 0 : 2 a , and =0 : 2 a . using ou mode-ma ching code. The ini e-di e ence ime-do- main (FDTD) esul s epo ed in [41] o a e in- cluded o compa ison pu poses (ou da a ep oduced accu a ely all o he esul s in [41]). Ac ually, he accu a e mode-ma ching compu a ion o he lowe ansmission peak equency e eals ha he maximum ansmission occu s a equencies clea ly below he onse equency o he mode (see, o ins ance, he case co esponding o in Fig. 5). This ac can be explained in e ms o he appa en la ge leng h o he slo esona o due o end e ec s. As s a ed in he analysis o he ze o- hickness case, he o dina y ansmission peak can be in- e p e ed in e ms o he esonance o he undamen al mode o he sho -ci cui ed ini e-leng h sec ion o slo line, in ou case, slo wid h , slo leng h , me alliza ion hickness , and wi h he esonance condi ion gi en by , whe e is he p opaga ion cons an o he slo mode. In Fig. 5 some a ows ha e been included o ma k he abo e “ heo e - ical” o al ansmission equencies de i ed om he esonan slo model neglec ing end e ec s and aking as he acuum wa enumbe . No e ha , in his way, he abo e a ows also accoun o he cu o equency o he mode inside he hole. Since , he shi o lowe equencies o he ansmission peak (when compa ed wi h he onse equency o he mode in he hole) ha can be app ecia ed in Fig. 5 is mainly due o end e ec s a he wo sho -ci cui ed ends o he slo esona o . This shi o lowe equencies is mo e e iden o sho e slo s because o he la ge ela i e weigh o he end e ec . The shi is also mo e p onounced when he sc een hick- ness is small because is no longe so close o (see, o ins ance, he case 3 mm shown in Fig. 3). A e he discussion in he p e ious pa ag aph, i is clea ha he i s ansmission peak canno be conside ed “ex ao dina y” in any sense. The impo an obse a ion conce ning ex ao - dina y ansmission h ough nonze o- hickness sc eens is ha , apa om ha FSS-like peak, a second na ow ansmission Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. MEDINA e al.: EXTRAORDINARY TRANSMISSION THROUGH ARRAYS OF SMALL HOLES FROM A CIRCUIT THEORY PERSPECTIVE 3115 Fig. 6. T ansmission coe icien j S j h ough small holes p ac iced in se e al nonze o hickness sc eens (solid lines). The peaks app oach o each o he as he sc een hickness inc eases. Fo la ge hickness, he wo peaks collapse in o a single peak, and, inally, ansmission disappea s o e y hick sc eens. Fo hose cases whe e he wo peaks a e clea ly no iceable, he ci cui model p e- dic ion is included (dashed lines). The dimensions a e a = a = 5mm; w = 1.5 mm, and w = 0.5 mm. peak always appea s below and close o he onse equency o he i s g a ing lobe [41]. This is he ansmission peak ha can be p ope ly called “ex ao dina y” because, o he bes o he au ho s’ knowledge, i has no been epo ed and discussed be- o e he pape by Ebbesen [1]. F om Fig. 5, i is appa en ha he o dina y peak posi ion (i.e., he FSS-like peak wi h la ge bandwid h) depends on he size o he hole along he -di ec ion , bu he ex ao dina y peak is always close o . This obse a ion is in pe ec ag eemen wi h ou p e ious heo e - ical discussion. The ex ao dina y ansmission peak has been ela ed o su ace plasmons in p e ious li e a u e (genuine plas- mons, spoo plasmons, o bo h). Howe e , in ou heo y, o - dina y and ex ao dina y ansmission peaks a e bo h ela ed o impedance ma ching due o cancella ion o eac i e ene gy (imagina y pa o he Poyn ing ec o lux) by p ope balance o elec ic and magne ic ene gy s o ed in he cu o modes in ol ed in he discon inui y p oblem. This condi ion can be ul illed in he p oximi y o he onse o spoo plasmons bu , in ou opinion, i is he impedance ma ching ha should be conside ed as he ele an cause. As will be b ie ly discussed in Sec ion VI, his kind o impedance ma ching can also be a ained in closed wa e- guide sys ems whe e su ace plasmons a e absen o simply ha e no sense. B. Sc eens Wi h Small Holes Nex , he case o a hick sc een wi h small (subwa eleng h) holes ( ue ex ao dina y ansmission si ua ion) will be consid- e ed. In Fig. 6, we plo o he same small ec angula holes p ac iced on a ious sc eens wi h di e en hickness. Since he holes a