Asymptotics for metamaterials and photonic crystals.

1.2 MB·English

by  Antonakakis, T.

Asymptotics for metamaterials
and photonic crystals
T. Antonakakis1,2,R.V.Craster1andS.Guenneau3
rspa.royalsocietypublishing.org 1DepartmentofMathematics,ImperialCollegeLondon,
LondonSW72AZ,UK
2EuropeanOrganizationforNuclearResearch,CERN1211,Geneva23,
Switzerland
Research 3FresnelInstitute,UMRCNRS7249,Aix-MarseilleUniversity,
Marseille,France
Citethisarticle:AntonakakisT,CrasterRV,
GuenneauS.2013Asymptoticsfor
Metamaterial and photonic crystal structures are
metamaterialsandphotoniccrystals.ProcR central to modern optics and are typically created
SocA469:20120533. frommultipleelementaryrepeatingcells.Wedemon-
http://dx.doi.org/10.1098/rspa.2012.0533 stratehowonereplacessuchstructuresasymptotically
by a continuum, and therefore by a set of equations,
that captures the behaviour of potentially high-
Received:7September2012
frequency waves propagating through a periodic
Accepted:14January2013 medium.Thehigh-frequencyhomogenizationthatwe
userecoverstheclassicalhomogenizationcoefficients
in the low-frequency long-wavelength limit. The
SubjectAreas:
theory is specifically developed in electromagnetics
appliedmathematics,wavemotion for two-dimensional square lattices where every cell
contains an arbitrary hole with Neumann boundary
Keywords: conditionsatitssurfaceandimplementednumerically
metamaterials,cloaking,negativerefraction, for cylinders and split-ring resonators. Illustrative
numerical examples include lensing via all-angle
homogenization
negativerefraction,aswellasomni-directiveantenna,
endoscopeandcloakingeffects.Wealsohighlightthe
Authorforcorrespondence: importanceofchoosingthecorrectBrillouinzoneand
R.V.Craster the potential of missing interesting physical effects
e-mail:[email protected] dependinguponthepathchosen.
1. Introduction
Photonic crystal (PC) media [1,2] and metamaterials [3]
are topical areas in optics, and both involve non-
resonant or resonant interactions created by waves
within periodic structures. Such structures are of
considerable current interest [4], with applications to
invisibilityandcloaking[5],amongothers.Inphotonics,
a typical structure may involve multiple cylindrical
holes [6], and in metamaterials, the peculiar properties
(cid:2)c 2013TheAuthors.PublishedbytheRoyalSocietyunderthetermsofthe
CreativeCommonsAttributionLicensehttp://creativecommons.org/licenses/
by/3.0/,whichpermitsunrestricteduse,providedtheoriginalauthorand
sourcearecredited.
(a) (b)
2
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Brillouin zone .a
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.3
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of split-ring resonators (SRRs) [7] are typical building blocks. In both cases, the physics is
neatly encapsulated and displayed by dispersion diagrams that relate the Bloch wavenumber
in the irreducible Brillouin zone to the frequency; these illustrate essential features such as
band gaps of frequencies where propagation in an infinite periodic structure is disallowed.
Such dispersion diagrams for infinite periodic media can then be used to design the size and
geometry of structural elements within a cell to create particular optical features. We turn to
arraysofcylindricalholesandtoSRRsinordertoillustratetheversatilityofanewtechniquethat
creates effective homogenized models, even at high frequencies, and in doing so also uncover
details of how the SRR geometry affects the metamaterial properties. In this article, we treat a
specific polarization in electromagnetism, transverse electric (TE), whereby the magnetic field
is perpendicular to the plane of periodicity, and we consider perfectly conducting structural
elements,suchthattheholeshaveaNeumannconditionuponthem;thisisofparticularinterest
astherehasbeendiscussionthathomogenizationtheoryisinvalidforthiscase.
