Me ama e ial uning by manipula ion o nea - ield in e ac ion
Da id A. Powell,1,*Mikhail Lapine,2,1 Maxim V. Go kuno ,3Ilya V. Shad i o ,1and Yu i S. Ki sha 1
1Nonlinea Physics Cen e, Resea ch School o Physics and Enginee ing, Aus alian Na ional Uni e si y, Canbe a,
Aus alian Capi al Te i o y 0200, Aus alia
2Depa men Elec onics and Elec omagne ics, Facul y o Physics, Uni e si y o Se ille, A da. Reina Me cedes s/n, 41015 Se ille, Spain
3A. V. Shubniko Ins i u e o C ys allog aphy, Russian Academy o Sciences, Lenin A e. 59, 119333 Moscow, Russia
共Recei ed 6 Decembe 2009; e ised manusc ip ecei ed 14 June 2010; published 19 Oc obe 2010兲
We analyze he nea - ield in e ac ion be ween he esonan subwa eleng h elemen s o a me ama e ial and
p esen a me hod o calcula e he elec ic and magne ic in e ac ion coe icien s. We show ha by adjus ing he
ela i e con igu a ion o he neighbo ing spli ing esona o s i becomes possible o manipula e his nea - ield
in e ac ion, and hus une he esponse o me ama e ials. We use he esul s o his analysis o explain he
expe imen ally obse ed uning o mic owa e me ama e ials.
DOI: 10.1103/PhysRe B.82.155128 PACS numbe 共s兲: 41.20.Jb, 78.70.Gq, 42.70.Qs
I. INTRODUCTION
Me ama e ials, which a e ypically egula a ays o sub-
wa eleng h esonan pa icles, o e us a new deg ee o ee-
dom in con olling he elec omagne ic esponse o ma e .
Thus we a e no longe comple ely cons ained by he p op-
e ies o exis ing ma e ials bu can ailo he esponse in an
almos a bi a y ashion, o example, achie ing e y high,1
e y low,2and nega i e3 alues o e ac i e index, pe mi -
i i y and/o pe meabili y. Because o he inhe en ly s ong
dispe sion o esonan me ama e ials, hey mus be modi ied
in o de o ope a e in a di e en equency band. The e o e,
he e is a signi ican push o ha e a u he deg ee o con ol
o e hese ma e ials— unabili y o hei esponse.
Fo una ely, he enginee ed na u e o me ama e ials al-
lows hei p ope ies o be con olled ex e nally, ei he by
dynamically modi ying hei s uc u e o by adding some
nonlinea inclusion and con olling wi h ex e nal ields.4Ex-
amples o he la e app oach include he in oduc ion o a -
ac o diodes,5 e oelec ics6and pho oconduc i e
semiconduc o s.7On he o he hand, e en wi hou eso ing
o such exo ic 共and o en lossy兲cons i uen s, he e is a g ea
deal o eedom o manipula e he s uc u e i sel , and his is
he app oach we ake he e. We conside speci ically he spli
ing esona o 共SRR兲as one o he mos impo an me ama-
e ial elemen s, no ing ha while he ine de ails o nea - ield
in e ac ion a e s uc u ally speci ic, ou app oach can be ap-
plied o a wide a ie y o s uc u es.
An analy ical model o he magne ic esponse o a sub-
wa eleng h a ay o iden ically o ien ed wi e loops loaded
wi h a capaci ance8 akes in o accoun he mu ual in e ac ion
o all he elemen s in he la ice, which is essen ial o de i -
ing he e ec i e pe meabili y co ec ly. Al hough ha analy-
sis is limi ed o he quasis a ic case accoun ing only o mag-
ne ic nea - ield in e ac ions, i is c ucial o e ealing he
consequence o la ice changes. These end o be o e looked
by o he wise igo ous app oaches which include spa ial dis-
pe sion bu de elop Lo en z local ield app oaches, based on
nea es -neighbo in e ac ion9o poin -dipole
app oxima ion.10
In pa icula , i was poin ed ou in Re . 8 ha he esonan
equency o he me ama e ial pe meabili y can be al e ed by
a ying he la ice cons an s wi hou changing he s uc u al
uni s. This scheme is illus a ed in Fig. 1共a兲and has been
e i ied by expe imen s in Re . 11. Howe e , a p ac ical con-
sequence o his change in la ice cons an is ha he sample
size also changes co espondingly. Mo e ecen ly, an al e na-
i e app oach was sugges ed in Re . 12: in oducing a shi
be ween laye s in o de o c ea e a monoclinic la ice wi h
he shi inc easing linea ly be ween laye s, as shown in Fig.
1共b兲. This con igu a ion keeps he densi y o elemen s wi hin
he me ama e ial cons an and can une he coupling be ween
neighbo ing pa icles o modi y he esponse o he comple e
me ama e ial. Howe e , o ini e-size samples, his shi in-
e i ably esul s in a signi ican change in sample shape.
Thus, o p ac ical pu poses, we ha e p oposed a supe la ice
ype o geome y, whe eby only e e y second laye is shi ed
by he same amoun , as shown in Fig. 1共c兲. This uning
scheme p o ed o be obus and allows signi ican manipula-
ion o he esonan equency wi h only a small change in
he sample geome y.12 The e o e, he sample geome y and
i s e ec i e p ope ies can be enginee ed almos indepen-
den ly o achie e he desi ed manipula ion o elec omag-
ne ic wa es.
Howe e , as we demons a e below, his s uc u al uning
o me ama e ials depends e y s ongly on he na u e o he
nea - ield in e ac ions. Since me ama e ial elemen s such as
spli ing esona o s a e usually no highly symme ic, he
ela i e o ien a ion o pa icles wi hin he la ice is o key
impo ance. This e ec is no desc ibed by exis ing ci cui
heo y models and can gi e ise o some su p ising expe i-
men al esul s, which we p esen he e.
(
a
)
(b)
(
c
)
FIG. 1. 共Colo online兲Se e al app oaches o modi y he la ice
o me ama e ial unabili y: 共a兲a change in he la ice cons an 共Re .
11兲,共b兲a con inuous shi o he laye s 共Re . 12兲, and 共c兲a supe -
la ice o al e na ing shi s o laye s 共Re . 12兲.
PHYSICAL REVIEW B 82, 155128 共2010兲
1098-0121/2010/82共15兲/155128共8兲©2010 The Ame ican Physical Socie y155128-1
In o de o unde s and he coupling mechanisms and how
