Extension of Babinet's Principle for Plasmonic Metasurfaces

Ex ension o Babine ’s p inciple o plasmonic me asu aces Ci e as: Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 Submi ed: 4 Augus 2021 .Accep ed: 30 Sep embe 2021 . Published Online: 19 Oc obe 2021 J. D. O iz, 1 J. P. del Risco, 2,3 J. D. Baena, 2,a) and R. Ma qu es 4 AFFILIATIONS 1 Facul y o Enginee ing, Uni e sidad de San Buena en u a, Bogo  a 110141, Colombia 2 Depa men o Physics, Uni e sidad Nacional de Colombia, Bogo  a 111321, Colombia 3 Depa men o Ma hema ics, Uni e sidad Se gio A boleda, Bogo  a 111711, Colombia 4 Depa men o Elec onics and Elec omagne ism, Uni e sidad de Se illa, Se illa 41012, Spain a) Au ho o whom co espondence should be add essed: [email p o ec ed]du.co ABSTRACT Babine ’s p inciple is widely applied in op ics and has been use ul o designing me asu aces wi h dual beha io . Al hough his p inciple can be igo ously demons a ed o infini ely hin pe ec conduc ing sc eens, i is no exac o any eal sc een. In ac , me als used in plasmonic me asu aces a e a om good conduc o s, and he hickness o samples is no negligible in compa ison wi h he ypical size o he pa e ned s uc u e. In his pape , we p opose an ex ension o Babine ’s p inciple alid o plasmonic me asu aces by edefining he concep o complemen a y sc eens and finding impedance ela ions be ween such sc eens ha ul ima ely leads o a simple ela ion be ween he ansmission ma ices o wo complemen a y plasmonic me asu aces. The heo y is alid unde he assump ions o he elec oquasis a ic app oxima ion and plane wa es in he a field. I may find applica ions in he design o op ical plasmonic me asu aces, nanoci cui s, and nanoan ennas. Published unde an exclusi e license by AIP Publishing. h ps://doi.o g/10.1063/5.0065724 The ela ed opics o op ical plasmonic me asu aces, 1,2 nanoci - cui s, 3–6 and nanoan ennas 7 ha e a ac ed much a en ion du ing he las wo decades. Complemen a i y is a widely used concep in he di - ac ion heo y ha migh be use ul o new de elopmen s in hese esea ch a eas. I is closely ela ed o Babine ’s p inciple, which es ab- lishes ce ain duali y ela ions be ween he fields sca e ed by wo com- plemen a y sc eens. 8,9 In ac , Babine ’s p inciple has al eady been applied o designing complemen a y me asu aces a mic owa es, 10 e ahe z, 11 and op ical equencies. 12–18 In hose p e ious wo ks, i was demons a ed ha he ansmi ance o one me asu ace is like he eflec ance o i s complemen a y coun e pa , and ice e sa, excep o some deg ee o de ia ion. Aside om ha , he duali y be ween elec ic and magne ic fields has been confi med in he nea field o some specific complemen a y plasmonic s uc u es. 19,20 Howe e , while Babine ’s p inciple can be igo ously demons a ed o infini ely hin pe ec conduc ing sc eens, 8,9 i is no exac o op ical plasmonic sc eens o which nei he he hickness is negligible no he me als a e good conduc o s. Ne e heless, e y ecen wo ks 21–23 sugges ha Babine ’s p inciple is s ill quali a i ely alid o plasmonic s uc u es. Would i be possible o ex end Babine ’s p inciple o plasmonic sc eens, maybe by in oducing some co ec ion, and placing i s applica ion o hese s uc u es on mo e solid g ounds? Along his Le e , we will show ha his ex ension is possible. Namely, ou ex en- sion o Babine ’s p inciple will be alid o complemen a y