Many materials of abiding interest in physics are created from periodically repeating cells
such as those of figure 1a, which shows a square array of SRRs with a square cell as
the dashed square. Given such a medium, with, say, a defect or many hundreds of cells
within a macrocell, it is attractive to replace it with an effective medium on a macroscale;
naturally, one hopes that the effective replacement continuum model material captures the
behaviour created by the microscale. For long waves, in the quasi-static low-frequency limit,
there is an established theory, homogenization theory [4,8–11], that replaces a microstructured
medium with an averaged macroscale model; this is highly successful and an attractive
approachforlow-frequencywaveswithwavelengthsmanytimesthetypicalcellsize.However,
many of the features of interest in real PCs, or other periodic structures, such as all-angle
negative refraction (AANR) [12,13] or ultra-refraction [14] occur at high frequencies where the
wavelengthandmicrostructuredimensionareofsimilarorders.Therefore,theconventionallow-
frequencyclassicalhomogenizationclearlyfailstocapturetheessentialphysics,andadifferent
approachtodistillthephysicsintoaneffectivemodelisrequired.Fortunately,ahigh-frequency
homogenization (HFH) theory, developed in Craster et al. [15], is capable of capturing features
such as AANR and ultra-refraction for some model structures [16]. Somewhat tangentially,
there is existing literature in the analysis community on Bloch homogenization [17,18] and, in
particular,Hoefer&Weinstein[19],whichisrelatedtowhatwecallHFH.Thereisalsoliterature
on developing homogenized elastic media, with frequency-dependant effective parameters,
based upon periodic media [20]. There is therefore considerable interest in creating effective
3
continuummodelsofmicrostructuredmediathatbreakfreefromtheconventionallow-frequency
homogenization limitations. In this article, we turn our attention to microstructures of abiding
interest, perfectly conducting holes (in TE polarization) that have been much studied in the ...rspa
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Interesting details such as the behaviour at crossing points, and degenerate behaviour, where ..cie
thelocalbehaviourinthedispersioncurveswitchesfromquadratictolinearareallfoundfrom ...typ
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effectivemacroscaleequationsthathavethemicroscalecompletelycapturedwithincoefficients. ...blis
IntheSRRcase,thehomogenizationproceduremustbeperformednumerically,andthisisdone ...hin
hweirteh;inheancceel,l.with minor modifications, the methodology can now be applied to any geometry .....g.org
conOclnustihoenstoapboicutotfhqeupaossis-isbtailtiitcyhofoumsoinggenhiozmatoiogne,naizasteiroinesatoafllp,eavpeenrsfo[r23th–e25lo]wderastwascopuesrtpiclebxainndg ......ProcR
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in the dispersion diagram. For doubly periodic perfectly conducting inclusions with Dirichlet ....ocA
(patnerardinocsdovinecrivtseyen)mtbiooanguannledhtaiocrym(TocMogne)ndpiiztoialoatnirosizn,attfhiaoeirlnse,;iit.she.enrwoeiitinhstenarnoceecploetncottrrfoictvhefieresalydcowpueistrthpicetnhbdirsaicnducelhadruwtcoittihothnte.heNplooartnaiegbilonyf,, ........469:2012
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the HFH has no such failing, as shown in Craster et al. [15]. The discrepancy in the literature .5
.3
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iswith(cylindrical)perfectlyconductinginclusionswithNeumann(TEpolarization)boundary
conditions for which analytical multi-pole methods [24] produce linear asymptotics as the
acoustic branch approaches zero wavenumber and the precise behaviour is not replicated by
conventional homogenization; the slope of the asymptotics is important as it is related to the
effectiverefractiveindexofthemediumforlongwaves.Furthermore,ifthecylindershavefinite
dielectricproperties,thenthelimitsofzeroconductionandofzerowavenumberdonotcommute.
Complementarytothisareclaimsthathomogenizationtheorywillactuallyoperatecorrectly,see
the comments and responses by Halevi et al. [26] and Nicorovici et al. [27]. Here, we advance
thetheoryofhomogenizationtohigherfrequenciesand,asacorollary,areabletodemonstrate
conclusively,usingourapproach,thatonecanindeedhomogenizeperfectlyconductingcylinders
intheTEpolarizationforthequasi-staticlow-frequencylimit.
Parallel to the electromagnetic setting is a mathematically identical interest in, mainly
cylindrical, periodic inclusions in acoustics, water waves and anti-plane elasticity. Homogeni-
zation in those settings for long waves relative to the cell spacing leads to effective
equations [28,29] that do not appear to have any issues, and if, furthermore, the inclusions are
takentobesmall,thensingularperturbationtheorycanbeemployed[30],orasymptoticcoupled
modetheory[31],togoodeffect.
WegenerateanHFHtheoryforrepeatingcellscontainingNeumanninclusions;thislimiting
caseisnotcoveredwithinCrasteretal.[15]andisofindependentinterest,evenatlowfrequencies.