hey a e a ec ed by he la ice shi , i is use ul o s a wi h
he simples geome y—a pai o spli ing esona o s. Se -
e al au ho s ha e conduc ed nume ical and expe imen al in-
es iga ions o coupling be ween me ama e ial elemen s o
di e en ypes o elemen s and ela i e o ien a ion be ween
hem, in mic owa e13,14 and op ical equency anges. 共See
he o e iew in Re . 15 and e e ences he ein, as well as
Re s. 16–21.兲A de ailed s udy has p e iously been unde -
aken on ailo ing he geome ic a angemen o a pai o
coupled one-dimensional SRR a ays o enginee he dispe -
sion cu es o magne oinduc i e wa es.22
As we ha e shown ecen ly,12 a ough quali a i e unde -
s anding o he s uc u al uning can be achie ed by using
ci cui heo y wi h pu ely induc i e coupling be ween SRRs.
Howe e , his app oxima ion ails o p o ide a quan i a i e
unde s anding o he me ama e ial uning in ques ion and,
mos impo an ly, does no explain he obse ed s ong in lu-
ence o he ela i e SRR o ien a ion. The e o e, below we
de elop a dis inc model based on he calcula ion o he un-
damen al mode o a single esona o . The knowledge o he
cu en and cha ge dis ibu ions wi hin he mode allows cal-
cula ing he coupling cons an s o a pai o spli ings. These
cons an s a e hen used o explain he expe imen ally ob-
se ed uning esponse o ou me ama e ial samples.
In Sec. II, we de elop ou app oach o calcula e he
me ama e ial coupling, including a discussion o he limi a-
ions o o he pu ely analy ical me hods. In Sec. III, we apply
ou app oach o he s udy o in e ac ion be ween a pai o
spli ing esona o s which a e shi ed la e ally ela i e o
each o he and explain quan i a i ely how he shi a ec s
he posi ion o he undamen al esonance. In Sec. IV,we
apply hese esul s o a bulk me ama e ial and iden i y he
mechanisms a wo k in he expe imen al uning o a me ama-
e ial slab in a wa eguide. Finally, Sec. Vconcludes he
pape wi h u he discussions and ou look.
II. NEAR-FIELD INTERACTION IN METAMATERIALS
Conside ing a single SRR, i is known13 ha i possesses a
disc e e se o eigenmodes 共s anding wa es兲wi h co e-
sponding eigen equencies. In an a bi a ily exci ed SRR, he
cu en s and cha ges can be ep esen ed as a supe posi ion o
he eigenmodes. The undamen al mode wi h he lowes e-
quency is ele an o he magne ic esonance in SRRs. On
he equency scale his mode is well isola ed om he
highe -o de modes, and we can es ic ou sel es o he
single-mode app oxima ion neglec ing he exci a ion o
highe modes.
The ime-dependen cha ge densi y
and cu en densi y
Jin a esonan elemen wi h exci ed undamen al mode can
be w i en in he mos gene al o m o a s anding wa e
共x, 兲=Q共 兲q共x兲,共1兲
J共x, 兲=I共 兲j共x兲,共2兲
whe e qand Jdesc ibe he cha ge and cu en dis ibu ions
in space. In SRRs, he a ia ion in he cu en dis ibu ion
ac oss he wid h o he conduc i e ack could be neglec ed,
howe e , o gene ali y ou app oach akes in o accoun he
comple e h ee-dimensional su ace-cu en dis ibu ion.
To sa is y he conse a ion o cha ge
ⵜ·J=−
共x, 兲
共3兲
we imply ha
I共 兲=Q
˙共 兲,共4兲
ⵜ·j共x兲=−q共x兲.共5兲
Thus i he cu en is known, i is easy o ind he cha ge
dis ibu ion and ice e sa. The mode p o ile ob ained nu-
me ically o ou SRR geome y is shown in Fig. 2. The
cu en jis symme ic and eaches i s maximum a he poin
opposi e o he gap. In acco dance wi h Eq. 共5兲, he cha ge
dis ibu ion q共x兲is an isymme ic and goes h ough ze o
whe e j共x兲is maximal. We see ha q共x兲 eaches i s maxi-
mum magni ude nea he gap.
In he single-mode app oxima ion, he dynamics o he
SRR can be ully desc ibed by he ime-dependen ampli ude
Q共 兲, and we may w i e he SRR Lag angian as a sum o
e ms quad a ic in Qand Q
˙
L=AQ
˙2−BQ2,共6兲
whe e Aand Ba e cons an s which will be discussed below.
Acco dingly, he SRR ene gy eads
E=Q
˙
L
Q
˙−L=AI2+BQ2共7兲
and is nicely sepa a ed in o induc i e 共magne ic兲and capaci-
i e 共elec ic兲pa s. Clea ly, o a passi e SRR we equi e
Aⱖ0, Bⱖ0.
The Lag angian equa ion o mo ion
d
d
L
Q
˙=
L
Q共8兲
yields ha he dynamics o a single SRR is desc ibed by he
oscilla o equa ion o he cha ge ampli ude
2
1
0
1
2
x ( m m )
3
2
1
0
1
2
3
y ( m m )
( a )
2
1
0
1
2
x ( m m )
3
2
1
0
1
2
3
( b )
FIG. 2. 共Colo online兲Nume ically calcula ed 共a兲cu en and
共b兲cha ge dis ibu ion ac oss an SRR a esonance.
POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲
155128-2
Q
¨共 兲+
0
2Q共 兲=0, 共9兲
and he undamen al mode esonance occu s a he equency
0=冑B/A.
No e ha in con as o he known modi ica ions o he
Lag angian o malism o me ama e ials 共see, e.g., Re . 23兲,
he e we do no ely on an equi alen ci cui model. Al hough
one migh iden i y he pa ame e s 2Aand 1/2Bas e ec i e
induc ance and capaci ance, espec i ely, below we e alua e
hem explici ly om he known undamen al mode shape. In
ac , he s ongly inhomogeneous mode p o ile 共see Fig. 2兲
sugges s ha i is unlikely ha he co ec alues o he pa-
ame e s would ag ee wi h hose calcula ed om a ci cui
analysis. Addi ionally, o ind he esonan equency, we do
no need o calcula e Aand Bexplici ly, only hei a io.
Fo he pu poses o ou analysis, i is su icien o con-
side he case o a pai o SRRs, and i will subsequen ly be
shown ha his explains all o he impo an ea u es ob-
se ed in ou expe imen s wi h a bulk me ama e ial. In his
case, he Lag angian can be w i en as a sum o he single
SRR Lag angians and coupling e ms, which we also w i e
as quad a ic in cu en s and cha ges
L=A共Q
˙1
2+Q
˙2
2+2
␣
Q
˙1Q
˙2兲−B共Q1
2+Q2
2+2

