DTIC ADA531101: Optical Negative Refraction in Ferrofluids with Magnetocontrollability

0.42 MB·English

by  Defense Technical Information Center

week ending
PRL 104, 034501 (2010) PHYSICAL REVIEW LETTERS 22 JANUARY 2010
Optical Negative Refractionin Ferrofluids with Magnetocontrollability
Y.Gao,1,2 J.P. Huang,1,* Y.M. Liu,3L. Gao,4 K.W. Yu,2 and X. Zhang3
1Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai200433, China
2Department of Physics and Institute of Theoretical Physics, The Chinese University of Hong Kong,
Shatin, New Territories, HongKong
3NSF Nanoscale Science and Engineering Center (NSEC), 5130 Etcheverry Hall, University of California,
Berkeley, California 94720-1740, USA
4Department of Physics, Soochow University, Suzhou 215006, China
(Received 5 October 2009; published 20 January 2010)
We numerically demonstrate optical negative refraction in ferrofluids containing isotropic Fe3O4
nanoparticles, each having an isotropic Ag shell, in the presence of an external dc magnetic field H.
The all-angle broadband optical negative refraction with magnetocontrollability arises from H-induced
chains or columns. They result in hyperbolic equifrequency contour for transverse magnetic waves
propagatinginthesystem.Thefiniteelementsimulationsverifytheanalysesusingtheeffectivemedium
approximation. Experimental demonstration and potential applications are suggested and discussed.
DOI: 10.1103/PhysRevLett.104.034501 PACS numbers: 47.65.Cb, 78.20.(cid:1)e, 78.66.Sq
In1968,Veselago[1]theoreticallyinvestigatedtheelec- thanCoisduetoitsloweropticalabsorption.Inprinciple,
trodynamic consequences of a medium with simultane- the Ag used for making shells may be replaced by other
ouslynegativepermittivityandpermeability.Hepredicted metallicmaterialshavingapermittivitywithabignegative
that such a medium possesses a negative (phase) index, real part and small positive imaginary part. If water is
which can result in a reversed Snell’s law, i.e., negative replaced with kerosene, the qualitative results as to be
refraction. People have realized negative refraction of revealed would remain unchanged. Both experiments
optical waves or microwaves in different systems, includ- [10–12] and computer simulations [13,14] have shown
ing metamaterials [2,3], photonic crystals [4], plasmonic that nanoparticle chains can be formed in ferrofluids as
waveguides [5], chiral media [6], and superconductor- the magnetic dipolar interaction overwhelms the thermal
ferromagnet superlattices [7]. The significant application energy. For ferrofluids, the field-induced columnar phase
of negative refraction is the concept of a perfect lens, withequalspacingwasexperimentallyreported[11,12]in
which can lead to subwavelength imaging beyond the confined ferrofluid films subjected to an in-plane H, and
diffraction limit [8]. the dipolar interaction between the chains (which are lo-
However, almost all the existing methods for achieving cally displaced in a hexagonal fashion) gives rise to their
negative refraction were proposed or established in the arrangement in columns. In other words, nanoparticles in
realm of solid materials, in contrast to soft materials with ferrofluidscanexistintheformofboth(either)chainsand
the specific characteristic ‘‘softness’’. Literally softness (or) columns. In our work, the word ‘‘chain’’ denotes a
might offer an extra freedom of tailoring physical proper- single-particle-width z-directed array of touching Fe3O4
ties; hence, it encourages us to investigate optical refrac- nanoparticles with an Ag shell, and ‘‘column’’ represents
tionincertainsoftmaterials.Asaresult,wereveal,forthe the aggregation of such chains, namely, the chains aggre-
firsttime,anewclassofall-anglebroadbandopticalnega- gate together to form z-directed thick columns.
tive refraction in ferrofluids with magnetocontrollability. Incidentally, throughout this work both permittivities and
Its underlying mechanism arises from assembly metallic permeabilitiesdenote relative ones.
