Table Of ContentAbsorption losses in periodic arrays of thin metallic wires
∗
P. Markoˇs and C. M. Soukoulis
Research Center of Crete, 71110 Heraklion, Crete, Greece
and
Ames Laboratory and Department of Physics and Astronomy,
Iowa State University, Ames, Iowa 50011
3
0
We analyze the transmission and reflection of the electromagnetic wave calculated from transfer
0
matrix simulations of periodic arrangements of thin metallic wires. The effective permittivity and
2
the absorption is determined. Their dependence on the wire thickness and the conductance of the
n metallic wires is studied. The cutoff frequency or effective plasma frequency is obtained and is
a compared with analytical predictions. It is shown that the periodic arrangement of wires exhibits
J
a frequency region in which the real part of thepermittivity is negative while its imaginary part is
0 verysmall. Thisbehaviorisseen for wires with thicknessas small as17 µm with alatticeconstant
3 of 3.33 mm.
]
ci Rapidlyincreasinginterestinleft-handedmetamateri- LHM,9,10 seemed to agree with their pessimistic conclu-
s als(LHM)(forarecentreview,see1)raisedsomeinterest- sion. However, it is not clear why the transmission was
-
l ingquestionsabouttheelectromagnetic(EM)properties so low in the original experiments. Recent experimental
r of composites which contain very thin metallic compo- measurements11,12 established that the transmission of
t
m nents. The most simple example of such a composite is LHM could be as good as in the right-handed systems.
. a periodic array of thin metallic wires. Pendry el al.2 Our aim in this Letter is to study numerically how
t
a predicted that such system behaves as a high pass filter the effective permittivity of the periodic arrangement of
m with an effective permittivity metallic wires depends on the wire radius and on the
- conductance of the wires. We present results for the real
nd ǫeff =1− f2+fp22iγf. (1) i(tǫy′effo)fatnhde iwmiraegimnaerdyiu(mǫ′e′,ffe)sptiamrtatoeftthheeetffraenctsimveisspieornmliotstsives-
o
and the plasma frequency and compare our results with
c
In Eq. 1, f is the effective plasma frequency, or cutoff
[ p the analytical formulas given in Eqns. 1-4.
frequency3, and γ is the damping factor. Various theo- In our numerical simulations we use the transfer ma-
2 retical formulas were derived for the dependence of the
trix method (TMM). Details of the method are given
v plasma frequency on the lattice period a and the wire elsewhere13. Here we only point out the main advantage
3 radius r. Pendry el al.2 obtained that
4 of the TMM, namely that it gives directly the transmis-
3 c2 sion t, reflection r and absorption A = 1−|t|2−|r|2 of
2 f2 = light , (2) the EM plane wave passing through the system. Con-
1 p 2πa2ln(a/r) trary to this, in the FDTD method, which was used in
2 Ref.6, one obtains t and r form the time development of
0 Shalaev and Sarychev4 obtained that
the wave packet, which is much more complicated and
/
at c2 probably also less accurate. To be able to obtain ǫeff,
m f2 = light , (3) one also needs the phase of r and t, in addition to their
p 2πa2[ln(a/√2r)+π/2 3] amplitudes. This is also easily achieved in the TMM.
- −
d From the obtained data for the transmission and re-
while Maslovski et al.5 that
n flection, we calculate the effective permittivity of the
o c2 system.14 The refraction index is given by the equation
c f2 = light . (4)
: p 2πa2[lna2/4r(a r)] 1
v − cos(nkL)= (cid:2)1 r2+t2(cid:3). (5)
i 2t −
X In Eqs. 2-4, clight is the velocity of light in vacuum.
Periodic arrangements of thin metallic wires are Since we do not expect any magnetic response, we fixed
r
a used as a negative-ǫ medium7,8 in the left-handed the value of the permeability to be µ 1. The permit-
structures9,10,11. It is therefore important to understand tivity is then found as ǫ =n2. ≡
eff
how the electromagnetic response - not only the effec- ThewaywediscretethespaceintheTMMmightcon-
tive plasma frequency, but also the factor γ - depends trol the accuracy of our results. To test how discretiza-
on the structural parameters of the wire system. Re- tion influences our results, we repeated the numerical
cently, Ponizhovskaya el al.6 claimed that for small wire simulationfordifferentdiscretizations. Thewireisrepre-
radius,the absorptionin the wire systemis so largethat sentedasarectangularwithsquarecrosssection2r 2r,
×
the transmission losses do not allow any propagation r being the “wire radius”. The obtained results for the
of EM wave in left-handed structure. Very low trans- effective permittivity ǫ (both real and imaginarypart)
eff
mission, measured in the original experiments on the are almost independent of the discretization procedure.
