S uc u al unabili y in me ama e ials
Mikhail Lapine,1,2,a兲Da id Powell,1Maxim Go kuno ,3Ilya Shad i o ,1Rica do Ma qués,2
and Yu i 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
2Dep . 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, Leninski p osp. 59, 119333
Moscow, Russia
共Recei ed 14 July 2009; accep ed 2 Augus 2009; published online 27 Augus 2009兲
We p opose an e icien app oach o uning he ansmission cha ac e is ics o me ama e ials
h ough a con inuous adjus men o he la ice s uc u e and con i m i expe imen ally in he
mic owa e ange. The concep is a he gene al and applicable o a ious me ama e ials as long as
he e ec i e medium desc ip ion is alid. The demons a ed con inuous uning o a me ama e ial
esponse is highly desi able o a numbe o eme ging applica ions o me ama e ials, including
senso s, il e s, and swi ches, ealizable in a wide equency ange. © 2009 Ame ican Ins i u e o
Physics.关DOI: 10.1063/1.3211920兴
Me ama e ials a e p ominen o he excep ional oppo -
uni ies hey o e in ailo ing mac oscopic p ope ies
h ough app op ia e choice and a angemen o hei s uc-
u al elemen s.1,2In his way, i is no only possible o design
a me ama e ial o a equi ed pu pose, bu also o implemen
u he adjus men capabili ies a he le el o assembly. This
makes me ama e ials di e en om con en ional ma e ials
and opens exci ing oppo uni ies o mul i unc ionali y ia
unabili y.
Tunable me ama e ials imply he possibili y o con inu-
ously change hei p ope ies h ough an ex e nal in luence
o signal wi h he in insic mechanism o unabili y. The key
means o uning esonan me ama e ials, na u ally, lies in a -
ec ing he sys em so as o change he pa ame e s o he
esonance. As a consequence, he cha ac e is ics o me ama-
e ial can be a ied, enabling, o ins ance, unable ansmis-
sion.
The ini ial app oach o ealize unable me ama e ials
based on nonlinea p ope ies3has al eady been p o en
expe imen ally,4,5and u he me hods ha e been sugges ed,
e.g., based on he econ igu abili y o liquid c ys als.6How-
e e , such me hods become inc easingly di icul o imple-
men a highe equencies.
In his le e , we pu o wa d an app oach ha elies on
he s uc u al uning o he en i e me ama e ial. This concep
is independen o he speci ic ealiza ion as well as scalable
o any equency p o ided ha he mac oscopic equi emen s
o me ama e ials a e obse ed.
The gene al p inciple o he p oposed uning me hod is
clea h ough a simple analogy. Indeed, he p ope ies o
c ys als a e known o be de e mined by he na u e o he
a oms as well as by he geome y o he c ys al la ice, so in
na u al ma e ials he collec i e esponse o a oms de e mines
he o e all esponse o ex e nal ields.7In na u al ma e ials,
howe e , he possibili ies o une hei p ope ies dynami-
cally a e limi ed o na u ally a ailable c ys als and yield ela-
i ely weak e ec s, such as elec o/magne os ic ion, pho o-
e ac ion, e c.
In con as , me ama e ials o e a unique oppo uni y o
design and a y he s uc u e enabling a desi ed esponse
unc ion and a con enien mechanism o unabili y. Mo e
impo an ly, he ange o unabili y o a gi en p ope y can
be much b oade han in na u al ma e ials, as he la ice e -
ec s can be made much s onge h ough highe e iciency o
collec i e e ec s in he la ice, achie ed by an app op ia e
design.
To demons a e he e iciency o his app oach, we con-
side an aniso opic me ama e ial based on esonan elemen s
sui able o p o iding a i icial magne ism, such as spli - ing
esona o s o a ious kind, as shown in Fig. 1. Fo su i-
cien ly dense a ays, he in e ac ion be ween such elemen s
di e s conside ably om a dipole app oxima ion, and he
speci ic p ocedu e o calcula e he e ec i e pe meabili y
was de eloped ea lie .8The la e con e ges co ec ly o a
Clausius–Mosso i app oxima ion in he limi o a spa se la -
ice. Consequen ly, he e ec o mu ual coupling is enhanced
d ama ically as compa ed o con en ional ma e ials, and
he e o e i is pa icula ly sui able o demons a e he e i-
ciency o la ice uning.