Q1Q2兲.
共10兲
The pa ame e s
␣
and

a e he dimensionless cons an s o
magne ic and elec ic nea - ield in e ac ion, espec i ely.
The co esponding Lag angian equa ions o mo ion yield
he sys em o ODEs o he ime-dependen ampli udes Q1,2
Q
¨1+
0
2Q1=−
␣
Q
¨2−

0
2Q2,共11兲
Q
¨2+
0
2Q2=−
␣
Q
¨1−

0
2Q1.共12兲
Sol ing hese equa ions one inds ha a pai o esona o s
exhibi s wo esonances: symme ic and an isymme ic. Fo
he symme ic esonance, Q1=Q2, which yields he esonan
equency
S=
0冑1+

1+
␣
共13兲
while he an isymme ic mode wi h Q1=−Q2has he
equency
AS =
0冑1−

1−
␣
.共14兲
The desc ibed esonance spli ing is well known in he
heo y o ha monic oscilla o s. Gene ally, b inging oge he
wo oscilla o s o he same esonan equency in oduces
coupling be ween hem, which esul s in spli ing in o wo
modes. Examples ha e been shown o SRR esonan e-
quency as a unc ion o some coupling pa ame e , e.g., mu-
ual o ien a ion24 o wis angle,25 and ypically demons a e
a spli ing o hyb idiza ion o modes.
As we see, he di ec ion and s eng h o he esonance
shi a e de e mined by he coupling cons an s
␣
and

.To
e alua e hem, we use he exp ession o he elec omagne ic
ene gy ollowing om Lag angian 共10兲:
E=A共I1
2+I2
2+2
␣
I1I2兲+B共Q1
2+Q2
2+2