chain or column structures induced by an external dc An effective medium approximation and a 2D finite
magneticfieldH.Thisworkpavesanewwayfordesigning elementmethod.—Letusstartbyconsideringtheferrofluid
tunable, activemetamaterials. system containing Fe3O4 nanoparticles coated by an Ag
In general, ferrofluids are colloidal suspensions com- shell, in the presence of z-directed H. The chains, each
posed of ferromagnetic or ferrimagnetic nanoparticles of having the same number n of nanoparticles [15], are in-
about 10 nm diameter dispersed in a carrier fluid, usually duced to appear by H. They are all directed along z axis,
water or kerosene [9]. Two typical kinds of materials for andareassumedtoberandomlysuspendedinthesystem.
fabricatingsuchnanoparticlesareCoandFe3O4[9].Inthis Inthequasistaticapproximation,we canutilizetheaniso-
workweshallconsideranaqueousferrofluidsystemwhich tropicformoftheeffectivemediumapproximation(EMA)
contains spherical isotropic Fe3O4 nanoparticles, each [16]tocalculatetheeffectivepermittivitytensorwiththree
coated witha sphericalisotropicAg shell,in thepresence nonzero diagonal components " , " ð¼" Þ, and "
xx yy xx zz
of H in z axis. The reason why we choose Fe3O4 rather given by
0031-9007=10=104(3)=034501(4) 034501-1 (cid:1) 2010 The American Physical Society
Report Documentation Page Form Approved
OMB No. 0704-0188
Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and
maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information,
including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington
VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it

does not display a currently valid OMB control number.
1. REPORT DATE 3. DATES COVERED
22 JAN 2010 2. REPORT TYPE 00-00-2010 to 00-00-2010
4. TITLE AND SUBTITLE 5a. CONTRACT NUMBER
Optical Negative Refraction in Ferrofluids with Magnetocontrollability
5b. GRANT NUMBER
5c. PROGRAM ELEMENT NUMBER
6. AUTHOR(S) 5d. PROJECT NUMBER
5e. TASK NUMBER
5f. WORK UNIT NUMBER
7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION
Department of Physics and Surface Physics Laboratory ,(National Key REPORT NUMBER
Laboratory),Fudan University, Shanghai 200433, China, ,
9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR’S ACRONYM(S)
11. SPONSOR/MONITOR’S REPORT
NUMBER(S)
12. DISTRIBUTION/AVAILABILITY STATEMENT
Approved for public release; distribution unlimited
13. SUPPLEMENTARY NOTES
This work was supported by the NNSFC under Grant Nos. 10604014 and 10874025, by CNKBRSF under
Grant No. 2006CB921706, and by the RGC General Research Fund of Hong Kong SAR Government. X. Z.
acknowledges funding support from U. S. Army Research Office (ARO) MURI program 50432-PH-MUR.
14. ABSTRACT
We numerically demonstrate optical negative refraction in ferrofluids containing isotropic Fe3O4
nanoparticles, each having an isotropic Ag shell, in the presence of an external dc magnetic field H. The
all-angle broadband optical negative refraction with magnetocontrollability arises from H-induced chains
or columns. They result in hyperbolic equifrequency contour for transverse magnetic waves propagating in
the system. The finite element simulations verify the analyses using the effective medium approximation.
Experimental demonstration and potential applications are suggested and discussed.
