Scattering of EM waves by many small perfectly conducting or impedance bodies

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by  A. G. Ramm

Scattering of EM waves by many small
6 perfectly conducting or impedance bodies
1
0
2
A.G.Ramm
n
a
KansasStateUniversity,Manhattan,KS66506-2602,USA
J

9 [email protected]

]
h
p
- Abstract
h
at Atheoryofelectromagnetic(EM)wavescatteringbymanysmallparti-
m cles of an arbitrary shape is developed. The particles are perfectly con-
[ ducting or impedance. For a small impedance particle of an arbitrary

shape an explicit analytical formula is derived for the scattering ampli-
1
v tude. Theformulaholdsas a 0, where a isacharacteristicsize ofthe
0 →
small particle and the wavelength is arbitrary but fixed. The scattering
6
0 amplitudeforasmallimpedanceparticleisshowntobeproportionalto
2 a2 κ, where κ [0,1) is a parameter which can be chosen by an experi-
−
0 ∈
menterashe/shewants. Theboundaryimpedanceofasmallparticleis
.
1 assumedtobeoftheformζ ha κ,whereh const,Reh 0. Thescat-
−
0 = = ≥
teringamplitudeforasmallperfectlyconductingparticleisproportional
6
1 toa3,itismuchsmallerthanthatforthesmallimpedanceparticle.
:
v The many-body scattering problem is solved under the physical as-
i sumptions a d λ, where d is the minimal distance between neigh-
X
≪ ≪
boring particles and λ is the wavelength. The distribution law for the
r
a smallimpedanceparticlesisN (δ) N(x)dxasa 0.HereN(x) 0is
∼ δ → ≥
anarbitrarycontinuousfunctionthatcanbechosenbytheexperimenter
R
and N (δ) is the number of particles in an arbitrary sub-domain ∆. It
is proved that the EM field in the medium where many small particles,
impedanceorperfectlyconducting,aredistributed,hasalimit,asa 0
→
andadifferentialequationisderivedforthelimitingfield.
Onthis basis the recipe is given for creatingmaterialswith a desired
refractioncoefficientbyembeddingmanysmallimpedanceparticlesinto
agivenmaterial.
1
Keywords: electromagnetic waves; scattering; impedance bodies; small bod-
ies.
MSC:78A45;78A25;Z8A40;78M40;35825;35J2;35J57.
1 Introduction
Electromagnetic(EM) wave scatteringis a classical area of research. Rayleigh
statedin1871.see[17],thatthemainpartofthefield,scatteredbyasmallbody,
ka 1,wherek isthewavenumberanda isthecharacteristicsizeofthebody,
≪
is thedipoleradiation,butdidnot giveformulasfor calculatingthisradiation
forbodiesofarbitraryshapes. ForsphericalbodiesMie(1908)gaveasolution
toEMwavescatteringproblemusingseparationofvariablesinthesphericalco-
ordinates. Thismethoddoesnotworkforbodiesofarbitraryshapes. Rayleigh
andMieconcludedthatEMfield,scatteredbyasmallbody,isproportionalto
O(a3).Weprovethatthefieldscatteredbyasmallimpedancebody(particle)of
anarbitraryshapeisproportionaltoa2 κ,whereκ [0,1)isaparameterwhich
−
∈
can be chosen by the experimenteras he/she wishes, see formula (1.3) below.
Since 2 κ 3, it follows, for a 0, that the scattering amplitude for small
− < →
impedanceparticleis much larger than the scatteringamplitudefor perfectly
conductingordielectricsmallparticle. Thisconclusionmaybeofpracticalim-
portance.
There is a large literature on low-frequency wave scattering and multiple
scattering,see[1],[3],[6],[7],[19].