2
Figure 1 shows how the effective permittivity depends radius decreases. For instance, the parameter γ is only
on the wire radius. We analyzed four different wire ar- 0.003GHzforr =100µm. Aswewillseebelow,(fig. 4)γ
rays with period a = 3.33 mm. For all of them the real increasesupto1.2GHzwhenthewireradiusdecreasesto
partoftheeffectivepermittivityisnegativeandcouldbe 15µm. Nevertheless, even for very thin wires, losses are
fitted by Eq. (1). This enables us to obtain easily the muchlessthanwhatwasclaimedinRef.6. Asitisshown
plasma frequency. Only for the smallest wire thickness in fig 1, also an array of wires with thickness 17 17µm
′′ ×
studied (17 17µm) one gets a relatively large ǫ for createsanegative-ǫmedium. Asthisisinstrongcontrast
small freque×ncies. In this regionrelation(1) is notevffalid. withtheresultsofRef.6,wedecidedtostudy exactlythe
Nevertheless, for frequencies larger than 5 GHz, ǫ′′ is same system as that of6. Results of our simulations are
eff
′
small and ǫ negative. shown in fig. 4. Although such systems are not used in
eff
Therightlowerpaneloffig. 1showsdataforwireswith experimentalarrangementsoftheLHM,ourresultsgivea
the cross section of 17 300µm. These parameters were comparison between two different numerical treatments.
used in the experiment×of Shelby et al.10. We again see Our data again clearly show that ǫ′ is negative for f <
eff
that the formula given by Eq. 1 qualitatively agree with f . As it is shown in the inset, the transmission losses
p
ourdata. Thus,thereisnodoubtthatthisarrayofwires are also small.
′
really produces a medium with negative ǫ , which then We believe that the present data are more accurate,
eff
can be used in the creation of the left handed systems. than those published previously in6, not only because
Figure2 comparesourdata forplasmafrequency with they agreewith the theoreticalanalysis but also because
the analytical formulas given by Eqns. 2-4. Accepting the TMM gives the reflection and its phase straightfor-
some uncertainty in the estimation of the plasma fre- ward. This is important because the main difference be-
quency from the numerical data, we can conclude that tweenourresultsandthoseof6 seemstobeintheestima-
forthinwiresourdataagreewiththetheoreticalformula tionofreflectionR= r2. Whencomparingourdatafor
(3) of Shalaev and Sarychev.4 For thicker wires, our re- absorption, given in t|he| inset of fig. 4 with those given
sults are in agreement with the formula (4) of Maslovski infig2bofRef.6,weseethatourabsorptionismuchless
et al.5 than that estimated in Ref.6.
Wealsostudytheroleoftheconductanceofthemetal-
lic wires. In the simulations shown in fig. 1, we consider
In conclusion, we analyzed numerically the transmis-
the metallic permittivity to be ǫ =( 3+588 i) 103.
m − × sionpropertiesofaperiodicarrangementsofthinmetallic
We are aware that this value of ǫ is smaller than the
m wires. From the transmissionand reflection data we cal-
permittivity of realistic metallic wires: for instance for
culate the effective permittivity and plasma frequency,
copper ǫ 5 107 i, as follows from the relation be-
tween thme p≈erm×ittivity and the conductance15 (the con- which agrees qualitatively with theoretical predictions.
ductivityofcopperisσ 5.9 107(Ωm)−1). Ourdatain Both the effective permittivity and the absorption data
≈ × confirmthat the arrayof thin metallic wires,used in the
fig. 1 therefore underestimate losses, because transmis-
recentexperimentsontheleft-handedmetamaterials,in-
sion losses are smaller for higher values of the metallic
deedbehavesasanegativepermittivitymediumwithlow
permittivity16. This is clearly shown in fig. 3 where we
present the ratio κ = ǫ′′ /ǫ′ vs frequency for two sys- losses.
| eff eff|
tems whichdiffer only in the value ofthe imaginarypart
of the metallic permittivity. Fig. 3 shows also the fre- Acknowledgments: This work was supported by
quency dependence of the absorption as obtained from Ames Laboratory (Contract. n. W-7405-Eng-82).
the numerical simulations. The absorption also exhibits Financial support of DARPA, NATO (Grant No.
a maximum in the neighbor of the plasma frequency. PST.CLG.978088), APVT (Project No. APVT-51-
′′
Fig. 1 also confirms that ǫ increases when the wire 021602)andEUprojectDALHMarealsoacknowledged.