15. SUBJECT TERMS
16. SECURITY CLASSIFICATION OF: 17. LIMITATION OF 18. NUMBER 19a. NAME OF
ABSTRACT OF PAGES RESPONSIBLE PERSON

a. REPORT b. ABSTRACT c. THIS PAGE Public Release 4
unclassified unclassified unclassified
Standard Form 298 (Rev. 8-98)
Prescribed by ANSI Std Z39-18
week ending
PRL 104, 034501 (2010) PHYSICAL REVIEW LETTERS 22 JANUARY 2010
" (cid:1)" " (cid:1)"
p 1 xx;zz þð1(cid:1)pÞ 2 xx;zz ¼0;
" þð 1 (cid:1)1Þ" " þð 1 (cid:1)1Þ"
1 g xx;zz 2 g xx;zz
x;z x;z
(1)
wherepisthevolumefractionofthecoatednanoparticles,
and"2thepermittivityofwater.ExceptfortheEMA,some
other theories which appear to be more complex can also
be adopted, e.g., the spectral representation approach
[16,17]. For Eq. (1) we have modeled an individual chain
asaspheroidwithg (shapefactorperpendiculartoH)and
x
g (shapefactoralongH)[18].Theysatisfyageometrical
z
sumruleg þ2g ¼1[19].Itisimportanttonotethatthe
z x
result calculated bythe isotropiccounterpart of theEMA,
namely, Eq. (1) with g ¼1=3, was demonstrated to
x;z
reasonablylieintheHashin-Shtrikmanboundswhichpro-
videthetightestconstraintsfortheeffectivepermittivityof
FIG. 1 (color online). 2D finite element simulations of the
a composite comprising an isotropic mixture of two iso-
distribution of the absolute value (denoted by the color or
tropicdielectricmaterials[20].Thusinviewofthephysi-
brightness) of electric field components of an incident TM
calreasonabilityandformalcompactness,weprefertouse wave with (cid:1)¼758 nm, in a ferrofluid system of thickness
the EMA [Eq. (1)] for treating the ferrofluid system. In 1 (cid:3)m containing chains of Fe3O4 nanoparticles coated by an
Eq.(1)"1denotestheequivalentpermittivityofthecoated Ag shell. The ferrofluid system is replaced by an effective
nanoparticle.ItcanbegivenbysolvingLaplace’sequation medium described by the parameters calculated according to
of electrostatics together with appropriate boundary con- Eq. (1). The direction and size of the blue or dark arrows
ditions[21],whichisvalidinourquasistaticsystemwhere indicate, respectively, the direction and magnitude of the local
the incident wavelength (cid:1) is much larger than the size of power flow or Poynting vector.
thecoatednanoparticles.Aftersomestraightforwardderi-
vations, we obtain
a y-directed magnetic field component is incident to the
" ð1þ2(cid:2)Þþ2" ð1(cid:1)(cid:2)Þ ferrofluid system at an angle (cid:4)¼30(cid:3) with respect to z
"1 ¼"a "ffð1(cid:1)(cid:2)Þþ"aða2þ(cid:2)Þ (2) aExqi.s.(1H),ewree(cid:4)obtjauisnt"den¼ote4s:4th3eþa0n:g1l5eioafndin"cide¼nc(cid:1)e.3U:2s9inþg
xx zz
with (cid:2)¼r3=ðrþdÞ3, where r is the radius of the core 0:19i according to the parameters: "1 ¼(cid:1)21:4þ0:9i,
s(Fheel3lO(4A)gw)itwhiptherpmeirtmtivitittiyvi"tyf,"an.dEdquisattihoenth(2ic)kinseassnoefxtahcet i"s21¼(cid:3)1m:7.7H,epre¼"01:h2a2s,nbe¼en5c0a,lcaunldattehdealecncgotrhdionfgetaochEqch.a(2in)
solution for an isolated Fe3Oa4 nanoparticle with an Ag 4w:i8thþ2t:h2ei (apsareaxmpeetreimrs:entarll¼y 5mnemas,uredd¼at5(cid:1)n¼m,758"fn¼m
shell, but it serves as the result of the first-order approxi-
for magnetite, a ferrimagnet and crystallites with the in-
mation forsuch ananoparticlein ourferrofluidsystemby
verse cubic spinel structure [23]), and " ¼(cid:1)20:61þ
neglecting localfieldeffectscausedbyalltheothernano- a
1:27i (which was measured for a Ag thin film at (cid:1)¼
particles.InEq.(1),gz isgivepnffiffiffibffiffiyffiffiffiffiaffiffiffisffiffiimpleapproximation 758 nm [24]). Notewe already take into account the con-
g ¼1=ð1(cid:1)n2Þþnlnðnþ n2(cid:1)1Þ=ðn2(cid:1)1Þ3=2 [22],
z ductivity property of the materials by using the complex
where n(cid:2)2. In reality, n is proportional to the ratio of
permittivityvalues since theconductivityand permittivity
magnetic energy to thermal energy (a lowest-order as-
are related [25]. Figure 1 plots the distribution of the
sumption). Thus, without loss of generality, n will be
absolute value of electric field components, and it clearly
used to equivalently represent the strength of H. This
demonstrates the appearance of negative refraction.