The general high-frequencymodel is developed in §2, with the low-frequencylimit covered in
a §2b. Armed with the general theory, we verify its efficacy upon the well-studied cylindrical
inclusion case (§3a), which allows us to demonstrate that homogenization does indeed work
despite arguments to the contrary, and for all frequencies, not just in the low-frequency limit.
ThecylindricalinclusionscontrastwiththeSRRs,whereadditionalbranchessplitfromthosein
thecylindricalcase,andforwhichinterestingasymptoticresultsemergein§3b.Section4givesan
overviewofcomplementarygeometricalasymptoticlimits.TheHFHasymptoticsgiveadditional
physicalinsightthatthenmotivatesustoexplorefurtherfeaturesoftheSRRsandcylindersin§5.
Finally,concludingcommentsandremarksarepresentedin§6.
2. Generaltheory
For infinite perfectly periodic media, consisting of elementary cells that repeat, one focuses
attentiononasingleelementarycell;quasi-periodicFloquet–Blochboundaryconditionsdescribe
the phase shift as a wave moves through the material, and dispersion relations then relate the
Blochwavenumber,thephaseshift,tofrequency.TheBlochwavenumberisavector,andfigure1b
4
shows the irreducible Brillouin zone [32] ΓXM associated with a single repeating elementary
squarecellcontaining,say,acircularhole,alsoshownisthesmallertriangleforagroupoffour
repeating cells. The dispersion diagrams we show are frequency versus wavenumber around ...rspa
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pathΓXM)andthatanalmostflatbandoccursforanarrayofcylinders;thisflatbandleadsto ...typ
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directionalstandingwavepatterns.Wealsonotethatthesymmetryoftheholeisimportant,and ...blis
forthetwothinligamentSRRoffigure1a,oneshouldusethesquareΓMXN. ...hin
andThweheeingetnhseosleuteioignesnstholauttieomnsergareeapreerftehcetlyBloinchpmhaosdeeosraotuthteofedpgheasseofactrhoessBrthilelouceinll,ztohneen, .....g.org
sAtasynmdipntgotwicatveecshenxiqisut,esanbdastehderaeroaurendstahnigdhin-fgrewquavenecfyrelqounegn-wcieasve(washyomsepftroetqicusehnacvieesrceacnenbtelyhbigehen). ......ProcR
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developed [15], and Schrödinger ordinary differential equations in one-dimensional periodic ....ocA
media (or partial differential equations in two dimensions) emerge; this approach also works .4
pfoerrfmecictlryosctorundctuucrteidngdihsoclreestea[n3d4]hoarvferammaet-elriikaelmpreodpiear[t3ie5s].vTahreysinegrepceernitotdhiecoalrliyesoanvtohidesthcaeliessoufethoef .......69:2012
.0
elementarycellandonlytreatmodelproblems,mainlyinonedimension,forwhichcompletely .5
.3
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analytic progress can be made. The key idea for periodic media is to replace the complicated
microstructuredmediumwithanequivalent,effective,continuumonamacroscale,thatis,one
wishes to homogenize the medium, even when the wavelength and microstructure may be of
similarscales.
Thetheoryisultimatelynotlimitedtojustreproducingdispersioncurvesasymptotically,and
it can be adjusted to treat localized defect modes and other features due to local non-periodic
materialchangesorboundaries,withtheseeffectscomingthroughinextraforcingtermswithin
thecontinuumpartialdifferentialequations.
Webeginwithatwo-dimensionalstructurecomposedofasquarelatticegeometryofidentical
cellswithidenticalholesinsideeachofthem.Thesidelengthofthedirectlatticebasevectors,i.e.
the side of each square cell, is taken as 2l. Note that for simplicity, equal length lattice vectors
and a square lattice are assumed, and both assumptions could be relaxed. These elementary
cellsdefinealengthscalethatisthemicroscaleofthestructure.Asnotedabove,realstructures
couldbecreatedfrommanyhundredsorthousandsofsuchelementarycells,andweintroducea
macroscalelengthdenotedbyLthatcouldbeviewedasacharacteristicoveralldimensionofthe
structure.Theratioofthesescales,(cid:3)≡l/L,isassumedsmall.