eff
* Present and permanent address: Institute of Physics, Slo- Appl.Phys. Lett. 81, 4470 (2002)
vak Academy of Sciences, 845 11 Bratislava, Slovakia. E- 7 J. Brown, Progr. Dielectrics 2, 195 (1960)
mail: [email protected] 8 W. Rotman, IRE Trans. Antennas Propag. AP10, 82
1 P.Markoˇs,andC.M.Soukoulis,e-printcond-mat/0212136 (1962)
2 J.B. Pendry, A.J. Holden, W.J. Stewart, and I. Youngs, 9 D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser,
Phys. Rev.Lett. 76, 4773 (1996) and S. Shultz,Phys. Rev.Lett. 84, 4184 (2000)
3 M.M. Sigalas, C.T. Chan, K.M. Ho, and C.M. Soukoulis, 10 R.A. Shelby, D.R. Smith, and S. Shultz, Science 292, 77
Phys. Rev.B 52, 11744 (1995) (2001)
4 A.K. Sarychev,and V.M. Shalaev, cond-mat/0103145 11 K.Li,S.J.McLean,R.B.Gregor,C.G.Parazzoli,andM.H.
5 S.I.Maslovski,S.A.Tretyakov,andP.A.Belov,Microwave Tanielian, preprint
and Opt.Techn.Lett. 35, 47 (2002) 12 E. Ozbay,K. Aydin,E. Cubukcu,and M. Bayindir, Com-
6 E.V. Ponizovskaya, M. Nieto-Vesperinas and N. Garcia, posite Metamerials, to bepublished (2002)
3
18 13 P. Markoˇs and C.M. Soukoulis, Phys. Rev. E 65 036622
(2002)
z] 16 14 D.Smith,S.Shultz,P.Markoˇs,andC.M.Soukoulis,Phys.
H Rev.B 65 195104 (2002)
y [G 15 J.D. Jackson, Classical Electrodynamics, 3rd. ed. John
c 14 Willey and Sons,New York,1999
en 16 J.B.Pendry,A.J.Holden,W.J.Stewart,andI.Youngs,J.
u
q Phys.: Condens. Matt. 10, 4785 (1998)
e
a fr 12
m
s
Pla 10
8
0 0.05 0.1 0.15 0.2
wire radius r [mm]
FIG. 1: Effective permittivity as a function of frequency for
various shapes of the metallic wires. The lattice period in
all cases is a = 3.33 mm. We used the metallic permittivity
ǫm =(−3+588 i)×103.
18
z] 16
H
G
y [
c 14
n
e
u
q
e
a fr 12
m
s
Pla 10
8
0 0.05 0.1 0.15 0.2
wire radius r [mm]
FIG. 2: Plasma frequency as a function of the wire radius.
The lattice constant is a = 5 mm. Solid, dashed and dot-
dashed line is the result of Pendry et al. (2), Shalaev and
Sarychev (3) and Maslovski et al. (4), respectively. Dot line
is a fit of our data to the function fp = a0/pln(a1/r) with
parameters a0=20.9 and a1=0.84.
4
100
10−1 n
o
pti
ε’| 10−2 bsor
ε’’/ A
|
10−3
10−4
0 5 10 15 20
Frequency [GHz]
′′ ′
FIG. 3: Ratio κ=|ǫeff/ǫeff| for a lattice of wires with radius
50µm. Metallicpermittivityisǫm =(−3+588i)×103 (open
circles) and ǫm =(−3+5 880 i)×103 (full circles). Dashed
(solid) line is absorption for the corresponding systems ob-
tainednumericallybytheTMM.Thesenumericalresultscon-
firm that losses are smaller for higher metallic permittivity
and that the value of the plasma frequency, estimated ap-
proximatelyfromthepositionofthemaximumofκ,doesnot
depend on the valueof themetallic permittivity.
10
y
mittivit 0
er 0.3
p
e on
ectiv −10 orpti 0.2
Eff bs 0.1
A
0
0 5 10 15
Frequency [GHz]
−20
4 6 8 10 12
Frequency [GHz]
FIG. 4: Effective permittivity for a lattice of thin metallic
wires. The wire radius is r = 15µm and the lattice constant
is a = 5 mm. The metallic permittivity is ǫm = −2000+
106i. Twodifferentdiscretizationsareusedwithmeshsizesof
30µm(opensymbols)and15µm(fullsymbols). Thesolidand
dashed lines arefit toEq. 1with fp =11.1 GHzand γ =1.2
GHz. The length of the system was up to 60 unit lengths
for open symbols and 10 unit lengths for full symbols. Inset
shows the numerically calculated absorption as a function of
frequency.
0
10
66 x 66 µm
f =17.9 GHz
p
0
−50
1 x 1 mm
−10
f =25.1 GHz
p
−20 −100
0 10 20 0 5 10 15 20
20 4
17 x 17 µm 0
f =22.67
10 f =14 GHz p
p −4
17 x 300 µm
−8
0
Re eps
−12
Im eps
−10 −16
0 5 10 15 20 4 6 8 10 12 14 16