Q1Q2兲.共15兲
The i s g oup o e ms gi es he magne ic ene gy and he
second g oup desc ibes he elec ic ene gy.
A possible ou e o calcula e
␣
and

is o app oxima e
he elec ic and magne ic esponse o each ing by a ew
e ms o he mul ipole expansion. The p oblem wi h his ap-
p oach is ha i is based on he assump ion ha he obse e
共i.e., he second SRR兲is a a la ge dis ance compa ed o he
dimensions o he sou ce. This equi emen is s ongly io-
la ed in ou me ama e ial samples, whe e he sepa a ion be-
ween ings is ac ually much smalle han he ou e ing di-
ame e . This is essen ial o achie ing s ong uning by la ice
manipula ion.
The e o e we ha e chosen o calcula e
␣
and

nume i-
cally om he known cha ge and cu en dis ibu ions, q共x兲
and j共x兲, o he undamen al mode in a single SRR. Indeed,
in he single-mode app oxima ion, he ene gy o a pai o
SRRs eads
E=Q1
2WE,11 +Q2
2WE,22 +2Q1Q2WE,12 +I1
2WH,11 +I2
2WH,22
+2I1I2WH,12,共16兲
whe e he pa ame e s
WE,mn =
冕
Vm
d3x
冕
Vn
d3x⬘q共x兲q共x⬘兲
4
⑀
0兩x−x⬘兩,共17兲
WH,mn =
冕
Vm
d3x
冕
Vn
d3x⬘
0j共x兲·j共x⬘兲
4
兩x−x⬘兩.共18兲
The in eg als can be easily e alua ed once he cha ge and
cu en dis ibu ions a e known. The in eg a ions o e xand
x⬘a e o e he same ing i m=no o e di e en ings
o he wise. Acco dingly, V1is a olume con aining only he
i s ing and V2is a olume con aining only he second. The
singula e ms a x=x⬘a e handled using he analy ical o -
mulas gi en in Re . 26.
Compa ing Eqs. 共15兲and 共16兲shows ha he coupling
pa ame e s can be e alua ed as
␣
=WH,12
WH,11
,

=WE,12
WE,11
.共19兲
Fo compa ison pu poses, when induc i e coupling is he
dominan in e ac ion mechanism be ween he SRRs in a
me ama e ial, we a e able o conside an a ay o spli ings
as an a ay o cu en loops wi h some mu ual induc ance
be ween hem. This app oach can hen be used o de ine he
e ec i e pe meabili y o a me ama e ial sample.8In pa icu-
la , o hin wi e loops wi h hei axes o ien ed in he same
di ec ion, he mu ual induc ance can be ound27 by nume ical
in eg a ion
Lnn⬘共 兲=
0 0
2
4
冕
0
2
冕
0
2
d
1d
2cos共
1−
2兲
⫻兵
2+z2+2 0
2关1 − cos共
1−
2兲兴
+2
0共cos
2− cos
1兲其−1/2,
whe e he dis ance ec o be ween he ing cen e s has
METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲
155128-3
been decomposed in o a adial componen
and axial com-
ponen z, 0is he ing adius, and
1and
2 ep esen he
angle abou each ing. We can use he calcula ed mu ual
induc ance o de ine an equi alen magne ic in e ac ion pa-
ame e
␣
L=Lnn⬘/L共20兲
共whe e Ldeno es he sel -induc ance o one elemen 兲which
should app oxima e he in e ac ion ene gy calcula ed om
Eq. 共19兲. Asymp o ically his in e ac ion decays as 1/ so in
a la ge a ay he nea es neighbo s p o ide he s onges con-
ibu ion bu do no necessa ily domina e o e all o he s.
Clea ly, his in e ac ion is highly aniso opic,22 being posi-
i e o ings on he same axis, bu nega i e o ings in he
same plane.
I is also possible o de elop equi alen ci cui models o
calcula e he elec ic in e ac ion be ween ings. Fo a pai o
coaxial ings, he o al in e ac ion can be eliably modeled as
a ci cula pa allel conduc o ansmission line, as o
b oadside-coupled 共bc兲spli ing esona o s28 o , al e na-
i ely, wi h an ex ended ci cui model accoun ing o he
dis ibu ed capaci ance and induc ance.29 Howe e , once we
in oduce some o se be ween he ings, hese app oaches
a e no applicable and so we do no conside hem he e.
III. TUNING INTERACTION BETWEEN A PAIR OF SPLIT
RING RESONATORS
Ha ing de eloped an app oach o calcula ing nea - ield
in e ac ion be ween a pai o ings, we now apply i o a
canonical sys em which has he basic p ope ies o ou ex-
pe imen al a angemen . We conside a pai o SRRs, ei he
iden ically 关gap- o-gap 共g g兲兴 o ien ed o o a ed by 180°
wi h espec o each o he 关b oadside-coupled 共bc兲兴 and sub-
jec o a la e al o se
␦
a. The geome y and inciden pola -
iza ion a e shown in Fig. 3. The ings ha e a e age adius
0=2.25 mm, ack wid h o 0.5 mm, me al hickness o 0.03
mm, gap wid h o 1 mm, and a e sepa a ed in he ans e se
di ec ion by 1.5 mm. The esul ing esonan equency is
10.6 GHz.
We plo he in e ac ion ene gy calcula ed om Eq. 共19兲in
Fig. 4 o di e en o se s be ween he ings. I can be seen
ha he elec ic coupling pa ame e

is nea ly symme ic
be ween he wo con igu a ions. This can be unde s ood om
Fig. 2, whe e we see ha he cha ge dis ibu ion has a s ong
dipole componen o ien ed in he xdi ec ion. Fo he sym-
me ic mode o he b oadside-coupled o ien a ion he cha ges
accumula ed on he closes sides o he SRRs ha e opposi e
signs, he o al-cha ge dis ibu ion has he na u e o a pai o
an ipa allel dipoles, hus WE,12⬍0, and

is also nega i e. In
con as , in he gap- o-gap o ien a ion, he closes cha ges a e
o he same sign, he o al-cha ge dis ibu ion becomes like a
pai o pa allel dipoles and he pa ame e