Each cell is identical in geometry and the material within each cell is characterized by
two periodic functions, in ξ≡(x1/l,x2/l), namely aˆ(ξ) and ρˆ(ξ). Depending on the application,
these could be stiffnesses and density for shear horizontal polarized elastic waves or inverse
permittivityandpermeabilityinelectromagnetism.Thegeometryisspecificinthesensethatit
containsanarbitraryhole,orsetofholes,andboundaryconditionshavetobeprescribedonthe
hole.Inthisstudy,Neumannconditionswillbeusedthatarethenaturalboundaryconditionsfor
perfectlyconductingholesinthetransverseTEpolarizationofelectromagnetismorforstress-free
holesinanti-planeshearelasticity.
A time-harmonic dependence of propagation exp(−iωt), with frequency ω, is assumed
throughout,andhencef(cid:2)orthsuppressed,andanon-dimensionalizationbysettingaˆ≡aˆ0a(ξ)and
ρˆ≡ρˆ0ρ(ξ),wherecˆ0= aˆ0/ρˆ0isthecharacteristicwavespeed,leadstotheresultingequationof
study,
ωl
l2∇x·[a(ξ)∇xu(x)]+Ω2ρ(ξ)u(x)=0, with Ω= cˆ , (2.1)
0
on−∞<x1,x2<∞;Ω isthenon-dimensionalfrequencyanduistheout-of-planedisplacement
inelasticityortheH3componentofthemagneticfieldinTEpolarization.
The two-scale natureof the problem is incorporatedusing the small and large lengthscales
5
to define two new independent coordinates, namely X=x/L, and ξ=x/l. Equation (2.1) then
becomes
..rsp
Sta∇nξd·in[ag(ξw)∇avξeus(Xo,cξc)u]r+wΩhe2nρ(tξh)eur(eXa,rξe)p+er(cid:3)i[o2da(icξ)(∇orξa+n∇ti-ξpae(ξri)o]d·i∇cX)ub(oXu,nξd)a+ry(cid:3)2cao(nξd)∇itiX2oun(sXa,ξcr)o=ss(02t..h2e) ..........a.royalsocie
elementary cell (in the ξ coordinates), and these standing waves encode the local information ..ty
.p
aboutthemultiplescatteringthatoccursbytheneighbouringcells.Theasymptotictechniqueis ..ub
then a perturbation about these standing wave solutions, as these are associated with periodic ...lish
andanti-periodicboundaryconditions,whichare,respectively,in-phaseandout-of-phasewaves ...ing
acrossthecell;theconditionsinξ ontheedgesofthecell,∂S1,areknown, .....org
u|ξi=1=±u|ξi=−1 and u,ξi|ξi=1=±u,ξi|ξi=−1, (2.3) ......ProcR
wfieiltdhathned+theorfr−eqfuoernpcey,riodicoranti-periodiccases,respectively.Wenowposeanansatzforthe .....SocA
u(X,ξ)=u0(X,ξ)+(cid:3)u1(X,ξ)+(cid:3)2u2(X,ξ)+···⎫⎬⎭ (2.4) ........469:2012
and Ω2=Ω02+(cid:3)Ω12+(cid:3)2Ω22+···. ...053
.3
The u(X,ξ)’s adopt the boundary conditions (2.3) on the edge of the cell. An ordered set of
i
equationsemergeindexedwiththeirrespectivepowerof(cid:3),andaretreatedinturn,
(au0,ξi),ξi+Ω02ρu0=0, (2.5)
(au1,ξi),ξi+Ω02ρu1=−(2au0,ξi+a,ξiu0),Xi−Ω12ρu0 (2.6)
and (au2,ξi),ξi+Ω02ρu2=−au0,XiXi−(2au1,ξi+a,ξiu1),Xi−Ω12ρu1−Ω22ρu0. (2.7)
The leading order equation (2.5) is independent of the long scale X and is a standing wave on
theelementarycellexcitedataspecificeigenfrequencyΩ0 andassociatedeigenmodeU0(ξ;Ω0),
modulatedbyalong-scalefunctionf0(X)andso
u0(X,ξ)=f0(X)U0(ξ;Ω0). (2.8)
Atthispoint,wewillassumeisolatedeigenfrequencies,butrepeatedeigenvaluesariseandare
discussedlater.Theentireaimistoarriveatapartialdifferentialequationforf0 posedentirely
uponthelongscale,butwiththemicroscaleincorporatedthroughcoefficientsthatareintegrated,
notnecessarilyaveraged,quantities.
