Materials with a desired refraction coefficient can be made by embedding small particles

A.G.Ramm

(MathematicsDepartment,KansasSt.University,

Manhattan,KS66506,USAandTUDarmstadt,Germany)

ramm@math.ksu.edu

Abstract

Amethodisproposedtocreatematerialswithadesiredrefractioncoef-ficient,possiblynegativeone.Themethodconsistsofembeddingintoagivenmaterialsmallparticles.Givenn0(x),therefractioncoefficientoftheoriginalmaterialinaboundeddomainD⊂R3,andadesiredrefractioncoefficientn(x),onecalculatesthenumberN(x)ofsmallparticles,tobeembeddedinDaroundapointx∈DperunitvolumeofD,inorderthattheresultingnewmaterialhasrefractioncoefficientn(x).PACS:03.04.Kf

MSC:35J05,35J10,70F10,74J25,81U40,81V05

Keywords:”smart”materials,wavescatteringbysmallbodies,many-bodyscatteringproblem,negativerefraction,nanotechnology

1Introduction

Thereisagrowinginteresttomaterialswiththedesiredproperties,inparticular,withnegativerefractioncoefficient(see[1]andreferencestherein).In[2]theroleofspatialdispersionsisemphasizedinexplainingunusualpropertiesofmaterials.In[3]theroleofdispersionforwavepropagationinsolidsisdescribed.In[4]boundary-valueproblemsindomainswithcomplicatedboundarieswerestudied.In[6],[7]wavescatteringbysmallbodiesofarbitraryshapesisstudiedandformulasfortheS-matrixareobtained.In[5]ageneralmethodforcreatingmaterialswithwave-focusingpropertiesisproposedandjustified.Ouraiminthispaperistouseasimilarapproachforcreationofthematerialswithadesiredrefractioncoefficientbyembeddingsmallparticlesintoagivenmaterialwith

1

knownrefractioncoefficientn0(x).TheacousticwavescatteringbythegivenmaterialisdescribedbytheHelmholtzequation

󰀉1inD′:=R3\\D,223

[∇+kn0(x)]u=0inR,n0(x)=(1)

n0(x)inD.Herek>0isthewavenumberinD′.Equation(1)canbewrittenasthe

Schr¨odingerequation

L0u:=[∇2+k2−q0(x)]u=0inR3,

q0:=k2−k2n0(x).

(2)

Weassumek>0fixedanddonotshowk-variableinq0.Clearly,q0=0inD′.Thescatteringsolutionto(2)isuniquelydefinedbytheradiationcondition:

u0=e

ikα·x

+A0(β,α)

eikr

r

),r:=|x|→∞,β:=

x

.(5)

rr

Weprovethatthesolutiontoproblem(4)–(5)convergesasM→∞tothesolutionoftheproblem

LU:=[∇2+k2−q(x)]U=0inR3,

󰀊eikrxikα·x

U=e+A(β,α),r=|x|→∞,β=

r

(6)

+o

󰀅1

andgiveaformulaforp(x).Itturnsoutthatp(x)canbemadeanarbitrarydesiredfunctionbychoosingthedensityofthenumberN(x)oftheembeddedparticlesaroundeachpointx∈Dandtheimpedancesζmproperly.Thus,q(x)canbemadeanarbitrarydesiredfunction.Thereforetherefractioncoefficient

n(x)=1−k−2q(x)=n0(x)−k−2p(x)

(9)

canbemadearbitrary,inparticular,negative.

Ifn0(x)isgivenandonewishestocreatethematerialwiththecoefficientn(x),thenonecalculates

p(x)=[n0(x)−n(x)]k2,

andembedsN(x)smallparticlesperunitvolumeofDaroundeachpointx∈Dandchoosestheirimpedancesζmsothatthefunctionp(x)isobtainedforthenewmaterial.InSection2wegiveanalyticalformulasforN(x)andζmandsufficientconditionsfortheconvergenceofthesolutionto(4)–(5)tothesolutionof(6)–(8)asM→∞insuchawaythatrelations(13)-(14)hold.