⬎0.
A an o se o abou one ing adius 共
␦
a⬇ 0兲, he cha ge
on one ing is app oxima ely equidis an om bo h posi i e
and nega i e cha ges on he opposi e ing, hus he ne cou-
pling passes h ough ze o. A la ge o se s, he elec ic cou-
pling changes i s sign bu emains smalle , since only he
nea es hal es o he SRRs a e e ec i ely in e ac ing wi h
his in e ac ion decaying owa d ze o as he o se inc eases.
The magne ic in e ac ion ene gy is also qui e di e en o
he wo o ien a ions wi h he magne ic in e ac ion
␣
Lcalcu-
la ed by Eq. 共20兲lying be ween he gap- o-gap and b oadside
cu es. Fo he b oadside-coupled case, he si ua ion is quali-
a i ely simila o he analy ical esul
␣
L. A low o se , he
magne ic ield o one ing cu s h ough he o he ing in he
same di ec ion o he su ace no mal, hus ein o cing he
magne ic ield and inc easing he o al ene gy. As he o se is
inc eased, he si ua ion g adually shi s o become like a pai
o loops in he same plane, whe e he ield om one ing cu s
h ough he o he in he opposi e di ec ion wi h espec o he
su ace no mal. Hence,
␣
bc unde goes a change in sign.
Howe e , in compa ison o he ing wi h uni o m cu en , he
coupling is subs an ially mo e nega i e. This is due o he
cu en maxima being on opposi e sides o he ings, and
hence u he away om each o he .
Fo he gap- o-gap o ien a ion he magne ic in e ac ion is
much s onge o low
␦
a. This is due o he cu en maxima
being loca ed nea each o he which p oduces a s onge con-
ibu ion o he in eg al in WH,12 hus inc easing
␣
. As he
ings a e u he sepa a ed om each o he , he in e ac ion
ene gy educes, bu does no unde go a change in sign.We
can in ui i ely unde s and his by neglec ing he small con-
ibu ions o he cu en in he egion nea he gaps, hus we
e ec i ely ha e wo linea cu en elemen s in he same
plane which always in e ac wi h he same sign. Howe e ,
his balance is no uni e sal and is de e mined by he speci ic
geome y and pa ame e s. To check his, we s udied a geom-
e y wi h a e y small gap so ha he cu en dis ibu ion was
δa
(a) (b)
E
H
k
δa
FIG. 3. Geome y o he pai o ings 共a兲b oadside-coupled and
共b兲gap- o-gap o ien a ion.
0 . 2 0 . 1 0 . 0 0 . 1 0 . 2 0 . 3
N o
m a l i z e d I n e a c i o n e n e g y
0 . 0
0 . 5
1 . 0
1 . 5
2 . 0
2 . 5
3 . 0
δ a /
0
β
b c
α
b
c
β
g g
α
g g
α
L
FIG. 4. 共Colo online兲Magne ic 共
␣
兲and elec ic 共

兲coupling
pa ame e s o b oadside coupled 共bc兲and gap- o-gap 共g g兲o ien-
a ion o a pai o shi ed SRRs. The magne ic in e ac ion calcula ed
om he mu ual induc ance is gi en as
␣
L.
POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲
155128-4
much mo e homogeneous wi h lowe esonance equency.
In his simula ion 共no shown兲 he magne ic coupling did
change sign and bo h alues o
␣
con e ged closely o
␣
L.
In o de o e i y ha he calcula ed coupling co ec ly
desc ibes he equency spli ing o his sys em, we compa e
he equency shi p edic ed by Eqs. 共13兲and 共14兲wi h ha
ob ained om he ull nume ic simula ions. Fo consis ency
wi h ou in e ac ion-ene gy app oach we assume a homoge-
neous ee-space backg ound. We use he equency domain
sol e o he comme cial so wa e package CST MICROWAVE
STUDIO 共Re . 30兲 o model a pai o ings, in a uni cell wi h
pe iodic bounda y condi ions in he di ec ions ans e se o
he p opaga ion di ec ion. This pe iodic sys em enables us o
de ine a ansmission coe icien , and he bounda ies a e 10
mm om he ings. This alue is chosen o be la ge enough
so ha he e is no signi ican in e ac ion wi h pe iodic neigh-
bo s, ye small enough o a oid signi ican sca e ing in o
highe o de di ac ion modes. Thus we can conside his he
limi ing case o a highly dilu e me ama e ial slab.
The ansmission spec um as a unc ion o o se is plo -
ed in Fig. 5. We see ha all he impo an ea u es o he
mode spli ing a e ep esen ed co ec ly by ou single-mode
heo y o coupled SRRs. Expansion o Eqs. 共13兲and 共14兲 o
small coupling p edic s equency spli ing o ⌬
=⫾共