Before we continue to the next order, equation (2.6), we define the Neumann boundary
conditionsontheholes,∂S2,
∂u
∂n=u,xini|∂S2=0, (2.9)
where n is the unit outward normal to ∂S2, and which in terms of the two scales and ui(X,ξ)
become
U0,ξini=0, (U0f0,Xi+u1,ξi)ni=0 and (u1,Xi+u2,ξi)ni=0. (2.10)
The leading order eigenfunction U0(ξ;Ω0) must satisfy the first of these conditions. Moving to
the first-order equation (2.6), we invoke a solvability condition by integrating over the cell the
productofequation(2.6)andU0minustheproductofequation(2.5)andu1/f0(X).Theeigenvalue
Ω1 iszero,andwecansolveforu1=f0,XiU1i(ξ),soU1 isavectorfield.Byre-invokingasimilar
solvabilityconditionforequation(2.7),weobtainthedesiredpartialdifferentialequationforf0
⎫
Tijf0,XiXj+Ω22f0=0, where ⎪⎪⎬
Tij=(cid:2)(cid:2) ρtUij2dS fori,j=1,2⎪⎪⎭ (2.11)
S 0
posedentirelyonthelongscaleX.Thetensort consistsofintegralsoverthemicrocellinξ and
ij 6
isultimatelyindependentofξ.Theformulationsfort read
ij
(cid:3)(cid:3) (cid:3)(cid:3)
and tii= SaU02d(cid:3)(cid:3)S+ Sa(U1i,ξiU0−U1iU0,ξi)dS fori=1,2 (2.12) ..........rspa.royalso
tij= Sa(U1j,ξiU0−U1jU0,ξi)dS fori(cid:7)=j. (2.13) ....ciety
.p
.u
Thereisnosummationoverrepeatedsufficesforthet .Notably,althoughthegeneralapproach .b
ii ..lis
followsCrasteretal.[15],therearesubtledifferencesinducedbytheNeumannconditions;the .h
nsoolulotinognetrhuerseeftohrethaeuxfiirlisat-royrdfuenrcetqiounatVio1n(ξis;Ωof0)t,hwehfoicrmh tUur1n(ξs;Ωou0t)=toVb1e(ξn;uΩm0)e−ricξaUlly0(ξa;wΩk0w).aWrde. .......ing.org
Icpfiononansnrittt∂deeiSaai-t2ledi.old,enTimsfshofeeaelvnrsenittnnhupgtemaiacldelekriaearidecqgaciuentlal[gys3tioo6oflr]nou,d,rtteih(orUaeneUr1qsei1bujoi,yaξsfit)aiUm,oξllin0ou+,wcoahnΩinn∂d0s2gSiρms1uUuspa1bnljtesod=er.tqw−ruUei(aet12hntiagttUilhsey0en,aξUesire+1sacjo,lolagnau,ξerdtieiUoobmc0noo)eu,motnwrfipdeituatshhr.teeytdhcnoeounnssdi-anhimtgoioemanbossogtouaefnnn(d2de.oaa1urr0dys) ...................ProcRSocA469:2012
(a) Repeatedeigenvalues ..05
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A potential limit case not treated above is that, for some standing wave frequencies, there is
more than one propagating mode, i.e. there are repeated eigenfrequencies with multiplicity p.
Thegeneralsolutiontotheleadingorderproblemthenbecomes
u0=f0(l)(X)U0(l)(ξ;Ω0), (2.14)
withsummationassumedoverrepeatedsuperscriptsl,and
(cid:7) (cid:8)
∂
A +Ω2B fˆ(l)=0 for m=1,2,...,p, (2.15)
∂X jml 1 ml 0
j
withΩ1notnecessarilyzero,and
(cid:3)(cid:3) (cid:3)(cid:3)
Ajml= Sa(U0(m)U0(l,)ξj−U0(m,ξj)U0(l))dS and Bml= ρU0(l)U0(m)dS. (2.16)
Thecoupledsystemofpartialdifferentialequations(2.15)forthef(l)(X)aresolved,butbecome
0
degenerateifΩ1=0.Formostoftheexamplestreatedinthisarticle,Ω1 isfoundtobezero.We
thenhavetoproceedinasimilarwayusedtoobtainequation(2.11).Wegetanotherdegenerate
casewherethecoupledpartialdifferentialequationsbecome
(cid:7) (cid:8)
∂2 ∂2
A + D +Ω2B fˆ(l)=0 form=1,2,...,p, (2.17)
∂X∂X ml ∂X ∂X kjml 2 ml 0
i i k j
withthefollowingcoefficients:
(cid:3)(cid:3) (cid:3)(cid:3)
A = aU(l)U(m)dS, B = ρU(l)U(m)dS (2.18)
ml 0 0 ml 0 0
and (cid:3)(cid:3)
Dkjml= Sa(U0(m)U1(lk),ξj−U0(m,ξj)U1(lk))dS, (2.19)
andtherangeofvariationoflisequaltothemultiplicityoftheeigenvalue.