e small, he wo pe ec ansmission peaks a e close o , as expec ed om ou heo y. Also, we can see ha he la ge he hickness o he sc een is, he close he wo peaks a e loca ed. How does ou model accoun o his ac ? I is igno ed in he model in Fig. 4(a) (as is ypical in he mod- eling o hin diaph agms in wa eguides), i would be possible o accoun o he hickness dependence o he odd-exci a ion peak close o bu no o he hickness dependence o he Fig. 7. Induc ances L and L accoun ing o he eac i e ields inside he ec angula hole as a unc ion o sc een hickness ( ) . These induc ances a e he induc ances in pa allel wi h L and C in Fig. 4(b). (  , ): Da a ex ac ed om he nume ical mode-ma ching analysis. Solid/dashed lines: heo e ical da a ollowing he discussion in Sec ion V. The dimensions a e he same as in Fig. 6. e en-exci a ion peak. In he la e case, he peak equency would be con olled exclusi ely by and , which a e no de- penden on he hickness o he sc een. Thus, mus be in- cluded in he model o accoun o he displacemen o he lowe equency peak when he hole hickness a ies. Quali a i ely, i is expec ed an inc ease o wi h ( o small alues o , should be p opo ional o ) and ha dec eases mono oni- cally wi h s a ing om a . F om ou ci cui model and he assumed equency dependence o he a ious in ol ed pa ame e s ( , , , , and a e equency-indepen- den , while in (2) is he only equency-dependen pa- ame e ), i is possible o ex ac he alues o all o he pa am- e e s o he equi alen ci cui om a ew da a compu ed wi h mode ma ching. Using easonable app oxima ions, he unc ion can be app oxima ed by a hi d-o de polynomial unc ion (some hing simila o (4) o ze o- hickness sc eens) a low equencies. The coe icien s o his polynomial and he alues o he esonance ( ansmission) equencies de e mine all o he pa ame e s o he ci cui model in Fig. 4(a). Using his simple i ing scheme, we ha e ob ained he alues o he in- duc ances appea ing in pa allel wi h and in Fig. 4(b), i.e., and , o se e al hick- nesses o he sc een. These esul s ha e been plo ed in Fig. 7 (disc e e ci cles and hombuses). Ou p e ious quali a i e dis- cussion abou he dependence o induc ances wi h espec o is clea ly suppo ed by hese esul s. As we can see om Fig. 7, is la ge (and goes o in ini y) o e y hin sc eens while becomes small and p opo ional o o hin sc eens. Howe e , as he hickness o he sc een inc eases, and end o ap- p oach each o he , as hey a e iden ical o elec ically e y hick sc eens. The e olu ion o he pai o peaks in Fig. 6 wi h e- spec o he sc een hickness can be easily explained in e ms o he esul s o and in Fig. 7. As he alues o and app oach each o he , he ansmission peaks a e close and close . No e ha he p edic ions o ou equi alen ci cui ha e also been plo ed in Fig. 6 (dashed lines). These p edic ions a e e y accu a e o he wo-peak cases. Howe e , i can be seen Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply. 3116 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 56, NO. 12, DECEMBER 2008 ha he wo ansmission peaks collapse in o a single peak when he hole hickness is su icien ly la ge, i.e., when and a e almos iden ical. (Fo his single-peak case, he ci cui model in Fig. 4 does no p o ide good esul s.) I is expec ed ha he ansmission peaks disappea o e y hick sc eens due o he e anescen beha io o he ields inside he hole ( he ields o he below-cu o modes exci ed a he le side o he sc een canno each he second in e ace due o s ong eac i e a enua ion). This would be in pe ec ag eemen wi h he nume ical esul s epo ed in [25, Fig. 8] o ci cula shaped holes in hick sc eens. The single peak cu es in Fig. 6 e eal a limi a ion o he lumped equi alen ci cui shown in Fig. 4. When he wo es- onance equencies collapse in o a single one, i is ound ha , and hen he equi alen ci cui in Fig. 4 p edic s no ansmission a all. The e o e, in spi e o he success o ou model up o his poin o explain ex ao dina y ansmission e- sul s, i would be con enien o ha e a mo e sophis ica ed model o he diaph agm discon inui y ha can accoun mo e accu a ely o he ac ual dependence o and wi h . Wi h his modi- ica ion, he equi alen ci cui model is expec ed o