Table Of ContentE(cid:27)e
tive medium theory of negative index
omposite metamaterials
A.I. C buz, D. Felba
q and D. Cassagne
Groupe d'Etude des Semi
ondu
teurs UMR CNRS-UM2 5650, CC074,
Université Montpellier II, Pl. E. Bataillon, F-34095 Montpellier Cedex 05, Fran
e
6
0 In the homogenization of
omposite metamaterials the role played by the relative positions of
0 the wires and resonators is not well understood, though essential. We present a general argument
2 whi
h shows that the homogenization of su
h metamaterials
an be seen as the homogenization of
1Dsinglenegativesta
ks. Thesensitivitytothegeometri
alparametersisduetothefa
tthatlight
n
seestheharmoni
meanofthediele
tri
onstantwhenpropagatingparalleltothelayers. Sin
ethe
a
diele
tri
onstant is not positive de(cid:28)nite the harmoni
mean is not bounded and presents abrupt
J
variations. We dis
uss appli
ations of this remarkablephenomenon.
9
1
Y
metamaterial
Meta-materials with a possibly negative index of re-
] One period
l fra
tion have been under intense study sin
e the results X
l ε<0
a of Pendry [1, 2℄ suggested that they might be fabri
ated d
h by interspersing metalli
wires (giving an e(cid:27)e
tive nega- µ<0 y
-
s tive permittivity) with magneti
resonators su
h as split
e ringsordiele
tri
(cid:28)bers(givingane(cid:27)e
tivenegativeper-
m
meability). Howeverthedetailsoftheexa
twayinwhi
h
. these two e(cid:27)e
ts mix and perturb ea
h other are not Figure 1: The stru
ture we study is a 1D periodi
medium
t ε
a well understood. In this letter we present an e(cid:27)e
tive with a unit
ell
omposed of two layers, one with negative
µ µ ε
m medium theory that a
ounts quantitatively for this be- (Hand positive ), one with negative (and positive ). The
- havior,providesanexpli
itgeometri
alre
ipeforobtain- (cid:28)eldis dire
ted in theZ dire
tion and thein
ident (cid:28)eld is
d ing true negativeindex media and opens the way to new dire
ted upwards.
n
appli
ations.
o
c
[ No general argument exists that by mixing magneti
sentially a resonan
e phenomenon and does not depend
resonators and wires one must obtain a negative index
riti
allyonthe
ouplingbetweenneighboringresonators
1
material at some frequen
y. Indeed, as the results of [5, 7℄.
v
Pokrovsky and Efros have shown [3℄, the negative ep- However, while some interesting results on 1D single
4
4 silon e(cid:27)e
t of metalli
gratings is fragile with respe
t to negative media have been published [8, 9, 10℄, the nat-
4 the brutal mixing of the metalli
wires with other ob- ural question of the pre
ise
onditions under whi
h su
h
1 je
ts. Negative epsilon is essentially an interferen
e phe- stru
tures
anbehomogenizedtoobtainanegativeindex
0 nomenon and anything that perturbs the interferen
e of medium has not been satisfa
torily answered.
6 the waves s
attered by neighboring wires is liable to de- Inthisworkweo(cid:27)erarigorousself-
ontainedhomoge-
0
stroy it. The wires need to (cid:16)see(cid:17) ea
h other. The results nization analysis that gives a
lear des
ription of the 1D
/
t of Pokrovsky and Efros are the (cid:28)rst indi
ation that the alternating single negative medium des
ribed above and
a
m e(cid:27)e
tive medium depends
riti
ally on the mi
ros
opi
pPi
·tkureds
hemati
allyinFigure1. Weshallusetheterm
geometry [4℄. They essentially tell us that resonators (cid:16) negative(cid:17) tounambiguouslyrefertowavesorhomo-
-
d should not be pla
ed in between the wires, or the ag- geneous materials that may support waves whi
h have a
n gregate may lose its negative permittivity. The regions dotprodu
tofthe Poyntingve
torand waveve
torthat
o o
upiedbythewiresmusthavehigher
onne
tivity,and is negative.