(b) Theclassicallong-wavezero-frequencylimit
The current theory simplifies if one enters the classical long-wave, low-frequency limit where
Ω2∼O((cid:3)2) as U0 becomes uniform, and without loss of generality is set to be unity, over the
elementarycell.Thefinalequationisagain(2.11)wherethetensort simplifiesto
ij 7
(cid:3)(cid:3) (cid:3)(cid:3) (cid:3)(cid:3)
tii= SadS+ SaU1i,ξidS and tij=(cid:2)(cid:2)SaU1j,ξidS fori(cid:7)=j (2.20) ...rspa
s(woliuthtionnoosufmmationoverrepeatedsuffices)andTij=tij/ SρdS.Intheaboveequations,U1i isa ........royalso
(aU1j,ξi),ξi=−a,ξj, (2.21) ....ciety
wciricthlebsoournSdRaRrsyicnoanndiottiohnersw(fi0s,Xeih+omu1o,ξgi)ennie=ou0somnetdhieuhmo,leabisocuonndsataryn.tIsnotehqeuilaltuiostnra(t2i.v2e1)eixsatmheplseasmoef .....publis
.h
aarsethatforU0,butwithdifferentboundaryconditions.ThespecificboundaryconditionsforU1j ......ing.org
.
where ni represents the normal vUec1tj,oξirnic=om−pnojnefnotrs.j=Th1e,2,role of U1 is to ensure Neum(2a.2n2n) .......ProcRS
boundary conditions hold and the tensor contains simple averages of the stiffness and density ....ocA
(cceloaqrsurseiicvctaaiollenenxttlpeyrrmeisnstivhoenartsfetoarkpetehsrmeinihttotoimvaicotycgoeuannniztdetdhpeecorbmeofuefiancbideialnirttyyincfooanrdsiTctaiEolanrmswoadtaev∂seS)2e.sqEuupqauptialoetnmionewn(itt2eh.d20p)beryisiottdhhieec ........469:2012
.0
coefficienta;(2.21)isthewell-knownannexproblemofelectrostatictypesetonaperiodiccell[8,9] ..53
.3
andalsoholdsforthehomogenizedvectorMaxwellsystem,whereU1nowhasthreecomponents
andi,j=1,2,3[37].
3. Illustrativeexamples
WenowillustratethetheoryusingarraysofcircularholesandSRRsandforperfectstructuresfor
whichfulldispersiondiagramscanbefoundnumerically.Tocomparewiththegeneraltheory,
wenowspecializeinBlochwaveswhereu(x+2lB)=u(x)exp(2liκ·B),whichtranslatesintwo-
scalecoordinatesasu(X+2(cid:3)B,ξ)=u(X,ξ)exp(2iκ·B),whereBiseitherb1,b2 orb1+b2 with
biastheorthonormalunitvectors.Floquet–Blochboundaryconditionsonthecellimplyf0(X)=
exp(iκX/(cid:3)).Inthisnotation,κ =K −d andd =0,π/2,−π/2dependingonthelocationinthe
j j j j j j
Brillouinzone.Equation(2.11)andthefrequencyexpansionofequation(2.4)leadto
T
Ω∼Ω0+ 2Ωij0κiκj, (3.1)
withsimilarresultsfor(2.17);thus,onecancomparedirectlywiththefullnumerics.Itisworth
whilenotingtheuseofT coefficients,astheirsignandabsolutevaluegiveinformationabout
ij
thegroupvelocityforthespecifiedfrequenciesandlocationsoftheBrillouinzone.