plasmonic me asu aces composed o ma e ials wi h high alues o he modulus o he pe mi i i y, which eal pa could be nega i e o egions filled wi h some plasmonic me al while posi i e in o he egions filled wi h high pe mi i i y dielec ics. We will no only p o ide a heo e ical o - mula ep esen ing his ex ension o Babine ’s p inciple bu also com- pa e i s pe o mance agains he p edic ions based on con en ional Babine ’s p inciple. We will show ha he ag eemen be ween heo y and nume ical simula ions using ou ex ension is g ea ly imp o ed o h ee di e en shapes. Be o e ex ending his p inciple, i is wo h o summa ize he con- en ional o m o Babine ’s p inciple which, unde ce ain conside - a ions, will p omp ly d i e us o a use ul co olla y linking he ansmission coe ficien s o wo complemen a y pe iodic sc eens whose pe iod is subwa eleng h (as usual o me asu aces). Le us s a assuming wo complemen a y hin conduc ing sc eens, one migh be called he o iginal one while he o he he complemen a y one, bo h sc eens o hogonal o he z-axis, and loca ed a z¼0.Unde hiscon- ex , complemen a y means ha bo h sc eens ha e been e ched wi h Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-1 Published unde an exclusi e license by AIP Publishing Applied Physics Le e s ARTICLE sci a ion.o g/jou nal/apl 04 June 2025 14:00:05 he same pa e n bu in e changing conduc o and ai egions. Mo eo e ,le hesou cebeplacedin hez<0 hal -space and emi ing a plane wa e ha impinges no mally on o he sc een. Le us call Einc and Binc he elec ic and magne ic inciden fields o he o iginal sc een, while o he complemen a y sc een he inciden fields E0 inc and B0 inc will be linked o he o me as ollows: E0 inc ¼cBinc and cB0 inc ¼Einc,beingc he speed o ligh in acuum. Then Babine ’s p inciple says 8,9 E a cB0 a ¼Einc; cB a þE0 a ¼cBinc;(1) whe e E a and B a a e he elec ic and magne ic fields ansmi ed h ough he o iginal sc een while E0 a and B0 a ha e he same meaning bu o he complemen a y p oblem. Al hough his is usually called a p inciple, i is an exac heo em when sc eens a e plana , infini ely hin, and made o pe ec conduc o . 8,9 Assuming he sou ce is e y a om he sc een, hen he inciden wa e can be app oxima ed as a plane wa e. In gene al, he ansmi ed field can be expanded as an infini y sum o plane wa es. Whene e he uni cell is much smalle han he wa eleng h, as usual o me asu aces, only a plane wa e emains a om he sc een. By defini ion, he inciden and ansmi - ed elec ic fields can be connec ed one ano he h ough E a ¼ Einc o he o iginal sc een and E0 a ¼ 0E0 inc o he complemen a y sc een, whe e  and  0a e he so-called ansmission ma ices. Simila ly, eflec ion ma ices could be defined by E e ¼ Einc and E0 e ¼ 0E0 inc. They a e jus 2 2 squa e ma ices, because hey a e linking fields ha a e ans e se o he z-axis and aking in o accoun bo h co-pola and c oss-pola e ec s (i.e., hey a e, in gene al, non- diagonal ma ices). Fu he mo e, as a as wa es a e app oxima ed by plane wa es, he magne ic fields can easily be ob ained om he elec- ic field by using he well-known o mula B¼ð1=kcÞkE,whe ek is he wa e ec o and kis he co esponding wa enumbe . Al e na i ely, when kis pa allel o he z-axis, his o mula can also be exp essed as ollows: cBinc cB a cB e cB0 inc cB0 a cB0 e 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ¼01 10 !  