c the simplest way of a
heiving this is a sta
k of alternat- The dis
riminating fa
tor, as in any homogenization
:
v ing rows of wires and resonators. Ea
h row of wires will problem, is the ratio between the wavelength and the
Xi itthyen(εa
=ta1s−ahωωoe22pmoagnedneωoeups=slaba2w2lnπi(tca20h/rn)e[g1a,ti5v℄e),paenrmdiettai
vh- pinetrriooddu
oefdthbeystthrue
teuxrter.emIendoiuspre
rasisoenaof
ormespolni
aattoironbeis-
r
havior, but we avoid it by (cid:28)xing the wavelength (and
a row of resonators will µa
=t a1s−a haoωm2ogeneousaslab wωith therefore the internal resonator geometry) and s
aling
negative permeability ( ω2−ωm2p, with and mp the relative pla
ement of wires and resonators. The dis-
determined by the geometry [2, 6℄). We thereby arrive
riminating geometri
al pararme=terλof the hλomogeneous
at the remarkable and quite general
on
lusion that the model is therefore the ratio d where is the free
d
homogenizationof
omplex wire/resonator
ompositesis spa
e wavelength and is the period. This parameter
thehomogenizationof1Dsta
ksofalternatingsingleneg- is, a priori, large, whi
h authorizes us to use the well-
ativehomogeneousslabs. This
on
lusionisadire
t
on- established two-s
ale exHpansion approa
h. In the follow-
sequen
eofthefa
tthatthenegativepermittivityofwire ing we tEreat only the polarization
ase. The resultµs
gratingsdepends
riti
allyon theirmutual interferen
es, hold in polarization as well, provided we ex
hange
ε
in
ontrasttothenegativepermeabilitye(cid:27)e
twhi
hises- and everywhere.
2
We note the ma
ros
opi
variables with
apital letters in mind as we pro
eed to average the equations (to ob-
X Y
and and the mi
ros
opi
variable with the small tain thehomogeneousbehaviorofthe (cid:28)elds)be
ausethe
y
letter asin Fig. 1. We(cid:28)rstneedtointrodu
ethesmall derivative of any
ontinuous periodi
fun
tion averages
η = d/λ y f
parameter whi
h will be used in tyhe=exYpansion. htfoiz=ero1. Wdfe(nyo)teythe average over of a quantity as
Next we de(cid:28)ne the mi
ros
opi
variable η whi
h d 0 d .
η y
varies fast, being small, so it des
ribes the system on TakingR the averageof 6 and 7 the derivatives disap-
the mi
ros
opi
s
ale, the s
ale of an individual layer. pear as explained above and we get
Sin
e we are interested in the homogeneous behavior of ∂U
0
hεiξ =
the system, at the end of the analysis this variable will 0
∂Y
be averaged over and will disappear. It
an be regarded
as an internal degree of freedom of the unit
ell.
ε(Y)
∂ξ 1
µ(YFo)ra1Dstru
turewithpie
ewise
ontinuous and 0 + k2hµi− k2 U =0.
Maxwell's equations
an be written as ∂Y ε X 0 (8)
(cid:18) (cid:28) (cid:29) (cid:19)
d 1 dU k2sin2(θ)
+ k2µ(Y)− U(Y)=0
dY ε(Y)dY ε(Y) whi
h we put ba
k together into one se
ond order equa-
(cid:18) (cid:19) (cid:18) (cid:19)
tion
H(Y)=U(Y,y) ikxsin(θ) k =2π/λ θ (1) ∂2U hεi
wkof2hienr
e2i(dθe)n
=e ikn the z =e 0 plane, and wher,e wiestwhiellannogtlee ∂Y20 + k2hεihµi− hε−1i−1kX2 !U0 =0.
X
sin below. This se
ond order di(cid:27)erential
equation
an be written as a system of two (cid:28)rst order
The quantYity in parenkth2eses is the propagation
on-
equations stant in the dire
tion, Y and we thus obtain the dis-
1dU
ξ = persion relation of the homogeneousmedium:
εdY (2)
k2 k2
X + Y =k2µ
dξ + k2µ− kX2 U =0 εY εX e(cid:27) where (9)
dY ε (3)
(cid:18) (cid:19)
ξ µ =hµi, ε =hεi, ε = ε−1 −1.