−
␣
兲/2, hence he cu es a e app oxima ely symme ic abou
0.
The s ong spli ing obse ed o he b oadside-coupled
o ien a ion is due o he opposi e signs o he elec ic and
magne ic coupling. A
␦
a/ 0⬇1.1 bo h
␣
bc and

bc change
signs, hence he c ossing o he symme ic and an isymme -
ic modes is obse ed. In con as , we see ha o he gap-
o-gap o ien a ion o small o se
␣
g g and

g g a e o he
same sign, and hus hey ha e an opposing e ec , esul ing in
small equency spli ing. Since

dec eases much as e and
changes sign, he esul in maximum equency spli ing o
␦
a/ 0be ween 1.1 and 1.5. We no e ha in Re . 12 he eso-
nan equency based on
␣
Lwas compa ed wi h expe imen al
esul s o he b oadside-coupled o ien a ion and s ong dis-
ag eemen was ound.
I can be seen ha he calcula ed ansmission h ough he
cell exhibi s di e en dep hs o he esonance o he sym-
me ic and an isymme ic modes. This is due o he di e en
e iciency o coupling be ween he modes and he inciden
plane wa e. Fo ins ance, i is impossible o exci e he an i-
symme ic mode wi h a no mally inciden plane wa e o
␦
a=0, since bo h ings a e exci ed in phase. As he o se is
inc eased, some e a da ion be ween he ings occu s and ex-
ci a ion o he an isymme ic mode is allowed.
The e a e se e al easons o he small quan i a i e dis-
ag eemen be ween he exac calcula ions and hose based on
he calcula ed in e ac ion ene gy. Fi s , he minimum o
ansmission occu s a a equency sligh ly di e en om he
esonan equency, due o coupling e ec s 共impedance
ma ching兲be ween he inciden wa e and he ing. Second,
he e may be some small con ibu ion o highe SRR eigen-
modes due o pe u ba ion o he cha ge and cu en dis ibu-
ions. Thi d, he e may s ill be some small in luence o he
pe iodic bounda ies. Finally, ou de eloped ela ions neglec
e a da ion, which is s ic ly alid in he subwa eleng h
limi , whe eas he ou e adius o he ings is 0.18a
0.
Re a da ion has p e iously been shown o modi y he in e -
ac ion be ween SRRs h ough i s in luence on he dispe sion
o magne oinduc i e wa es in a ays.31–33
We emphasize ha ou app oach de eloped in Sec. II is
ad an ageous o e he di ec nume ical calcula ion. Fi s ,
once he mode p o ile is known, calcula ion o he equen-
cies in Fig. 5 akes app oxima ely 30 s on a single CPU
whe eas he di ec calcula ion o he ull spec um akes se -
e al hou s on a mul ico e machine. Second, we a e clea ly
able o demons a e he na u e o he coupling, which yields
insigh in o he uning beha io .
IV. TUNING INTERACTION IN A BULK METAMATERIAL
We now apply ou app oach o explain expe imen al e-
sul s o uning he esponse o a slab o me ama e ial. The
me ama e ial is ab ica ed using pho oli hog aphy o e ch
coppe acks on o FR4 p in ed ci cui boa d, using he same
geome y as in ou nume ical simula ion o a pai o ings.
The ab ica ed sample has 30 laye s, each wi h i e ings in
he p opaga ion di ec ion and is only one ing in heigh 共i.e.,
0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 1 . 0 5 1 . 1 0
0 . 0
0 . 5
1 . 0
1 .
5
2 . 0
2 . 5
3
. 0
δ a /
0
(
a ) B
o a
d s i d
e c o
u p l e
d o i
e
n a
i o n
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 .
7
0 . 8
0
. 9
1 . 0
0 . 8 5 0 . 9 0 0 . 9 5 1 . 0 0 1 . 0 5 1 . 1 0
ω / ω
0
0 . 0
0 . 5
1 . 0
1 . 5
2 . 0
2 . 5
3 . 0
δ a /
0
( b )
G a p o g a p o i e n a i o n
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
1 . 0
FIG. 5. 共Colo online兲Nume ical esul s. T ansmission spec um
o a pai o 共a兲b oadside-coupled and 共b兲gap- o-gap o ien ed
ings. Solid line:
S om Eq. 共13兲, b oken line:
AS om Eq. 共14兲
METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲
155128-5
a5⫻30⫻1 a ay兲. The longi udinal pe iod o he sample is
7 mm, he ans e se pe iod is dic a ed by he sample hick-
ness 共1.5 mm兲as he e is no spacing be ween boa ds. As wi h
he pai o ings, we ha e assembled slabs wi h wo ela i e
o ien a ions o he spli ings in adjacen planes—gap- o-gap
and b oadside-coupled. The sample is placed in he cen e o
a WR229 ec angula me allic wa eguide, wi h dimensions
58.17 mm ⫻29.08 mm, exci ed a i s dominan TE10
mode. We emo ed he in luence o he coaxial adap e s and
eeding wa eguide sec ions by pe o ming a h ough- e lec -