correlation corresponds to the fact that higher H yields
To understand the finite element simulation results de-
larger n [13]. Apparently H ¼0 causes n¼1 or g ¼
z pictedinFig.1,weresorttoatheoreticalapproachforthe
1=3.Inthiscase,thenanoparticlesarerandomlysuspended
systemwithReð" Þ<0andReð" Þ>0.HereReð(cid:4)(cid:4)(cid:4)Þ
in the ferrofluid system, resulting in an isotropic effective zz xx;yy
means the real part of (cid:4)(cid:4)(cid:4). On the basis of Maxwell’s
permittivity.Itis worthnotingthattheEMA [Eq. (1)] has
equations,fortheTMwaveswiththemagneticfieldcom-
included energy dissipation related to absorption. On the
ponent polarized in the y axis and the electric field com-
other hand, the effect of scattering is assumed to be ne-
ponent located in the xz plane, the dispersion relation for
glectedduetothesmallscatteringcrosssectionofachain
thewave propagating in a general anisotropic medium is
with respect to an incident wavelength.
We use the software COMSOL Multiphysics 3.5 to per- ðk2=" Þþðk2=" Þ¼(cid:3) k2; (3)
x zz z xx yy 0
form finite element simulations of optical refraction in
ferrofluids, see Fig. 1. A transverse magnetic (TM) wherek (ork )isthex(orz)componentofwavevectork,
x z
Gaussian beam with a width of 1:6 (cid:3)m in xz plane with andk0 ¼2(cid:5)=(cid:1)thewavenumberinfreespace.Clearly,the
034501-2
week ending
PRL 104, 034501 (2010) PHYSICAL REVIEW LETTERS 22 JANUARY 2010
permittivitytensorterms(" and" )andthepermeability
xx zz
tensorterm((cid:3) )areinvolvedduetotheTMpolarization.
yy
Atoptical frequencies,thepermeabilityof naturalmateri-
als can be taken to unity [19], and therefore the effective
permeabilityoftheferrofluidsystemisaboutunity.Sothe
valueof(cid:3) istakenasunityinEq.(3)andthefollowing
yy
analyses. Negative refraction can be realized in the ferro-
fluid system because of the hyperbolic equifrequency sur-
faceandtheboundaryconditionforelectromagneticwaves
at the interface between a uniaxial medium and an iso-
tropic medium [26]. One might resort to Maxwell’s equa-
tionstocalculatethePoyntingvectordirectly.Thexandz
components of the time-averaged Poynting vector S are,
respectively, given by [27]
k H2 k H2 FIG. 2 (color online). Cross-sectional view of 3D finite ele-
S ¼ x y and S ¼ z y : (4)
x " 2!" z " 2!" ment simulations of the distribution of the absolute value of
zz 0 xx 0 electric field components of an incident TM wave with (cid:1)¼
Then the angles of refraction for the wave vector k and 758 nm, in a ferrofluid system of thickness 1 (cid:3)m containing
Poynting vector S are, respectively, givenby columnsaggregatedbychainsofFe3O4 nanoparticlescoatedby
an Ag shell. For convenience we replace each column with a
(cid:6) ¼tan(cid:1)1ðk =k Þ; (5) solidcylinderbyassumingthesolidcylindertopossessthesame
k x z
electromagnetic responses as the column. This assumption is
(cid:6) ¼tan(cid:1)1ðS =S Þ¼tan(cid:1)1½k " =ðk " Þ(cid:5): (6) reasonable,atleasttosomeextent,sincetheincidentwavelength
s x z x xx z zz
ismuchlargerthanthesizeofcoatednanoparticles.Wesetthe
For the system with Reð" Þ<0 and Reð" Þ>0, there cross sections of the cylinders to exist in a hexagonal lattice
zz xx
exists positive refraction for k, because of the hyperbolic since equal spacing between columns was experimentally ob-
equifrequency contour, the conservation of the tangential served[11].Theradiusofthecylinderis30nm,andthecenter-
to-center separation between two adjacent cylinders is 122 nm.