X Y
where we have introdu
ed the new variable de(cid:28)ned by (10)
e(cid:27)
equation 2 and where spe
ial
are must be taken with (cid:10) (cid:11)
the derivative operator as it now takes the form The relations 10 summarize the e(cid:27)e
tive medium
d ∂ 1 ∂ model and the rest of the letter is
on
erned with ex-
= + .
ploringtheir
onsequen
es. The mostimportantofthem
dY ∂Y η∂y (4)
are: 1). The medium is anisotropi
, 2). Its anisotropy
an be extreme due to the unboundedness of the har-
We develop the quantities of interest in multi-s
ale se-
η moni
mean, 3). This behavior is strongly dependent on
ries with parameter
the mi
ros
opi
geometryof the
omposite. We o(cid:27)er ex-
U(Y)=U0(Y,y)+ηU1(Y,y)+··· pli
it expressionsillustratingthe third pointbelowwhile
ξ(Y)=ξ0(Y,y)+ηξ1(Y,y)+··· (5) the (cid:28)rst two are dis
ussed after the presentation of the
numeri
al results validating our model. We should note
and insert them into equations 2 and 3. Note that all that we have not made any assumption about the losses
quantities on the right side of the equations must be pe- so, in general, all quantities in H10 are
omplex.
y d
riodi
in with period . By identifying same order Eqs. 9 and 10 show that for polarization, light sees
η
terms in we obtain the followingrelations. From Eq. 2 tYhe arithmeti
mean of epsilon when propagating in the
we obtain diXre
tionbuttheharmoni
meanwhen propagatingin
∂U0 =0 εξ = ∂U0 + ∂U1 the hdi1re
tionh.2Letusdenoteh1th+ehth2i
=kn1essesofthetwo
0 layers and (we assume for simpli
ity)
∂y and ∂Y ∂y (6) ε = a +ib
1 1 1
and the
orresponding permittivities and
ε =a +ib
2 2 2
. Rewriting Eq. 10 we have:
while from Eq. 3 we obtain ε ε
1 2
∂ξ ∂ξ ∂ξ k2 εX =h1ε1+h2ε2, εY =
0 =0 0 + 1 + k2µ− X U =0. h1ε2+h2ε1 (11)
0
∂y and ∂Y ∂y ε (7)
(cid:18) (cid:19)
Inthisformitiseasiesttoseehowthethreemain
onse-
U ξ
We noti
e that the zeroth order terms of the and quen
es
omeabout. The(cid:28)rstoneisobtainedbysetting
y ε = ε ε = ε
X Y 1 2
expansionsdonotdependon ,themi
ros
opi
variable. . One qui
kly arrives at whi
h
ontra-
However the (cid:28)rst order terms do depend on it and are di
ts ourassumptions. The se
ondone is seen by noting
d ε
Y
periodi
of period as noted above. This must be kept that there is nothing preventing the denominator of
3
||T ||2−||T||2
fromapproa
hingzero,ignoringlosses. Thethird results a) H A
from the prominent role played by the geometri
al pa- 0.3
h1 h2 θ=0°
rameters and . Their importan
e be
omes evident 0.2
as we explorethe
onditions under whi
h the material is
P ·k 0.1
negative. P ·k = kX2 + kY2
Maxwell's equations tell us thatH εY εX in 0
a uniaxial anisotropi
materiaPl i·nk =pko2laµrization. Com- 10Scal2e 0param3e0ter r [4di0mensi5o0nless]60
paring to Eq. 10 we obtaiPn·k e(cid:27). The su(cid:30)H-
ient
ondition to have a negative material in b)0.05 ||TH||2−||TA||2
polarization is therefore that the e(cid:27)e
tive permeability
k k
X Y 0.04
be negative. If in addition we require and to be
real (ignoring absorption) for all angles of in
iden
e we 0.03
ε < 0 ε < 0
X Y
must also impose and . Writing it out 0.02
r=35
expli
itly using Eq. 11 we obtain the rather restri
tive: 0.01
h2 < ε2 < h1 ε <0 h2 < h1 0
h1 ε1 h2 when 1 and h1 h2 0 30 60 90