Acco dingly, i all he cha ac e is ic dimensions 共la ice
cons an s and elemen size兲a e much smalle han he wa e-
leng h, we can desc ibe a egula la ice o such elemen s by
he esonan e ec i e pe meabili y,
a兲Au ho o whom co espondence should be add essed. Elec onic mail:
[email p o ec ed].
y
x
z
a
a
b
FIG. 1. 共Colo online兲Schema ic o he s agge ed la ice shi wi h a la e al
displacemen o e e y second me ama e ial laye .
APPLIED PHYSICS LETTERS 95, 084105 共2009兲
0003-6951/2009/95共8兲/084105/3/$25.00 © 2009 Ame ican Ins i u e o Physics95, 084105-1
09 June 2025 17:24:32
共
兲=1− A
2
2−
2+i⌫
共1兲
wi h he esonan equency
=
o
冉
L⌺
L+
o
S2
3L
冊
−1/2
共2兲
de e mined by he p ope ies o indi idual elemen s, such as
he esonance equency o a single elemen
o, hei geom-
e y 共which de ines sel -induc ance Land e ec i e c oss-
sec ion S兲, concen a ion
, as well as hei a angemen . The
la e e ec is de e mined by mu ual in e ac ion be ween he
elemen s, which in mos p ac ically ealizable cases is de-
ined by
L⌺=L+
o ⌺,共3兲
whe e he so-called la ice sum ⌺can be calcula ed o a
gi en geome y o elemen s and hei a angemen h ough
mu ual induc ance
兺
n⬘⫽n
Lnn⬘共
兲=−i
o ·⌺,共4兲
be ween all he elemen s in a physically small olume whe e
he a e age mac oscopic ield is e alua ed.8
No e ha he collec i e beha io o a numbe o ele-
men s in he la ice plays a c ucial ole, so ha in aniso opic
a ays mu ual in e ac ions canno be educed o he nea es
neighbo s app oxima ion as is easible in iso opic models
accoun ing o spa ial dispe sion.9
The mos s aigh o wa d la ice uning app oach is o
a y he la ice cons an b. We ha e shown8 ha he eso-
nance equency can be ema kably shi ed his way, and
con i med his wi h mic owa e expe imen s.10 Acco dingly,
a slab o me ama e ial can be uned be ween ansmission,
abso p ion and e lec ion back o ansmission. A clea dis-
ad an age o his me hod is ha a ying bimplies a signi i-
can change in he co esponding dimension o he me ama-
e ial, which is undesi able o applica ions.
He e we p opose ano he me hod o s uc u al uning, by
means o a pe iodic la e al displacemen o laye s in he xy
plane, so ha he esona o s become shi ed along x共y,o
bo h兲by a ac ion o he la ice cons an
␦
ape each b
dis ance om a e e ence laye wi h espec o he o iginal
posi ion. This dec eases he o e all mu ual induc ance in he
sys em 关Eq. 共3兲兴and leads o a g adual inc ease in esonan
equency, wi h a maximal e ec a chi ed o a displacemen
o 0.5a关see Fig. 2共a兲兴. Clea ly, u he shi is equi alen o
smalle shi alues un il he la ice exac ly ep oduces i sel
o he shi by a. As a consequence, he esonance o he
medium can be “mo ed” ac oss a signal equency, leading
o a d as ic change in ansmission cha ac e is ics 关see Fig.
2共b兲兴. I is clea ha o p ac ical applica ions i is no e en
necessa y o exploi he whole ange o la e al shi —in he
abo e example i is su icien o ope a e be ween 0.1aand
0.3awhe e mos o he ansi ion occu s.