Einc E a E e E0 inc E0 a E0 e 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ¼  R Einc E a E e E0 inc E0 a E0 e 8 > > > > > > > > > < > > > > > > > > > : 9 > > > > > > > > > = > > > > > > > > > ; ;(2) whe e   R ep esen s a 90coun e clockwise o a ion a ound he z-axis. F om (1) and (2), hedefini ionso  and  0in oduced abo e and wi h he 2 2 iden i y ma ix  1, i is a s aigh o wa d ask o demon- s a e he ollowing co olla y:  þ  R 0  R1¼  1:(3) I is wo h no ing ha , al hough a simila esul has been epo ed in many pape s in he scala o m þ 0¼1, his ma ix o m is mo e gene al because i also accoun s o cases p esen ing c oss-pola iza ion e ec s. In p inciple, he ansmission h ough one sc een could be guessed om he ansmission o i s complemen a y sc een by using (3). Howe e , le us emind ha (3) is only alid o sc eens o negligi- ble hickness and made o pe ec conduc o , which, clea ly, is no he case o plasmonic me asu aces. Then, we need o eplace i wi h ano he equa ion sui able o complemen a y plasmonic me asu aces. Now, le us conside he s uc u e shown in Fig. 1(a), which is a wo-dimensional (2D) piecewise homogeneous s uc u e filled wi h se e al media wi h iso opic ela i e pe mi i i ies ei; so, he s uc- u e is cha ac e ized by he ela i e pe mi i i y piecewise unc ion eðx;yÞ. I all egions a e filled wi h high pe mi i i ies hen, due o he con inui y o he no mal componen o eEon he bounda y, he no mal componen o Ecan be neglec ed inside he me asu ace, and hus, Eðx;yÞ¼Exðx;yÞ^ xþEyðx;yÞ^ y. Conside ing he uni cell is much smalle han he wa eleng h and i s hickness is also smalle han he skin dep h o plasmonic egions, i is also possible o use he elec oquasis a ic (EQS) app oxima ion: 24,25 he magne ic induc- ion in Fa aday’s law can be neglec ed o low equencies while he displacemen cu en densi y in he Ampe` e–Maxwell’s law is s ill kep because he low equencies a e compensa ed wi h he high al- ues o pe mi i i y. In his way, he p oblem is simplified o a 2D p oblem whose elec ic field sa isfies $ ðeEÞ¼0 and $ E¼0, whe e $ ¼^ x@=@xþ^ y@=@y. In o de o define he complemen a y p oblem, le us eplace he pe mi i i y eo he o iginal s uc u e wi h a new complemen a y pe mi i i y, as shown in Fig. 1(b), defined by e0ðx;yÞ¼ C1 eðx;yÞ;(4) whe e C 1 is an a bi a y cons an . I is wo h no ing ha his defini ion o “complemen a i y” includes he con en ional one as a pa icula case: when only wo ma e ials a e used his cons an can be chosen as C1¼e1e2, which means jus an in e change o ma e ials. Unde his ans o ma ion, he fields inside he complemen a y s uc u e can be ob ained om hose o he o iginal p oblem as ollows: E0¼C2e^ zE¼C2e   RE;(5) whe e C 2 is an a bi a y cons an associa ed wi h he eedom o scal- ing solu ions om linea equa ions. I can be s aigh o wa dly dem- ons a ed ha $ ðe0E0Þ¼0and$ E0¼0, by jus eplacing (4) and (5).This esul includesasapa icula case he esul sp e iously epo ed in Re s. 26–28 on he e ec i e conduc i i y o 2D wo phase composi es, which can be ob ained by simply eplacing he ela i e pe mi i i y eby i =ðe0xÞ,whe e is he spa ial dis ibu ion o conduc i i y. FIG. 1. Illus a ion o he uni cells o wo complemen a y me asu aces: he “o iginal” one (a) and he “complemen a y” one (b). Complemen a y pe mi i i ies and elec ic fields a e ela ed o Eqs. (4) and (5), espec i ely. Applied Physics Le e s ARTICLE sci a ion.o