gi e app o- p ia e esul s e en when he wo peaks collapse. As we will show in Sec ion V, his can be done, bu he lumped-elemen ci cui model should be abandoned in a o o a dis ibu ed model. V. SIMPLIFIED DISTRIBUTED MODEL FOR EXTRAORDINARY TRANSMISSION The complex dependence o he ansmission peak equen- cies in e ms o he sc een hickness has been conside ed in he li e a u e using a model ha uses elec ic and magne ic dipoles o accoun o he e ec s o small holes [25]. A sophis ica ed explana ion o his beha io based on he o ma ion o a “su - ace plasmon molecule” was p o ided yea s ago in [6]. I is hen an in e es ing challenge o ou equi alen ci cui model o ac- coun o his complex dependence on he sc een hickness using simple a gumen s. We ha e ound a ela i ely simple solu ion based again on well-known wa eguide concep s. The key poin is ha changing he hickness o he sc een does no app e- ciably a ec wha happens ou side he hole (i.e., and a e no dependen on ). The e o e, he pa ame e a ec s ansmission equencies h ough he alues o and (o , equi alen ly, and ). Al hough we al eady ha e a quali a i e idea abou he dependence o hese pa ame e s on om he discussions in connec ion wi h Fig. 7, a much mo e accu a e es ima ion o he dependence o and wi h can be achie ed a e consid- e ing ha , o hick sc eens, he dis ance can be compa able o he longi udinal a ia ion o elec omagne ic ields inside he hole. This means ha elec omagne ic ields inside he hole de- pend on in he speci ic manne gi en by wa eguide heo y, and his speci ic a ia ion a e can be easily included in he model. In pa icula , since he dominan mode (below cu o o ex a- o dina y ansmission condi ions) inside he hole is he mode, we p opose he app oxima e dis ibu ed equi alen ci cui shown in Fig. 8. In his equi alen ci cui , he hole is subs i u ed by a sec ion o an e anescen ansmission line o leng h . This ansmission line is cha ac e ized by he known imagina y cha - ac e is ic impedance and he a enua ion ac o co esponding o he mode o he small ec angula wa e- guide o dimensions and . The pa ame e in Fig. 8(a) Fig. 8. (a) New ci cui model accoun ing o dis ibu ed e ec s inside he hole o nonze o- hickness sc eens. (b) E en- and odd-mode equi alen ci cui s. The alues o L and L a e gi en in he ex . co esponds o he exci a ion ac o o he mode, which will be ob ained he e om mode-ma ching simula ion (al hough i can be oughly es ima ed om he geome ies o he la ge and small wa eguides in ol ed in he p oblem). A simple e en/odd exci- a ion analysis o he s uc u e in Fig. 8 yields (9) (10) No e ha he dis ibu ed model p o ides he explici dependence on o he pa ame e s o he model in Fig. 4(b). The ci cui pa ame e s o Fig. 8 ( , , and ) can be ob ained om a ew mode-ma ching simula ions, as was done wi h p e- ious equi alen ci cui models. Using (9) and (10), we ha e ob ained he da a co esponding o he solid and dashed lines plo ed in Fig. 7. As he dependence o and wi h is ex- plici ly known a p io i, he alues o and ha e o be com- pu ed only o a single alue o . I we choose a la ge alue o so ha , i is ound ha (11) (12) This means ha he equency esponse o any sc een hickness can be known wi hou pe o ming mode-ma ching simula ions o di e en hicknesses. I should be highligh ed ha he pe ec ma ching o he cu es in Fig. 7 wi h he disc e e poin s (ci cles and hombuses) con i ms all o ou assump ions ( o ins ance, ha he only ele an mode inside he hole is he mode). The use o he dis ibu ed model adds ano he signi ican ad an age when compa ed wi h he lumped model. The a ail- abili y o he explici exp essions in (9) and (10) allows he equi alen -ci cui model o accoun o wha happens when he wo peaks o ex ao dina y ansmission collapse. Thus, in Fig. 9, we compa e he ull-wa e mode-ma ching esul s (lines) wi h he dis ibu ed equi alen ci cui p edic ions (ci cles) o se e al cases o hick sc eens. Now, ou equi alen ci cui model shows an excellen ag eemen wi h he nume ical esul s e en when a single peak occu s and only pa ial ansmission Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on June 09,2020 a 16:45:29 UTC om IEEE Xplo e. Res ic ions apply.