line calib a ion.34 In Fig. 6we show he expe imen ally ob-
ained ansmission h ough each slab o closely coupled
SRRs wi h he co esponding e lec ion shown in Fig. 7.
The shi o esonan equency shows good quali a i e
ag eemen wi h he esul s o a pai o ings p esen ed in
Sec. III wi h e y simila changes in he spec um obse ed
共no ing ha
␦
a/a=0.5 co esponds o
␦
a/ 0=1.56兲. How-
e e , o he gap- o-gap o ien a ion, nume ical simula ion o
a sys em o wo boa ds wi h i e ings each, and pe iodic
bounda ies in bo h ans e se di ec ions 共no shown兲,is
quali a i ely simila o he expe imen ally obse ed esul s
bu quan i a i ely highly inaccu a e. The eason u ns ou o
be he loss o symme y when he sys em is placed inside he
wa eguide because he uppe and lowe wa eguide walls do
no co espond o pe iodic bounda ies bu ins ead ep esen
planes o mi o e lec ion. The e o e, his sys em mus be
desc ibed as ha ing a supe la ice a angemen in he e ical
as well as ho izon al planes wi h each supe cell consis ing o
ou SRRs. This cell has al e na ing o ien a ion o he SRRs
in he e ical di ec ion co esponding o he planes o mi o
symme y, as shown in Fig. 8共b兲. Once his uni cell is aken
in o accoun , nume ical simula ions a e in a good ag eemen
wi h he expe imen 关Fig. 9共b兲兴.
Na u ally, nume ical simula ions o he b oadside-
coupled o ien a ion also ag ee well wi h he expe imen 关Fig.
9共a兲兴. In his case a e y simila esul is p o ided wi h
simple pe iodic bounda y condi ions 共no shown兲. Fo his
o ien a ion he supe la ice e ec i ely o med by he wa e-
guide shown in Fig. 8共a兲does no ha e an essen ially di e -
en symme y o he o iginal supe la ice.
We can conclude ha in bo h cases he dominan mode o
he slab co esponds o he dominan symme ic mode o a
pai o ings wi h a simila pa e n o esonan equency s
o se occu ing. The weake coupling o he modes o he
gap- o-gap o ien a ion is due o he shape o he symme ic
mode. I s magne ic ield has a la ge componen pa allel o
ha o he wa eguide mode, howe e , i s elec ic ield is
p ima ily longi udinal, in con as o he ans e se elec ic
ield o he wa eguide mode.
In Fig. 9共b兲, we see wo highe o de modes, which mos
likely co espond o he highe o de esonances obse ed in
he expe imen al esul s in Fig. 6共b兲. F om he nume ical
3
. 0
3
. 5
4
. 0
4 . 5
5
. 0
0 . 0
0 . 2
0
. 4
0
. 6
0 . 8
1 . 0
T a n s m i s s i o n
δ a / a
( a ) B o a d s i d e - c o u p l e d o i e n a i o n
0 . 0
0
. 1
0
. 2
0 . 3
0 . 4
0 . 5
3 . 0 3 . 5 4 . 0
4 . 5
5 . 0
( G H z )
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1
. 0
T a n s m i s s i o n
( b ) G a p - o - g a p o i e n a i o n
FIG. 6. 共Colo online兲Expe imen al ansmission while uning
␦
a o spli ing esona o slab in wa eguide, o 共a兲b oadside-
coupled and 共b兲gap- o-gap o ien a ion o adjacen laye s.
3
. 0
3
. 5
4
. 0
4 . 5
5
. 0
0 . 0
0 . 2
0
. 4
0
. 6
0 . 8
1 . 0
R e l e c i o n
δ a / a
( a ) B o a d s i d e - c o u p l e d o i e n a i o n
0 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
3 . 0 3 . 5 4 . 0
4 . 5
5 . 0
( G H z )
0 . 0
0 . 2
0 . 4
0 . 6
0 . 8
1
. 0
R e l e c i o n
( b ) G a p - o - g a p o i e n a i o n
FIG. 7. 共Colo online兲Expe imen al e lec ion while uning
␦
a
o spli ing esona o slab in wa eguide, o 共a兲b oadside-coupled
and 共b兲gap- o-gap o ien a ion o adjacen laye s.
(
a)
(
b
)
FIG. 8. 共Colo online兲Schema ic o he e ec i e supe la ice
geome y co esponding o he wa eguide measu emen o 共a兲
b oadside-coupled and 共b兲gap- o-gap o ien a ion. Dashed lines
show planes o e lec ion symme y and he shaded egion shows
he supe cell wi h ou SRRs.
POWELL e al. PHYSICAL REVIEW B 82, 155128 共2010兲
155128-6
simula ions we obse e ha he cu en dis ibu ions o hese
modes a e symme ic, hus hey co espond o highe -o de
modes o he me ama e ial slab and no o he an isymme ic
mode o a pai o ings. In con as , in Fig. 9共a兲 he e is a
weakly coupled an isymme ic mode, which we e i ied by
inspec ion o he cu en s. This mode also quali a i ely
ag ees wi h he co esponding mode o he pai o ings wi h
somewha weake coupling due o he inc eased misma ch o
he inciden wa eguide mode. This mode may co espond o
some o he smalle ea u es obse able in Fig. 6共a兲, how-
e e , due o he size o hese ea u es his canno be eliably
de e mined.