g/jou nal/apl Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-2 Published unde an exclusi e license by AIP Publishing 04 June 2025 14:00:05 In o de o desc ibe he esponse o he s uc u e om a mac o- scopic poin o iew, le us define he a e age elec ic field Ea e as well as he su ace cu en densi y (coming om he displacemen cu en ) Jsinside he uni cell (u.c.) h ough he in eg als Ea e ¼1 Aððu:c: Eðx;yÞdx dy;(6) Js¼ixe0h Aððu:c: eðx;yÞEðx;yÞdx dy;(7) whe e Ais he a ea o he uni cell, his he hickness o he sample, and ha monic a ia ion wi h ime eix has been assumed. We can now define he su ace impedance o he s uc u e,   Zs,as hema ix ha ela es he ec o s in (6) and (7) in he ollowing way: Ea e ¼Ea e;x Ea e;y  ¼Zs;xx Zs;xy Zs;yx Zs;yy  Js;x Js;y  ¼  ZsJs:(8) By applying iden ical defini ions (6) and (7) in he complemen a y me asu ace, and using (4) and (5), i is eadily shown ha he comple- men a y a e age elec ic field E0 a e and he complemen a y su ace cu - en densi y J0 sa e ela ed o he o iginal ones as E0 a e ¼ C2 ixe0h   RJs;(9) J0 s¼ixe0hC 1C2   REa e:(10) Ye , E0 a e and J0 sa e ela ed by he complemen a y su ace impedance   Z0 sas: E0 a e ¼  Z0 sJ0 s, hen, when we pu oge he his defini ion wi h (8)–(10), we di ec ly ela e bo h su ace impedances. Wi h he acuum wa e numbe k¼xffiffiffiffiffiffiffiffiffi e0l0 pand he acuum impedance Z0¼ffiffiffiffiffiffiffiffiffiffiffi l0=e0 p, he final esul is   Z0 s  R  Zs  R1¼Z2 0 4K   1;(11) whe e K¼h2k2C1=4:(12) I ecalls a p e ious heo em epo ed in Re s. 26–28 o he e ec i e conduc i i y o 2D la ices o conduc i e cylinde s, which can be deduced om (11) as a pa icula case by using C1¼e1e2and ei i=ðixe0Þ. I is also closely ela ed o he p ope y ZZ 0¼Z2 0=4 epo ed by Booke 29 and Deschamps 30 o pe ec ly conduc ing complemen a y sc eens, which can be ob ained om (11) when K¼1, and he o diagonal componen s o he su ace impedance ma ix a e ze o. Now, le us use s anda d bounda y condi ions a he me asu ace o sol e he ansmission ma ix om he su ace impedance ma ix. Fi s , by imposing he con inui y o he angen ial componen o he elec ic field hen Einc þE e ¼Ea e ¼E a,whichisequi alen o   1þ ¼ . Second, due o he ound su ace elec ic cu en , he mag- ne ic field mus sa is y he ollowing discon inui y ela ion: ^ zðBinc þB e B aÞ¼l0Js, which is equi alen o   1   ¼Z0   Z1 s . In his way, we ha e go a sys em o wo linea equa ions wi h wo unknown ma ices,   and  , ha can be easily sol ed o ge he ans- mission ma ix  ¼  1þZ0 2   Z1 s  1 :(13) Recip ocally, he su ace impedance is   Zs¼ðZ0=2Þ½ 1  11.Since i is an a bi a y choice wha is he o iginal o complemen a y sc een, i is clea ha we can w i e a ela ion analogous o (13) o  0and   Z0 s. The e o e, by using (11), we can es ablish a di ec ela ion be ween bo h ansmission ma ices, which is  0¼  R1  1 ½   1ð1KÞ  1  R:(14) I ep oduces he co olla y p e iously ob ained om con en ional Babine ’s p inciple, exp essed in (3),onlywhenK¼1(see hesupple- men a y ma e ial o examples wi h K1) bu , in gene al, K6¼ 1 and equency dependen as indica ed in (12). The e o e, Eq. (14) can be conside ed as an ex ension o Babine ’s p inciple o complemen- a y plasmonic me asu aces. I is wo h o emind, howe e , ha his o mula was ob ained unde he assump ion o he EQS limi , which is alid o sc eens wi h subwa eleng h pe iodici y, low a ia ion