InthispaperatheoryofEMwavescatteringbyperfectlyconductingandby
impedance small bodies of arbitraryshapes is developed. For one-body scat-
tering problem explicit formulas for the scattering amplitudes are derived for
perfectlyconductingand for impedancesmallbodiesof arbitraryshapes. For
many-bodyscatteringproblemthesolutionisgivenasasumofexplicitterms
withthecoefficientsthatsolvealinearalgebraicsystem. Ifthesizeofthesmall
bodies a 0 and their number M M(a) , a limiting integral equation
→ = → ∞
is derived for thefield in thelimitingmedium. This equationallows us to ob-
tainalocaldifferentialequationforthefieldinthelimitingmediumandtogive
explicitanalyticformulasfortherefractioncoefficientofthelimitingmedium.
As a result we formulate a recipe for creating materials with a desired re-
fraction coefficient by embedding many small impedance particlesin a given
material.
Themethodsdevelopedinthispaperwereappliedtoacousticproblemsin
2
[11], to heat transfer in themedium where many small bodies are distributed
in[13],towavescatteringbymanynano-wiresin[14].
In Section 2 the theory of EM wave scattering is developed for small per-
fectlyconductingbodies(particles)ofarbitraryshapes.
InSection3thetheoryisdevelopedforEMwavescatteringbyoneimpedance
particleofanarbitraryshape.
InSection4thetheoryisdevelopedforEMwavescatteringbymanysmall
impedanceparticlesofanarbitraryshape.
In Section 5 a recipe for creating materialswith a desired refraction coeffi-
cient is given is given. The problem of creating materials with a desired mag-
neticpermeabilityissolved.
Physicalassumptionsinthispapercanbedescribedbytheinequalities:
a d λ, (1.1)
≪ ≪
where λ is the wavelength in R3\Ω, Ω is a bounded domain in which many
small particles D are distributed, 1 m M M(a), d is the minimal dis-
m
≤ ≤ =
tancebetweenneighboringparticles.
Theboundaryimpedanceisassumedtobe
h(x )
m
ζ , (1.2)
m = aκ
wherex D isanarbitrarypointinsideD ,h(x)isanarbitrarycontinuous
m m m
∈
functioninΩsuchthatReh 0,κ [0,1)isaparameter. Onecanchooseh and
≥ ∈
κasonewishes.
The distributionof the small impedance particles in D is given by the for-
mula
1
N (∆): N(x)dx(1 o(1)), a 0, (1.3)
= a2 κ ∆ + →
− Z
where∆ Ωisanarbitraryopenset,N (∆)isthenumberofsmallparticlesin
⊂
theset∆,andN(x) 0isanarbitrarycontinuousfunctioninΩ.
≥
TheexperimentercanchoosethefunctionN(x) 0ashe/shewishes.
≥
Onehas
N (∆) 1. (1.4)
=
xm ∆
X∈
By ω the frequency is denoted, k ω is the wave number, c is the velocity of
= c
lightintheair.
3
2 Scattering by perfectly conducting particles.
2.1 Scatteringbyoneparticle
TheproblemistofindthesolutiontoMaxwell’sequations
E iωµH, H iωǫE, inD : R3\D, (2.1)
′
∇× = ∇× =− =
where D is the small body, ka 1, a 0.5diamD, ǫ and µ are dielectric and
≪ =
magneticconstantsofthemediumin D′, k ωpǫµ, andtheboundarycondi-
=
tionis:
[N,[E,N]] 0 onS: ∂D. (2.2)
= =
HereandbelowN : N istheunitnormaltoS pointingintoD ,[E,N] E N
s ′
= = ×
isthevectorproductoftwovectors,E N (E,N)isthescalarproduct, S isthe
· = | |
surfacearea.
TheincidentfieldE is:
0
E
E Eeikαx, H ∇× 0, (2.3)
0 · 0
= = iωµ
whereα S2 isaunitvector,thedirectionoftheincidentplanewave,anditis
∈
assumedthatE α 0.Thisassumptionimpliesthat
· =
E 0, H 0. (2.4)
0 0