(a) Latticeofsquarecellswithcircularinclusions
The dispersion curves for arrays of cylindrical holes have been treated by many authors [38]
amongothers,andweproceedbycomputingthemnumericallyusingCOMSOLMULTIPHYSICS
andbyusingtheasymptoticsdevelopedin§2.Wechoosetoillustratethemfortwoholeradii,
withthegeometryasasquareofsidelength2andtheinclusion’sradiusiseither0.4or0.8,in
figures2and3,respectivelyaroundtheedgesoftheirreducibleBrillouinzoneoffigure1b.These
dispersion diagrams illustrate interesting features such as a Bragg stop band (due to multiple
scatteringoflightbetweenthecircularinclusions)thatisabsentforthesmallholes(figure2),but
whichdevelopsforlargerholes(figure3)asthemultiplescatteringbecomesmorepronounced.
Inbothfigures2and3,itisclearthattheasymptoticscapturethefinedetailsofthedispersion
curves near each standing wave frequency. The classical long-wave limit near the origin has
a linear asymptotic dispersion curve, from §2b, and these capture the gradient of numerically
calculated paths near the origin perfectly in both figures, showing that homogenization can
indeed be used in TE polarization. Figure 2b,c shows expanded regions near repeated roots
(a)5.0
(b) 3.3 8
4.5
3.1 ..rsp
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W 2.5 ....org
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2.0 (c) 2.0 ......ProcR
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1.0 W 1.9 ........469:2012
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0.5 ..33
0 1.8
–1 0 1 2 3 2 X 2.5
M G X M wavenumber
wavenumber
Figure2.Thedispersiondiagramforanarrayofsquarecells(side2)withcircularinclusionsofradius0.4.(a)Dispersion
curvesfromnumerics(solidlines),theasymptoticsolutionsfromHFHtheory(dashedlines)andthelinearlong-waveclassical
homogenizationasymptoticsarethedottedlinesemergingfromtheorigin.(b,c)Enlargementsnearrepeatedeigenvalues
wheretheasymptoticsfrom(2.17)areused.
usingtheasymptoticsof§2awiththetheorycapturingfinedetailssuchaschangesincurvature;
notably, all of this behaviour is encapsulated in the tensor T that is independent of the
ij
microscalecoordinates.
Astheholeradiusincreases,astopbandopensupwiththeacousticbranchisolatedfromthe
others(figure3a),andthechangesinthefielduareconcentratedalongtherelativelythinpieces
ofremainingmaterial;atypicaleigensolutionisshowninfigure3c.Thismotivatesonetoreplace
thecirculararraybyanarrayofsimplestringsorthinligamentsintheformofasquareframe,for
whichthedispersionrelation
2cos(2Ω)=cos(2κ1)+cos(2κ2) (3.2)
iseasilyfound[35,39].Thisdispersionrelation,fortheacousticbranch,isshowninfigure3band
isagoodapproximationtothatoflargeholescapturingthemainfeatures.Ifoneconsidersthe
(cid:3)
pathMX intheBrillouinzone,onenoticesthat(3.2)givesacompletelyflatband;thisisrelated
tothestrikingoccurrenceofdirectionalstandingwaves[33,40–42]thatareofcurrentinterestin
discreteorframe-likestructures.Notably,thecontinuumsystemofcylinderssharesthisfeature,
althoughthepathisnolongerperfectlyflat,suggestingthatdirectionalstandingwavesforming
cross-likevibrationswillexistherealso;thisisexploredin§5.
(b) Split-ringresonator
Wenowmodifythesimplecircularholebyinsertingasmallercircularinclusionwithinitattached
tothehole’swallsbyligaments.Theratioofwidthtolengthoftheligamentsused,aswellastheir
number, play a major role in the underlying physics. Again, there have been many numerical
(a)
9
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3.0 ...typ
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W 2.5 ..lis
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(b) (c) .3
1.5
1.0
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Figure3.Thedispersiondiagramisshownforsquarecells(ofside2)withcircularinclusionsofradius0.8.(a)Followsfigure2a,
butforthislargerradius.(b)Theacousticbranch(solidlines)withthedispersioncurvesfromaframeofthinstringsalsoshown
(dashedlines).(c)TheeigensolutionatMontheacousticbranchthatisinphaseverticallyandoutofphasehorizontallywith
thevariationconcentratedalongthehorizontalpieces.