o fields along he z-axis inside he me asu aces, and high pe mi i i ies. Since hese cons ains a e usually ulfilled by plasmonic me asu aces ope a - ing a op ical equencies, his heo y is expec ed o be use ul o he analysis o hese s uc u es. To alida e ou heo y, h ee couples o complemen a y plas- monic me asu aces using di e en shapes will be nume ically simu- la ed. We will ocus ou a en ion in he in a ed ange o equencies. Fo all cases, we will use silicon (Si) and sil e (Ag) no only because hey a e ex ensi ely used in plasmonics bu also because hey bo h ha e high pe mi i i y in he in a ed ange and, a he same ime, hey p o ide a big con as o pe mi i i y, which migh be desi able o designing complemen a y s uc u es. The ela i e pe mi i i y o Si is almos cons an wi hin he conside ed equency ange, wi h a negli- gible imagina y pa , and can be app oxima ed as eSi 11:9. 31–33 On he o he hand, he pe mi i i y o Ag depends s ongly on he e- quency wi h a D ude–Lo en z beha io desc ibed by eAg ¼1x2 p= ðx2þixcÞ, whe e he plasma equency is xp¼1:375 1016 ad/s and he collision equency is c¼3:12 1013 s1; hese pa ame e s ha e been ob ained om Re . 34. As a fi s example, we will analyze he complemen a y sc eens shown in Figs. 2(a) and 2(b),whicha e made o al e na ing Si and Ag ba s. Pe iodic bounda y condi ions a e used a he edges o he s uc u es. A bi a ily, (a) may co espond wi h he o iginal p oblem while (b) wi h he complemen a y one. Following he same a gumen s o Re . 6, o y-pola ized (x-pola ized) inciden wa es, i can be in e p e ed as a plana nanoci cui wi h se ies (pa allel) connec ions o nanoinduc o s ep esen ing he Ag egions (nega i e pe mi i i y below he plasma equency) and nanocapaci- o s ep esen ing he Si egions (posi i e pe mi i i y). By passing om he o iginal o he complemen a y s uc u e, each egion swi ches om he nanoinduc o o he nanocapaci o and ice e sa. The com- plemen a y p oblem is defined no only by he in e change o ma e i- als bu also by he 90 o a ion o he pola iza ion s a e o he inciden wa e, which means ha se ies (pa allel) connec ions swi ch o a pa allel (se ies) connec ion in he complemen a y p oblem. Unde hese assump ions, i can be shown ha (11) is sa isfied. In ac , we ha e nume ically compu ed he ansmission coe ficien s h ough bo h sc eens by using he comme cial so wa e CST Mic owa e S udio om 50 o 300 THz o , equi alen ly, om 1 o 6lm. The plo s o Applied Physics Le e s ARTICLE sci a ion.o g/jou nal/apl Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-3 Published unde an exclusi e license by AIP Publishing 04 June 2025 14:00:05 Fig. 2 show simula ed ansmission coe ficien s (solid lines) as well as he cu es de i ed om (3) (dashed lines) and (14) (do ed lines) by using he nume ical esul o he complemen a y sc een. A e y good ag eemen be ween ou heo y, i.e., (14), and he elec omagne ic sim- ula ions has been ound while he esul s coming om con en ional Babine ’s p inciple, i.e., (3), show a significan de ia ion om he sim- ula ed esul s. Al hough Fig. 2(a) shows a big de ia ion in he phase abo e 250 THz, i is no ac ually ele an i we conside ha he co e- sponding magni ude is nea ze o. (The phase is indefini e a ze o.) I is also wo h no ing ha , due o he mi o symme ies o he s uc u e espec o he x-plane and y-plane, he o diagonal elemen s o he ansmission ma ices