Wealsoprovethattherelativevolumeoftheembeddedparticlesisnegligible.Moreprecisely,if|Dm|thevolumeofDm,then

lim

󰀇M

m=1

|Dm|

M→∞

Jm

,Jm:=

󰀄

Sm

󰀄

dsdt

Sm

M

3

󰀊

.

LetM→∞andassumethatthefollowinglimitexists:

Mlim

Cm(0)(ζm|Sm|)−1:=h(x).

(13)

|x→∞m−x|≤d

Hereandbelowxm∈DmisanarbitrarypointinDm.BecauseDmissmall,the

choiceofthispointinDmisnotimportant.UndertheassumedrelationsbetweenaanddonehaslimM→∞a

quantityNd3

󰀊

d3=O=O(a2)→0asM→∞󰀊

.On󰀅1theotherhand,theM(x)Cmζm,whichhasphysicalmeaningoftheaveragequantityCmζmperunitvolumeofDaroundpointx,hasalimit:

lim

NM(x)Cx)

|xM→∞mζm=

C(m−x|≤d

a

andC(0)M

=O(a),andtheexistenceofthefirstlimitin(14)followsfromformula(13)andfromthesecondformula(14).Ourbasicresultistheformula:

󰀊

󰀁n0(x)−n(x)󰀃k2

:=p(x)=

C(x)

a

,M=O

󰀅1

smallacousticallysoftballsofradiusaperunitvolumeofDaroundeachpointa

x∈D,andtheresultingmaterialwillhaven(x)=n0(x)−k−2p(x).Inparticular,n(x)<0ifp(x)>k2n0(x).

Example2.Chooseanarbitraryfunctionp(x)=p1(x)+ip2(x),p2(x)≤0.Theconditionp2≤0guaranteesuniquenessofthesolutiontoproblem(6)-(7)withq(x)=q0(x)+p(x).Physicallythisconditionmeansthatthemedium,correspondington(x)=1−k−2q(x)hasnonnegativeabsorption.Lettheparticlesbeballsofradiusaandζm=ζm(x)=1

|Sm|=4πa2.ChooseN=N(x)andh(x)=h1+ih2fromthefirstequation(14)using(12):

pNa

1+p2=

(1+h.1)2+h22Thus,

pNa(1+h)

1=

1(1+h1)2+h2.

(16)

2

Wehavethreefunctions:N=N(x)>0,h1andh2,tosatisfytwoequations

(16).Thiscanbedonebyinfinitelymanyways.Forinstance,onecanh1=0,h2=−p2

p21

thedesiredn(x)=n0(x)−k−2p(x),where󰀊take

.Thus,togetthematerialwithp(x)=p1(x)+ip2(x),oneembeds

N(x)=a−1(p21+p2

2)/p1smallballsofradiusaperunitvolumearoundeachpointxandchoosestheimpedanceζm(x)=󰀅4πah(x)󰀊−1

,whereh=h1+ih2,h2=−p2/p1,h1=0.

2Derivationoftheresults.

Weseektheuniquesolutionto(4)–(5)oftheform

u=u0+󰀂

M󰀄G(x,t)σm(t)dt

(17)

m=1Sm

=u0+

󰀂

MG(x,xm)Qm+

m=1󰀂

Mm=1

󰀄Sm

󰀁

G(x,t)−G(x,xm)󰀃

σmdt.

HereL0G1=δ(x−y)inR3,Gsatisfiestheradiationcondition,σmaretobechosensothattheboundarycondition(4)issatisfied,Qm:=≫aforallm.󰀈

Smσmdt,xm∈Dm.InthegenericcaseQm=0onecanneglectthelasttermin(17)comparedwiththeprecedingtermif|x−xm|>dIndeed,underonehas|G(x,t)−G(x,xm)|≤|∇yG(x,y˜)·(t−xm)|=O

󰀅󰀈

|Qm|󰀊

≪|Qm|,wherewealsoassumethat󰀅thisassumptiona|Qm|Od=Sm|σm|dt󰀊

.Wewillseethatthisassumptionisjustified.Forexample,ifu|Sm=0,thenσmdoesnotchangesignonSm.Thus,genericallyonecanwriteu=u0(x)+