Wi hin he e ec i e medium pa adigm, a con inuous
shi o each laye wi h espec o he p e ious one appea s o
p o ide maximal e iciency. Fo ini e samples, howe e , his
poses ce ain disad an ages. Indeed, o small b/a a io
共which p oduces a s onge e ec 兲e en a small
␦
ashi
would imply a ema kable inclina ion o he sample in e -
ace. This would lead o undesi able shape dis o ion and
cause exci a ion o addi ional s anding wa es. P elimina y
expe imen s wi h his kind o uning ha e shown ha he
sys em gene ally ea u es he p edic ed beha io , howe e is
a he un epea able wi h ega ds o exci a ion and measu e-
men me hods, so he pe o mance canno be eliably as-
sessed.
To o e come his di icul y, we conside a s agge ed la -
e al shi as shown in Fig. 1, when e e y second laye is
shi ed while he es o he s uc u e emains a he o iginal
posi ion. This con igu a ion leads o a sligh ly di e en e i-
ciency pa e n 关compa e he wo cu es in Fig. 2共a兲兴, bu is
equally use ul; ob iously, he wo uning s a egies con e ge
o he iden ical esul o 0.5ashi , as he la ice pa e ns
shi ed wi h ei he me hod coincide in his case. Fo ini e
samples, he s agge ed shi is ad an ageous as i keeps he
sample in e ace s aigh and pa allel o he axis o esona o s
a all imes, while sligh egula dis o ion o he in e ace
shape is no expec ed o de e io a e he pe o mance, p o-
ided ha he o e all numbe o elemen s is su icien ly la ge
so ha su ace e ec s a e negligible.
Fo he expe imen al e i ica ion, we op ed o a small
econ igu able sys em, buil up o single-spli ings 共2.25 mm
mean adius, 0.5 mm s ip wid h, and 1 mm gap兲p in ed
wi h a pe iod a=7 mm on 1.5 mm hick ci cui boa ds. We
ha e i e esona o s in he p opaga ion di ec ion xand only
one pe iod along y; 30 boa ds a e s acked oge he in zdi-
ec ion wi h he minimum possible la ice cons an b
0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1
0.75
0.8
0.85
0.9
0.95
1
F equency,ω/ω
o
Magne ic ield
(
no malized
)
FIG. 3. 共Colo online兲Nume ically calcula ed magne ic ield benea h a i-
ni e me ama e ial slab 共5⫻1⫻30 elemen s兲 o a plane-wa e incidence.
Cu es om le o igh co espond o inc easing la ice shi om 0 o 0.5a
共wi h a 0.1ainc emen 兲.
0 0.1 0.2 0.3 0.4 0.5
0.76
0.78
0.8
0.82
0.84
La ice shi , δa
Resonance (ω/ω
o
)
(a)
0 0.1 0.2 0.3 0.4 0.
5
0
0.2
0.4
0.6
0.8
1
La ice shi , δa
T ansmission, Re lec ion
(b)
FIG. 2. 共Colo online兲共a兲Theo e ical shi o he esonance equency o
con inuous 共dashed兲and s agge ed 共solid兲la ice shi s a egy; 共b兲Calcu-
la ed ansmission 共g een兲and e lec ion 共 ed兲 h ough a me ama e ial slab
共one wa eleng h hick兲depending on la ice shi 共s agge ed兲a
=0.96
o.
084105-2 Lapine e al. Appl. Phys. Le . 95, 084105 共2009兲
09 June 2025 17:24:32
=1.5 mm used o he measu emen s. The es ima ed eso-
nance equency o a single esona o is abou 4.9 GHz, how-
e e he esonance o he dense me ama e ial is signi ican ly
shi ed o lowe equencies. To minimize he undesi able
bianiso opic e ec s occu ing in single-spli ings, he
boa ds a e assembled so ha he gaps a e opposi ely o ien ed
in adjacen laye s 共Fig. 1兲, esembling he design o
b oadside-coupled spli - ing esona o s.11 T ansmission mea-
su emen s 共Rohde and Schwa z ZVB ne wo k analyze 兲we e
pe o med o a ious la ice shi s in WR-229 ec angula
wa eguide.