cancel ou and hen only he diagonal compo- nen s a e depic ed in Fig. 2. The esponse o he o me s uc u e was poo ly dispe si e and dissipa i e. To check ou heo y o s uc u es p esen ing s ong dis- pe sion and dissipa ion, which is he common si ua ion o me asu a- ces as well as me ama e ials based on esona ing pa icles, he ansmission coe ficien s h ough a sc een made o spli ings and i s complemen a y sc een ha e been compu ed in he ange om 15 o 65 THz. Thei uni cells and hei geome ical pa ame e s a e ully specified in Fig. 3. Fo (a), a spli ing o Ag is embedded in a laye o Si while ma e ials appea in e changed in (b). The plo s o Fig. 3 show he simula ed ansmission coe ficien s (solid lines) as well as he cu es de i ed om (3) (dashed lines) and (14) (do ed lines). I can be app ecia ed how he cu es ob ained wi h ou heo y by using (14) ag ee wi h he simula ions much be e han (3), which comes om con en ional Babine ’s p inciple. Once again, we can a gue ha o diagonal componen s o  , i.e., xy and yx , a e canceled ou because o he mi o symme y espec o he y-plane. As a las example, in o de o demons a e he alidi y o ou he- o y o cases ha p esen c oss-pola iza ion e ec s, we decided o o a e he p e ious spli ings by 45as shown a he op o Fig. 4.This figu e also shows he simula ed esul s (solid lines) as well as he cu es de i ed om (3) (dashed lines) and (14) (do ed lines) wi hin he ange be ween 15 and 65 THz. The diagonal componen s o he ans- mission ma ix xx and yy exac ly ma ch each o he because o he mi - o symme y espec o he diagonal di ec ion. Howe e , nei he he x-plane no he y-plane a e mi o planes, hus, in his case, a non- ze o esponse is obse ed o he o diagonal componen s xy and yx which, by he way, a e also equal one each o he . As in he p e ious FIG. 3. T ansmission coe ficien s (magni ude and phase) h ough wo di e en spli ing me asu aces made o sil e (Ag) and silicon (Si). The geome ical pa ame e s a e a¼250 nm, 0¼100 nm, g¼10 nm, w¼30 nm, and h¼25 nm. Plo s show simula ed esul s (solid lines) as well as cu es de i ed om (3) (dashed lines) and (14) (do ed lines). FIG. 4. T ansmission coe ficien s (magni ude and phase) h ough wo di e en spli ing me asu aces made o sil e (Ag) and silicon (Si), whe e he spli ings a e o i- en ed a 45 espec o he la ice axes. The geome ical pa ame e s a e a¼250 nm, 0¼100 nm, g¼10 nm, w¼30 nm, and h¼25 nm. Plo s show sim- ula ed esul s (solid lines) as well as cu es de i ed om (3) (dashed lines) and (14) (do ed lines). FIG. 2. T ansmission coe ficien s (magni ude and phase) h ough wo di e en pa - allel s ip g a ings made o sil e (Ag) and silicon (Si). Thei uni cells a e shown on op whe e he geome ical pa ame e s a e w1¼50 nm, w2¼10 nm, and h¼25 nm. Plo s show simula ed esul s (solid lines) as well as cu es de i ed om (3) (dashed lines) and (14) (do ed lines). Applied Physics Le e s ARTICLE sci a ion.o g/jou nal/apl Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-4 Published unde an exclusi e license by AIP Publishing 04 June 2025 14:00:05 examples, ou heo y summa ized in (14) app oaches he simula ions be e han (3) (con en ional Babine ’s p inciple). In conclusion, we ha e ex ensi ely analyzed he concep o com- plemen a i y and he applicabili y o Babine ’s p inciple o plasmonic me asu aces. The defini ion o complemen a y sc eens is gene