h1 < (cid:12)ε2 (cid:12)< h2 ε <0 h1 < h2. (12) Angle of incidence θ [degrees]
h2 (cid:12)(cid:12)(cid:12)ε1(cid:12)(cid:12)(cid:12) h1 when 2 and h2 h1 c) 7 arg( T H )−arg(T A )
qTnuohtaeendt
iratuyb
oihav1(cid:12)(cid:12)le/rthoo(cid:12)(cid:12)2lee
xoppmllaabyiinendetshhewer
ietrhitbity
haelthdueenpbpeuonurdeneldyne
dgeenooefmsstehoterfib
εeaY-l hift [degrees] 3456
haviorofthe
ompositeontherelativespatialpla
ement hase s 12
of its building blo
ks [11, 12℄. Moreover, these relations P
0
give,forthe(cid:28)rsttimetoourknowledge,apre
isegeomet- 0 30 60 90
Angle of incidence θ [degrees]
ri
al re
ipe for the
onditions under whi
h a
omposite
will exhibit true negative index properties. We believe
that, along with Eqs. 10 from whi
h they are derived,
they will be
ome useful in the theoreti
al and espe
ially Figure2: O(cid:27)set betweenthep2ower transmission
oe(cid:30)
ients
kTAk
the experimental study of negative index
omposites. of single-negativkeTsHtak
2k ( ) and the equivalent homoge-
We now study the e(cid:27)e
t of losses on the eb(cid:27)>e
ti0ve nre=ouλs/mdedium ( ) as a fun
tion of the s
ale parameter
medium. We assume the media are passive ( ), r = 35 fornormalin
iden
e(a)andofangleofin
idern
>ef3o0r
(b). They are within 0.05 of one another for
and that the imaginary parts are
omparatively small
b≪|a| b for all angles of in
iden
e.
) Phase mismat
h between the
( )sothatwe
anignorequadrati
termsin su
h
as b1b2, et
. Rearranging10 we get transmission
oe(cid:30)
ientsofsinglenegativestra
=k3a5nde(cid:27)e
tive
medium◦asafun
tionofin
iden
eangle for . Theyare
εX =hai+ihbi within 7 of ea
h otherfor all angles of in
iden
e.
ε = a−1 −1+ihba−2i.
Y ha−1i2
(cid:10) (cid:11)
ε The stru
ture under study is
omposed of 200 alter-
Y h = 0.4 h = 0.6
Let us take a
loser look at . The real part is the 1 2
nating layers of thi
knesses , , with
ε = 2 ε = −2.8
harmoni
mean of the real parts of the permittivities of 1 2
permittivities , and permeabilities
µ = −3.295 µ = 0.53
the layers. These permittivities have di(cid:27)erent signs, and 1 2
, . The equivalent parameters
ε = −0.4 ε = −70
therefore this quantity is not bounded a priori. Let us X Y
A ε resulting from Eqs. 10 are , and
Y µ = −1
all it . Rewriting we get
. The total thi
kness of the stru
ture is 100
e(cid:27)
εY =A+i ab2 A2 (13) periods. We now present numeri
al simulations
om-
paring the transmission
oe(cid:30)
ient of the 1D alternating
(cid:10) (cid:11) TA
The unboundednessoftheharmoni
meanhasamu
h layer stru
ture, T , to that of the eqruivalent homoge-
H
stronger impa
t on the imaginary part (tbha,nbo)n≪theAr−ea1l neous stru
ture, , as a fun
tion of and of angle of
part. Eq. 13 is only valid when max 1 2 . in
iden
e.
Other
ases are of lesser interest as losses dominate. Figure 2.a) showsktTheko2(cid:27)−setkTbetkw2een the power trans-
H A
We now validate our model as formulated in Eq. 10 mission
oe(cid:30)
ients, , as a fun
tion of the
r
with numeri
al simulations. The transmission of the ho- s
ale parameter for normal in
iden
e. The same o(cid:27)-
mogeneous anisotropi
slab was
omputed using a 1D set as a fun
tion of angle of in
iden
e appears in Fig.
r = 35
transfer (T) matrix method, while for the alternating 2.b) for . Finally, in Fig. 2.