No e ha he abo e sys em o 5⫻1⫻30 esona o s can-
no be desc ibed by an e ec i e medium app oach, as he
numbe o elemen s is small. Also, he sys em is no su i-
cien ly subwa eleng h o a quasis a ic app oach o be used
and spa ial dispe sion becomes ema kable.12,13 Fo his ea-
son, we also pe o m semianaly ical calcula ions o he co -
esponding ini e s uc u es, wi h all he mu ual induc ances
included 关Eq. 共4兲兴, aking e a da ion e ec s in o accoun .
Al hough he pa icula esonance alues ob ained his way
共Fig. 3兲, a e di e en om hose which would be obse ed in
a medium, he o e all e ec o la ice uning was quali a-
i ely he same and p edic s excellen pe o mance 共Fig. 4兲.
The expe imen al ansmission spec a a e shown in Fig.
5, demons a ing d ama ic uning o he esonance equency.
Fu he mo e, compa ison o he expe imen al esonance shi
wi h he heo e ical p edic ions shows 共Fig. 4兲 ha he ex-
pe imen al sys em demons a es e en highe e iciency. This
e ec can be explained by accoun ing o he mu ual capaci-
ance be ween esona o s, neglec ed in he heo e ical calcu-
la ions. Indeed, o he b oadside-like con igu a ion o ings,
mu ual capaci ance be ween hem is dis ibu ed along he
whole ci cum e ence.11 Clea ly, when he esona o s a e la -
e ally displaced, he mu ual capaci ance dec eases, so ha
his e ec is added up o he inc ease o esonance equency
imposed by dec eased induc i e coupling.
The examples analyzed abo e illus a e he p ac ical ea-
sibili y o he p oposed uning concep . Pa icula de ails and
uning pa e ns may di e depending on he speci ic s uc-
u al elemen s used o c ea e me ama e ials, howe e i is
clea ha a ema kable esonance shi can be ealized o e a
wide ange o al e na i e geome ies, including nume ous
esona o a ie ies and e en ishne s uc u es which a e
mo e popula o highe equencies. Rema kably, he p o-
posed uning mechanism is no speci ic o he mic owa e
ange used in he abo e examples: his can be scaled in size
and equency as long as he me ama e ial desc ip ion in
e ms o e ec i e medium is applicable. And on he p ac ical
side, emendous e iciency o he s uc u al uning can be
used in a hos o applica ions such as senso s, il e s,
swi ches, and all kinds o de ices whe e p omp and sensi-
i e esponse o changing condi ions is equi ed.
In conclusion, we ha e p oposed and con i med expe i-
men ally an e icien concep o unabili y o me ama e ials
h ough a con inuous adjus men o he la ice s uc u e.
This wo k was suppo ed by he Aus alian Resea ch
Council. M.L. acknowledges hospi ali y o Nonlinea Phys-
ics Cen e and a suppo o he Spanish Jun a de Andalusia
共P ojec P06-TIC-01368兲. M.G. acknowledges suppo om
he Russian Academy o Sciences 共OFN P og amm “Physics
o new ma e ials and s uc u es”兲.
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0 0.1 0.2 0.3 0.4 0.
5
0.75
0.8
0.85
0.9
0.95
1
La ice shi
,
δa
Resonance
(
ω
/
ωo
)
FIG. 4. 共Colo online兲Compa ison be ween heo e ical esul s and expe i-
men al da a o he esonance equency shi : E ec i e medium app oach
共solid兲; ini e model 共ci cles兲; expe imen al esul s 共squa es兲.
3 3.5 4 4.5
5
0
0.2
0.4
0.6
0.8
1
F equency, GHz
T
ansm
i
ss
i
on
0.0
0.2
0.3
0.4
0.5
FIG. 5. 共Colo online兲Expe imen al ansmission in a wa eguide wi h
me ama e ial slab a di e en shi s. Cu es wi h dips om le o igh
co espond o inc easing la ice shi .
084105-3 Lapine e al. Appl. Phys. Le . 95, 084105 共2009兲
09 June 2025 17:24:32