󰀂

MG(x,xm)Qm,|x−xm|≥d≫a,(18)

m=1

withtheerrorO

󰀅a

ThefunctionsG(x,y)andu0(x)areknownbecausen0(x)isknown.LetusderiveanequationforfindingQm.IfQmarefoundthenthescatteringproblem(4)–(5)issolvedbyformula(18)foranyxawayfromanimmediateneighborhoodofthesmallparticles.ToderiveanequationforQmweneedsomepreparations.ThefunctionG(x,y)solvestheequation:

󰀄

eik|x−y|

G(x,y)=g(x,y)−g(x,z)q(z)G(z,y)dz,g(x,y):=

D

∂Ns

Itisknown([6],p.91)that

󰀄󰀄

Ajσjdt=−σj(t)dt,

Sj

Sj

σj(t)dt.(21)

∂(Tjσj)

2

,(22)

whereAj(k)istheoperatorsimilarto(21)withg(s,t)inplaceofg0(s,t),Ns:=N

istheouternormaltoSjatthepoints∈Sj.OnthesurfaceSjwehave

u=ue(s)+Tjσj,

ue:=u0+

M󰀂

G(s,xm)Qm.(23)

m=j

Usingboundarycondition(4)andformulas(22),(23),onegets

ueN(s)−ζjue(s)+

Ajσj−σj

4π|s−t|

6

.

Wereplacethelastintegralbyitsmeanvalue

1

4π|s−t|

4π|Sj|

:=

Jj

,and(25)yields:

Qj=−

ζj|Sj|

andget

d

󰀊

wherep(x)isdefinedin(15).Applyingto(29)theoperatorL0,definedin(2),andusingtherelationL0G(x,y)=−δ(x−y)yieldsequation(6)withqdefinedin(8).TheradiationconditionforUissatisfied:

A(β,α)=A0(β,α)+A1(β,α),

where

A1(β,α)=limAM(β,α)=−

M→∞

.Formulas(13)–(14)allowonetopasstothelimitM→∞in(28)

󰀄

U(x)=u0(x)−G(x,y)p(y)U(y)dy,(29)

D

(30)

1

4π|x|

u0(y,−β)+o

Inourderivationsitwasassumedthatζm=0.Ifζm=0forallm,thatis,the

smallparticlesareacousticallyhard,thenQm=0inthefirstorderwithrespecttoka.OnecanshowthatinthiscaseQm=O(k2a3),andthatthelastsum

7

󰀅1

|x|

=β.(32)

in(17)isofthesameorderofmagnitudeastheprecedingsum.Consequently,thetheoryinthiscaseisquitedifferent:theeffectivefieldinthemediumisnotdescribedbyequation(29),whichisequivalenttoalocalequation(6).Infact,theeffectivefieldinthiscaseisdescribedbyanintegrodifferentialequationwhichisnotequivalenttoalocaldifferentialequation.

LetusexplaintherelationQm=O(k2a3),mentionedabove.Write(24)withζj=0,integrateoverSjandusethefirstformula(22)toget

󰀄󰀄Qj=ueNds=∆uedx=O(k2a3).

Sj

Dj

References

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andexcitons,Springer-Verlag,Berlin,1984.[3]Landau,L.D.,Lifshitz,E.M.,Electrodynamicsofcontinuousmedia,Perga-monPress,Oxford,1984.[4]Marchenko,V.,Khruslov,E.,Boundary-valueproblemsindomainswith

fine-grainedboundary,NaukovaDumea,Kiev,1974.[5]Ramm,A.G.,Distributionofparticleswhichproducesa”smart”material,

Jour.Stat.Phys.,127,N5,(2007),915-934.[6]Ramm,A.G.,Wavescatteringbysmallbodiesofarbitraryshapes,World

Sci.Publ.,Singapore,2005.[7]Ramm,A.G.,Wavescatteringbysmallparticlesinamedium,Phys.Lett.

A.,(2007)(toappear)[8]Ramm,A.G.,Inverseproblems,Springer,Berlin,2005.

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