alized in such a way ha he con en ional defini ion, in ol ing only he in e change o wo ma e ials, is con ained as a pa icula case. This analysis leads o an ex ension o Babine ’s p inciple summa ized in (14). We nume ically demons a ed ha ou heo y ep esen s a sub- s an ial imp o emen o h ee di e en geome ies. Aside om ha , we eel ha i is qui e gene al and can be applied o o he complemen- a y plasmonic me asu aces ope a ing in he op ical ange. Fu he mo e, he ela ion be ween complemen a y impedances shown in (11) may find applica ions o designing complemen a y nanoci - cui s and nanoan ennas ope a ing in he in a ed ange. This will be explo ed in an upcoming wo k. See he supplemen a y ma e ial o in o ma ion ela ed o he chance o eco e ing he alidi y o con en ional Babine ’s p inciple o o mula (3) by choosing a sui able hickness o which K1ina wide ange o equencies. This wo k has been suppo ed by he Minis e io de Ciencia e Inno aci on wi h EU FEDER Funds (P ojec No. TEC2017-84724- P), by he Colombian Go e nmen h ough COLCIENCIAS (P ojec No. 1101-521-29389), and he Uni e sidad de San Buena en u a (P ojec No. CBI013-012-2018). DATA AVAILABILITY The da a ha suppo he findings o his s udy a e a ailable om he co esponding au ho upon easonable eques . REFERENCES 1 Y. Zhao and A. Alu, “Manipula ing ligh pola iza ion wi h ul a hin plasmonic me asu aces,” Phys. Re . B 84, 205428 (2011). 2 N. Meinze , W. L. Ba nes, and I. R. Hoope , “Plasmonic me a-a oms and me a- su aces,” Na . Pho onics 8, 889 (2014). 3 N. Enghe a, A. Saland ino, and A. Alu, “Ci cui elemen s a op ical equencies: Nanoinduc o s, nanocapaci o s, and nano esis o s,” Phys. Re . Le . 95, 095504 (2005). 4 M. Sil ei inha, A. Alu, J. Li, and N. Enghe a, “Nanoinsula o s and nanoconnec- o s o op ical nanoci cui s,” J. Appl. Phys. 103, 064305 (2008). 5 A. Alu and N. Enghe a, “All op ical me ama e ial ci cui boa d a he nano- scale,” Phys. Re . Le . 103, 143902 (2009). 6 Y. Sun, B. Edwa ds, A. Alu, and N. Enghe a, “Expe imen al ealiza ion o op ical lumped nanoci cui s a in a ed wa eleng hs,” Na . Ma e . 11, 208 (2012). 7 P. Biagioni, J.-S. Huang, and B. Hech , “Nanoan ennas o isible and in a ed adia ion,” Rep. P og. Phys. 75, 024402 (2012). 8 J. D. Jackson, Classical Elec odynamics (Wiley, 1999). 9 R. E. Collin, Field Theo y o Guided Wa es (Wiley-IEEE P ess, 1999). 10 F. Falcone, T. Lope egui, M. A. G. Laso, J. D. Baena, J. Bonache, M. Be ue e, R. Ma qu es, F. Ma  ın, and M. So olla, “Babine p inciple applied o he design o me asu aces and me ama e ials,” Phys. Re . Le . 93, 197401 (2004). 11 H.-T. Chen, J. F. O’Ha a, A. J. Taylo , R. D. A e i , C. Highs e e, M. Lee, and W. J. Padilla, “Complemen a y plana e ahe z me ama e ials,” Op . Exp ess 15, 1084 (2007). 12 T. Zen g a , T. P. Mey a h, A. Seidel, S. Kaise , H. Giessen, C. Rocks uhl, and F. Lede e , “Babine ’s p inciple o op ical equency me ama e ials and nano- an ennas,” Phys. Re . B 76, 033407 (2007). 13 C. Rocks uhl and F. Lede e , “Nega i e-index me ama e ials om nano- ape u es,” Phys. Re . B 76, 125426 (2007). 14 N. Liu, S. Kaise , and H. Giessen, “Magne oinduc i e and elec oinduc i e cou- pling in plasmonic me ama e ial molecules,” Ad . Ma e . 20, 4521 (2008). 15 L. Y. Wu, B. M. Ross, and L. P. Lee, “Op ical p ope ies o he c escen -shaped nanohole an enna,” Nano Le . 