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.
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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,
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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
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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)
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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
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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
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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,
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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-
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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)
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