wavevector, and the causality theorem [27]. On the basis
of Eqs. (3) and (6), the effectiverefractive index
0:9i. Also, a TM Gaussian beam with a width of 1:6 (cid:3)m
(cid:7)¼sin(cid:4)=sin(cid:6)s (7) isincidentat(cid:4)¼30(cid:3).Figure2showstheresultsimulated
can be computed. When Reð" Þ<0 and Reð" Þ>0, by the 3D finite element method based on the software
zz xx
Eq. (6) [or Eq. (7)] can yield Reð(cid:6) Þ<0 [or Reð(cid:7)Þ<0], COMSOL Multiphysics 3.5. It clearly demonstrates the ap-
s
namely, negative refraction for the Poynting vector, as pearanceofopticalnegativerefraction,whichechoeswith
shown in Fig.1. Fig. 1. The excellent agreement is expected, since the
A 3D finite element method beyond the EMA.—In the EMA is valid considering the size and spacing of nano-
aboveanalyses,weusetheEMA[Eq.(1)]tocalculatethe particles are much smaller than the wavelength (cid:1). The
effective permittivity of the ferrofluid system containing local periodic structures of the chains or columns will
chains. As a result, negative refraction has been revealed. not significantly influence the optical property, but can be
Experimentalresults[11,12]demonstratedthatthereexist incorporated as a high-order approximation based on the
columnswithequalspacinginconfinedferrofluidfilmsin Ewald-Kornfeld formulation [16].
thepresenceofanin-planeH.Accordingly,wededucethat Numerical results.—The agreement between Fig. 1 and
there also exist such columns with equal spacing in the 2 gives us confidence to further investigate the optical
ferrofluid system of our interest. This should be a reason- properties of the ferrofluid system by using the EMA.
able deduction because similar to the confined ferrofluid Figure 3 shows the real part of (cid:7) as a function of n, p,
films, our system possesses the same dipolar interaction and(cid:4)at(cid:1)¼758 nm.ItisworthremindingthathigherH
between(locallydisplaced)chainsthatcreatescolumns.In leadstolargern.EvidentlyhighHtogetherwithsuitablep
order to show thevalidity of the above analyses using the can lead to negative refraction, Reð(cid:7)Þ<0. Namely, one
EMA, we conduct 3D finite element simulations for a canachievethetransitionfrompositiverefractiontonega-
ferrofluid system containing columns with equal spacing. tiverefraction for the Poynting vectorby tuning H and/or
Certainly if we investigate the same system with chains p. Further the negative refraction Reð(cid:7)Þ<0 is shown to
rather than columns, in principle the present 3D simula- existwithinthefullrangeof(cid:4)(namely,all-anglenegative
tionswouldworkthesame.However,theyarequitetime- refraction). Figure 4 displays the imaginary part of (cid:7),
consuming. For the comparison with Fig. 1 (2D simula- namely extinction coefficient (which is proportional to
tions), in Fig. 2 we also take "2 ¼1:77 and the volume optical absorption). We find it varies with respect to H,
fraction of the columns p¼0:22. Each column has a p,and/or(cid:4).Inotherwords,oneisallowedtoachievelow
length of 1:0 (cid:3)m and a permittivity of "1 ¼(cid:1)21:4þ opticalabsorptionbychoosingH,p,and(cid:4)appropriately.
034501-3
week ending
PRL 104, 034501 (2010) PHYSICAL REVIEW LETTERS 22 JANUARY 2010
(sterically) or decorating them with charged groups (elec-
trostatically).Themagnetocontrollablenegativerefraction
inferrofluidspresentedherepromisesanewregimeanda
number of potential applications of metamaterials. The
field-induced-assembly system of nanoparticles may real-
ize a large specimen which is difficult to fabricate using
top-down techniques. Moreover, we can engineer the spa-
tial dielectric constant by the external magnetic fields for
reconfigurable optical devices, such as lenses [27], invis-
ible cloaks [28], and waveguides.
This work was supported by the NNSFC under Grant
FIG. 3(coloronline). Realpartoftheeffectiverefractiveindex
Reð(cid:7)ÞcalculatedaccordingtoEq.(7)asafunctionofn,p,and Nos.10604014and10874025,byCNKBRSFunderGrant
(cid:4) (degree) at (cid:1)¼758 nm. Three cross sections correspond to No. 2006CB921706, and by the RGC General Research
n¼10, p¼0:25, and (cid:4)¼70(cid:3), respectively. Fund of Hong Kong SAR Government. X.Z. acknowl-
edges funding support from U.S. Army Research Office
(ARO) MURI program50432-PH-MUR.