) we plot the phase
single negative sta
k we
hose a s
attering (S) matrix mismat
h between the two transmission
oe(cid:30)
ients as a
r = 35
methodforitsnumeri
alstability. Inwhatfollowsallan- fun
tion of angle of in
iden
e for . These results
gles and phase di(cid:27)eren
es are measured in degrees while demonstrate
learly that the meta-material behaves in
r
the s
ale parameter is dimensionless. all respe
ts as ahomogeneousmedium with the e(cid:27)e
tive
4
parametersgiven by Eqs. 10forperiods morethan 20 to treating them as 1D air/metal sta
ks, providing a gen-
30 times smaller than the wavelength. eral theoreti
al framework for the results of Shen et. al
The behavior at shorter wavelengths
an be des
ribed [19℄. Another promising appli
ation is related to plas-
by pushing the developments of Eq. 5 to higher orders. monlasing[20,21℄. Ourpreliminaryresultsshowthatby
Thesimplee(cid:27)e
tivemediumdes
riptionisthennolonger sta
king thin layers of a
tive material with thi
k metal-
quantitatively validbut,interestingly,negativeindex-like li
layers it may be possible to use the divergen
e of the
behaviors remain [9℄. More pre
ise results in this dire
- harmoni
mean to dramati
ally enhan
e the gain of the
tion will be presented elsewhere. a
tive material. More pre
ise results and examples will
Wehavethisfardetailedonlythelastofthethreemain be published elsewhere.
onsequen
es of Eq. 10. It is expressed quantitatively
In
on
lusion, we have argued that, be
ause of the
in Eqs 11 and 12. We now dis
uss the (cid:28)rst two: that
physi
alnatureofthenegativepermittivityphenomenon
negativeindexresonator/wire
ompositesareanisotropi
in metalli
gratings,
omposites employing them may be
andthattheiranisotropy
anbe
omeextremeduetothe
simply modeled as 1D single negative sta
ks. Rows of
divergen
e of the harmoni
mean.
wiresmakeup e(cid:27)e
tive negativepermittivity slabswhile
The signi(cid:28)
an
e of the anisotropy of the e(cid:27)e
tive
rows of resonators (metalli
or diele
tri
) provide the
medium resides in the di(cid:30)
ulty of interpreting experi-
negative permeability. We have shown using a rigorous
mentaldatafromasingleangleofin
iden
e. Themethod
self-
ontained homogenization theory that light sees the
ofextra
tinge(cid:27)e
tiveparametersthatismost
ommonly
harmoni
mean of the diele
tri
onstant when propa-
usedin theliterature[13, 14,15, 16℄mayfailwhen fa
ed
gating parallel to the layers. Sin
e the diele
tri
on-
with an anisotropi
material, parti
ularly when it is in
stant is not positive de(cid:28)nite the harmoni
mean is not
addition inde(cid:28)nite. This aspe
t points to the need for a
bounded. This remarkable phenomenon, never before
new extra
tion pro
edure, one that
an unambiguously
pointedout,explainsquantitativelythesensitivityofthe
pla
e the sample under study within one of the
ases
omposites' behavior to their mi
ros
opi
geometry and
listed by Smith and S
hurig [17℄ at a given frequen
y.
provides
lear
riteriafor
onstru
tingatruenegativein-
Su
h a pro
edure would ne
essarily involve transmission
dex metamaterial. Furthermore, it tells us that it may
and phase measurements over multiple angles of in
i-
now be possible to fabri
atemedia with arbitrarily large
den
e.
positiveornegativee(cid:27)e
tive parametersusing
on
eptu-
The impli
ations of the se
ond point stret
h beyond
ally simple and te
hnologi
ally straightforward 1D sin-
the anisotropy of resonator/wire
omposites. Among
gle negative sta
ks. More in depth investigations of the
otherthings,ittellsusthatitispossibletorealizemate-
possible appli
ations of this phenomenon are promising
rialswithverylargepositive or negative permittivitiesor
dire
tions for further resear
h.
permeabilities usinglowvalued single negativesta
ks. It
also enables a new understanding of enhan
ed transmis- ThisworkwassupportedbytheFren
hRNRTproje
t
sionthroughverythinslitsin thi
kmetals
reens[18℄ by CRISTEL and by the EU-IST proje
t PHAT 510162.
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