9, 1956 (2009). 16 N. Liu, T. Weiss, M. Mesch, L. Langgu h, U. Eigen hale , M. Hi sche , C. S€ onnichsen, and H. Giessen, “Plana me ama e ial analogue o elec omagne i- cally induced anspa ency o plasmonic sensing,” Nano Le . 10, 1103 (2010). 17 M. Hen schel, T. Weiss, S. Baghe i, and H. Giessen, “Babine o he hal : Coupling o solid and in e se plasmonic s uc u es,” Nano Le . 13, 4428 (2013). 18 F. Qin, L. Ding, F. Mon icone, C. C. Chum, J. Deng, S. Mei, S. Mei, Y. Li, J. Teng, M. Hong, S. Zhang, A. Al u, and C.-W. Qiu, “Hyb id bilaye plasmonic me asu ace e ficien ly manipula es isible ligh ,” Sci. Ad . 2, e1501168 (2016). 19 M. Schnell, A. Ga c ıa-E xa i, J. Aizpu ua, and R. Hillenb and, “Phase- esol ed mapping o he nea -field ec o and pola iza ion s a e in nanoscale an enna gaps,” Nano Le . 10, 3524 (2010). 20 A. Bi ze , A. O ne , H. Me bold, T. Feu e , and M. Wal he , “Te ahe z nea - field mic oscopy o complemen a y plana me ama e ials: Babine ’s p inciple,” Op . Exp ess 19, 2537 (2011). 21 M. Ho  ak, V. K  apek, M. H o n, A. Konecˇn a, F. Ligmaje , M. S € oge -Pollach, T.  Samo il, A. Pa  ak, Z.  Edes, O. Me elka, J. Babocky, and T.  Sikola, “Limi s o Babine ’s p inciple o solid and hollow plasmonic an ennas,” Sci. Rep. 9, 4004 (2019). 22 M. H o n, A. Konecˇn a, M. Ho  ak, T.  Sikola, and V. K  apek, “Plasmonic an en- nas wi h elec ic, magne ic, and elec omagne ic ho spo s based on Babine ’s p inciple,” Phys. Re . Appl. 13, 054045 (2020). 23 J. D. O iz, J. D. Baena, R. Ma qu es, A. N. Enemuo, J. Gollub, R. Akhmeche , B. Penko , C. Sa an os, and D. T. C ouse, “Babine ’s p inciple and sa u a ion o he esonance equency o scaled-down complemen a y me asu aces,” Appl. Phys. Le . 118, 221901 (2021). 24 H. A. Haus and J. R. Melche , Elec omagne ic Fields and Ene gy (P en ice- Hall, 1989). 25 J. La sson, “Elec omagne ics om a quasis a ic pe spec i e,” Am. J. Phys. 75, 230 (2006). 26 J. B. Kelle , “A heo em on he conduc i i y o a composi e medium,” J. Ma h. Phys. 5, 548 (1964). 27 A. M. Dykhne, “Conduc i i y o a wo-dimensional wo-phase sys em,” So . Phys. JETP 32, 63 (1971), a ailable a h p://je p. as. u/cgi-bin/e/index/e/32/1/p63?a=lis . 28 G. W. Mil on, The Theo y o Composi es (Camb idge Uni e si y P ess, Camb idge, 2002). 29 H. G. Booke , “Slo ae ials and hei ela ion o complemen a y wi e ae ials (Babine ’s p inciple),” J. Ins . Elec . Eng. Pa III A 93, 620–626 (1946). 30 G. A. Deschamps, “Impedance p ope ies o complemen a y mul i e minal pla- na s uc u es,” IRE T ans. An ennas P opag. 7, 371 (1959). 31 D. F. Edwa ds and E. Ochoa, “In a ed e ac i e indexes o silicon,” Appl. Op . 19, 4130 (1980). 32 H. H. Li, “Re ac i e index o silicon and ge manium and i s wa eleng h and empe a u e de i a i es,” J. Phys. Chem. Re . Da a 9, 561 (1980). 33 M. N. Polyanskiy, see h ps:// e ac i eindex.in o o “Re ac i e Index Da abase” (las accessed July 8, 2021). 34 L. J. Mendoza, D. Mu~ ne  on, D. Schinca, and L. B. Sca a di, “De e mina ion o plasma equency, damping cons an , and size dis ibu ion om he complex dielec ic unc ion o noble me al nanopa icles,” J. Appl. Phys. 116, 233105 (2014). Applied Physics Le e s ARTICLE sci a ion.o g/jou nal/apl Appl. Phys. Le . 119, 161103 (2021); doi: 10.1063/5.0065724 119, 161103-5 Published unde an exclusi e license by AIP Publishing 04 June 2025 14:00:05