We also confirmed that (all-angle) broadband negative
refraction can be achieved in such a ferrofluid system by
investigatingtwootherincidentwavelengths(cid:1)¼661 nm
and 706 nm at n¼50 and p¼0:22 (no figures shown *[email protected]
[1] V.G. Veselago, Sov. Phys. Usp. 10, 509 (1968).
herein).Forthecalculations,weadoptedtheexperimental
data: " ¼(cid:1)20:09þ0:45i[24]and " ¼4:8þ2:8i[23] [2] R.A. Shelby et al., Science 292, 77 (2001).
a f [3] J. Yao et al., Science 321, 930 (2008).
at (cid:1)¼661 nm, and " ¼(cid:1)23:40þ0:39i [24] and " ¼
a f [4] E. Cubukcu et al., Nature (London) 423, 604 (2003).
5:0þ2:4i [23] at (cid:1)¼706 nm. Clearly, the wavelength [5] H.ShinandS.H.Fan,Phys.Rev.Lett.96,073907(2006).
bandwidth canbewider for the system. [6] J.B. Pendry, Science 306, 1353 (2004).
In summary, for the first time, we have demonstrated [7] A. Pimenov et al., Phys. Rev. Lett. 95, 247009 (2005).
that magneto controllable all-angle broadband negative [8] J.B. Pendry, Phys. Rev. Lett. 85, 3966 (2000).
refractionatopticalfrequenciescanberealizedinaqueous [9] S. Odenbach, Magnetoviscous Effects in Ferrofluids
ferrofluids,whicharemadeofFe3O4 nanoparticlescoated (Springer, Berlin, 2002).
[10] K. Butteret al., Nature Mater. 2, 88 (2003).
byanAgshell.Inlookingforexperimentalrealization,the
[11] M.Klokkenburgetal.,Phys.Rev.Lett.97,185702(2006).
most important point is that the ratio between magnetic
[12] M. Klokkenburget al., Phys. Rev. E 75, 051408 (2007).
energy and thermal energy must be high enough, which
[13] Z.W. Wang et al., Phys. Rev. E 66, 021405 (2002).
causes the formation of chains and/or columns. Since the [14] R.A. Trasca and S.H.L. Klapp, J. Chem. Phys. 129,
permanent magnetic moment of Fe3O4 nanoparticles is 084702 (2008).
about 2:4(cid:6)104(cid:3) [9] ((cid:3) : the Bohr magneton), we can [15] If the chains have different n, a volume average may be
B B
estimate the threshold magnetic field 14:8 kA=m above adopted for Eq. (1) instead.
which the corresponding magnetic energy can overwhelm [16] J.P. Huang and K.W. Yu,Phys. Rep. 431, 87 (2006).
the thermal energy 1=40 eV at room temperature so as to [17] D.J. Bergman, Phys. Rep. 43, 377 (1978).
[18] The shape factors are the same as depolarization or
obtain appreciable H-induced formation of chains/col-
demagnetization factors for spheroids.
umns. Practically one might choose a higher H together
[19] L.D.Landau,E.M.Lifshitz,andL.P.Pitaevskii,Electro-
withalowertemperature.Ontheotherhand,thesystemat
dynamics of Continuous Media (Pergamon, New York,
zerofield(H ¼0)maybestabilizedagainstagglomeration 1984), 2nd ed.
by coating nanoparticles with long polymer molecules [20] D.J. Bergman and D. Stroud, Solid State Phys. 46, 147
(1992).
[21] T.B. Jones, Electromechanics of Particles (Cambridge
University Press, Cambridge, England, 1995).
[22] C.Z. Fan and J.P. Huang, Appl. Phys. Lett. 89, 141906
(2006).
[23] A. Schlegel et al., J. Phys. C 12, 1157 (1979).
[24] P.B. Johnson and R.W. Christy, Phys. Rev. B 6, 4370
(1972).
[25] J.D. Jackson, Classical Electrodynamics (John Wiley &
Sons, New York, 2001), 3rd ed.
[26] P.A. Belov, Microwave Opt. Technol. Lett. 37, 259
(2003).
[27] Y.M. Liu et al., Opt. Express 16, 15439 (2008).
FIG. 4(coloronline). SameasFig.3,butforimaginarypartof
[28] J.B. Pendry et al., Science 312, 1780 (2006).
the effective refractiveindexImð(cid:7)Þ.
034501-4