We do no conside o se s g ea e han 0.5a, since in an
in ini e la ice only shi s be ween 0 and 0.5 a e unique,
while in a ini e s uc u e, la ge shi s esul in e y i egula
bounda ies. No e ha in he simula ions we ha e neglec ed
he e ec o he mode p o ile o he ec angula wa eguide,
which would co espond o an e ec i e a ia ion in he angle
o incidence o he plane wa e as a unc ion o equency,
which can esul in a di e en esponse due o he aniso opy
and non-negligible spa ial dispe sion o he medium.10
Clea ly he coupling in he comple e la ice is much mo e
complica ed han in he simple wo- ing sys em, as he in e -
ac ions be ween a la ge numbe o ings mus be aken in o
accoun . In p inciple i is possible o ex end he analysis o
Sec. III o an a bi a y numbe o ings. Howe e , he quali-
a i e ag eemen be ween he expe imen al esul s and he
modeled pai o ings sugges s ha he phenomenology de-
eloped o he wo ings is gene ally applicable and leads o
co ec p edic ions.
Al hough an accu a e gene aliza ion o ou modeling ap-
p oach o a bulk sys em lies beyond he scope o his pape ,
i is clea ha he esul ing homogenized e ec i e me ama-
e ial pa ame e s will exhibi simila uning pa e n due o he
esonance shi . No e ha he in oduc ion o he e ec i e
pa ame e s is jus i ied when he a io o he uni -cell size o
he inciden wa eleng h is small. The e o e, when conside -
ing modi ica ions o he la ice which c ea e a supe la ice
s uc u e, he size o he supe cell should be smalle han he
wa eleng h. The e a e homogeniza ion app oaches in he li -
e a u e 共e.g., Re . 35兲which include unknown pa ame e s o
in e ac ion be ween esonan elemen s and ou semianaly i-
cal app oach would make an ideal ool o e alua ing hese
cons an s.
V. CONCLUSION
We ha e analyzed he nea - ield coupling wi hin me ama-
e ials, conside ing bo h he ela i e o ien a ion and he o -
se be ween he cen e s o wo neighbo ing esona o s. Using
a pai o spli ing esona o s as a simple model, we ha e
shown he coupling mechanisms a wo k in ou ecen ly p o-
posed uning scheme, based on he di ec calcula ion o he
in e ac ion ene gy. We ha e con i med ha hese mecha-
nisms can p edic quali a i ely he pe o mance o a ealis ic
me ama e ial s uc u e. This pa es a oad owa d a eliable
design and de elopmen o unable me ama e ials o a ious
applica ions.
We no e ha he speci ic geome y o he spli ings can
ha e a e y signi ican in luence on he quali a i e na u e o
he coupling, including cases which un coun e o ou in ui-
i e unde s anding o cu en loops in e ac ing magne ically.
Finally, we poin ou ha he app oach de eloped he e o
modeling nea - ield e ec s is pa icula ly p omising o
me ama e ials scaled down o ope a e a op ical equencies.
In he isible, he pa adigm o ideally conduc ing me al ails
and he a ea o applicabili y o ci cui models is a he lim-
i ed. In con as , he conside a ion in e ms o exci a ion and
in e ac ion o plasmonic s anding wa es will p o ide a clea
physical pic u e.
ACKNOWLEDGMENTS
The au ho s a e g a e ul o Lukas Jelinek and Rica do
Ma qués 共Uni e si y o Se ille兲 o help ul discussions. This
wo k was suppo ed by he Aus alian Resea ch Council.
M.L. acknowledges hospi ali y o Nonlinea Physics Cen e
and suppo o he Spanish Jun a de Andalusia unde P ojec
No. P06-TIC-01368. M.G. acknowledges suppo om he
Russian Academy o Sciences, BPS P og am “Physics o
new ma e ials and s uc u es.”
3 . 0 3 . 5 4 . 0
4 . 5
5 . 0
5 . 5
6 . 0
( G H z )
0 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
δ a / a
( a ) B o a d s i d e c o u p l e d o i e n a i o n
0
. 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
3 . 0 3 . 5 4 . 0
4 . 5
5 . 0
5 . 5
6 . 0
( G H z )
0 . 0
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
δ a / a
( b ) G a p o g a p o i e n a i o n
0 . 1
0 . 2
0 . 3
0 . 4
0 . 5
0 . 6
0 . 7
0 . 8
0 . 9
FIG. 9. 共Colo online兲Nume ically ob ained ansmission spec-
um o me ama e ial in wa eguide, 共a兲b oadside-coupled and 共b兲
gap- o-gap o ien a ion. The whi e line indica es expe imen ally ob-
ained esonan equencies.
METAMATERIAL TUNING BY MANIPULATION OF NEAR-…PHYSICAL REVIEW B 82, 155128 共2010兲
155128-7
*[email p o ec ed]
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