studiesfordispersiondiagrams,forinstanceinGuenneauetal.[43],andsemi-analyticalworkfor
narrowgapsasinLlewellyn-Smith&Davisetal.[44].Figure4a–cshowsthedispersioncurves
together with the asymptotics obtained from §4a for ‘long’ ligaments with a width to length
ratio η=h/l of 0.2. The new feature in comparison with figure 3 is the appearance of a low-
frequencystopbandbelowtheBraggstopband,whoseupperedgeremainsvirtuallyunaffected
bytheinsertionoftheresonatorineachcircularinclusionofthearray.Thelow-frequencystop
bandisassociatedwithalocalizedmodeuponresonanceoftheresonator,andisresponsiblefor
artificial magnetism in metamaterials in TE polarization. Physically, an array of cylinders with
capacitive splits such as in figure 4a–c respond resonantly to radiation with the magnetic field
whenitisorientedalongthecylindricalaxes[22].Theoscillatingmagneticfieldinducescurrents
to run around the perfectly conducting rings. These currents feel a finite inductive impedance
duetothefinitesizeoftheconductingloops,whiletheyfeelacapacitiveimpedanceduetothe
capacitivegapswithintheconductingloops.Thisgivesrisetoaresonantresponseofthesystem
wheretheresonanceisdrivenbythemagneticfieldoftheradiationwithaconsequentresonant
enhancement of the magnetic polarizability of each cylinder. If the array period is sufficiently
small compared to the wavelength of the applied magnetic field, then the metamaterial is
describedbyaneffectivemagneticpermeability[7].Thiseffectivemagneticpermeabilitydisplays
(a) (b) (c)
4.5 10
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2.5 ...lish
W ...ing
2.0 ....org
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1.5 ......ProcR
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01..50 ........469:2012
.0
.5
.3
.3
0
−1 0 1 2 3 −1 0 1 2 3 −1 0 1 2 3
M G X M M G X M M G X M
Figure4.Thedispersiondiagramshownforanarrayofsquarecells(side2)containingcircularholesofradius0.8,andwith
circularinclusionsofradius0.5,attachedtotherestofthecellbytwo,fourandeightthinligamentsin(a–c),respectively.
Thedispersioncurvesfromnumericsareshownassolidlines,theasymptoticHFHresultsareshownasdashedlines,withthe
low-frequencylinearclassicalhomogenizationshowndottedemergingfromtheorigin.Thethinligamentapproachof§4a
givesestimatesforthefirstnon-zeroeigenvalueatΓ,whicharethecrossesonthefrequencyaxis.Numericalvaluesforthese
estimatesare0.7082,0.9941,1.3851versusfinite-elementsimulationsof0.7058,1.0281,1.4845fortwo,fourandeightligaments,
respectively.Thesixthmodeoffigure4bisflatatfrequency3.741andcorrespondstothedipolemodedescribedin§4with
approximatefrequency3.6824.
a strong dispersion near the resonance frequency of the SRR, and it can become negative in
the frequency band just above the resonance frequency, i.e. in the low-frequency stop band.
Sadly,thelow-frequencystopbandappearsatfrequenciesalreadybeyondthescopeofclassical
homogenization,butfortunatelyHFHcapturesitsfinerdetails,asinfigure4a–c,andthusunveils
thefascinatingphysicsofartificialmagnetism.Forinstance,theinvertedcurvatureofthesecond
dispersioncurvearoundtheΓ pointinfigure4a,whichisahallmarkofaMieresonancedriving
theartificialmagnetism[22]iscapturedbytheHFH,asistheflatbandalongtheXMpath,which
isassociatedwithalocalizedmodeintheSRR(whichisthereforeinsensitivetoanyvariationof
theFloquet–Blochphaseshiftacrosstheunitcellalongthispath).Thehighlydispersivephysics
ofthelow-frequencystopbandwillrevealtheultra-refractionandAANReffectsshownin§5.
(c) Athinannuluswithholes
For contrast, we investigate the effect of shortening the ligaments so h/l is of order 1; adding
morecutsinthethinannuluspreservesthelowestresonantfrequency,butitbecomeslesssharp,
whichmakesthelow-frequencybandgapwider.Aphysicalsideeffectisthatartificialmagnetism
weakenswhentheresonanceislesssharp,i.e.effectivepermeabilityislessdispersiveandmight
notreachlargeenoughnegativevaluesforpotentialmetamaterialapplications,suchaslensing
via negative refraction. On the other hand, the metamaterial might work over a broader range
offrequenciesifthestopbandwidens,andthiscouldbeadesignrequirement.Thisleadstoa
subtlebalancebetweenhavingasharpresonanceandawidelow-frequencystopband;HFHcan