∇· = ∇· =
ThefieldE tobefoundis:
E E v , (2.5)
0 E
= +
wherethescatteredfieldv satisfiestheradiationcondition
E
∂v
E
r ikv o(1), r : x . (2.6)
E
∂r − = =| |→∞
µ ¶
Inequation(2.6)theo(1)isuniformwithrespecttothedirectionβ: x ofthe
= r
scatteredfieldasr .
→∞
ThescatteringamplitudeA(β,α,k)isdefinedasusual:
eikr 1 x
v A(β,α,k) o , r x , β . (2.7)
E
= r + r =| |→∞ = r
µ ¶
ThemagneticfieldH H v ,
0 H
= +
E v
E
H ∇× , v ∇× . (2.8)
H
= iωµ = iωµ
4
Letuslookforthesolutiontothescatteringproblem(2.1)-(2.6)oftheform:
eikx t
| − |
E E g(x,t)J(t)dt, g(x,t) , (2.9)
0
= +∇× = 4π x t
ZS | − |
where J is a tangential field to S. We assume that S C2, that is, S is twice
∈
continuouslydifferentiable.
Equations(2.1)aresatifiedif
E
E k2E, H ∇× . (2.10)
∇×∇× = = iωµ
SinceE satisfiesequations(2.10),theseequationsareequivalentto
0
v
v k2v , v ∇× E. (2.11)
E E E
∇×∇× = = iωµ
Equationforv isequivalenttotheequations:
E
( 2 k2)v 0, v 0inD , (2.12)
E E ′
∇ + = ∇· =
because v v 2v and v 0.Conversely,equations(2.12)
E E E E
∇×∇× =∇∇· −∇ ∇· =
areequivalentto(2.10)andto(2.1).
Theradiationconditionissatisfiedby
v g(x,t)J(t)dt
E
=∇×
ZS
foranyvector-function J(t).
Theboundarycondition(2.2)yields
J J
TJ : [N ,[ g(s,t),J(t)]]dt [N ,E ], (2.13)
s s s 0
2+ = 2+ ∇ =−
ZS
wheretheformula
J(s)
lim[N, g(x,t)J(t)dt] TJ, (2.14)
x→s− ∇×ZS = 2 +
was used, see [15]. Let us prove that equation (2.13) has a solution and this
solutionisuniqueinthespaceC(S)ofcontinuousonS functions. Thisproves
thatthescatteringproblemcanbesolvedbyformula(2.9)with J solving(2.13).
5
Theorem2.1. IfD issufficientlysmall,thenequation(2.13)isuniquelysolvable
inC(S)anditssolution J istangentialtoS.
Proof. Notethatanysolutiontoequation(2.13)isatangentialtoS field.Tosee
this,justtakethescalarproductof N withbothsidesofequation(2.13). This
s
yieldsN J(s) 0.Inotherwords, J isatangentialtoS field.
s
· =
Let uscheckthattheoperatorT iscompact inC(S). Thisfollows fromthe
formula
∂g(s,t)
TJ g(s,t)N J(t) J(t) dt. (2.15)
S s
= ∇ · − ∂N
ZSµ s ¶
Indeed,if J isatangentialtoS fieldthen
N J(s) 0. (2.16)
s
· =
SinceS C2,relation(2.16)implies
∈
1
N J(t) O( s t ) J(t), g(s,t)N J(t) O J(t). (2.17)
s s s
| · |= | − | | | |∇ · |≤ s t | |
µ| − |¶
Thus, thefirst integral in (2.15) is a weakly singularcompact operator inC(S).
Thesecondintegralin(2.15)isalsoaweaklysingularcompactoperatorinC(S)
because
∂g(s,t) 1
O , (2.18)
∂N = s t
¯ s ¯ µ| − |¶
¯ ¯
ifS C2. ¯ ¯
∈ ¯ ¯
Consequently,equation(2.13)isofFredholmtypeinC(S).Thecorrespond-
inghomogeneousequationhasonlythetrivialsolutionifDissufficientlysmall.
Thisfollowsfromthefollowingargument. Thehomogeneousversionofequa-
tion(2.13)meansthatthefunction
v g(x,t)J(t)dt
E
=∇×
ZS
solvesequations(2.12),satisfiestheradiationcondition(2.6),and
[N,v ] 0 onS. (2.19)
E
=
Thisimpliesthatv 0inD .
E ′
=
Lemma2.1(seebelow)impliesthatif v 0inD then J 0. Thisconclu-
E ′
= =
sion and the Fredholm alternativeprove the existence and uniqueness of the
solutiontoequation(2.13). ThesmallnessofthebodyD guaranteesthatk2 is
notaDirichleteigenvalueoftheLaplacianinD. Theorem2.1isproved.
6
Lemma2.1. Assumethatthefollowingconditionshold:
a)v 0inD ,
E ′
=
b) J istangentialtoS,
and
c)k2isnotaDirichleteigenvalueoftheLaplacianinD.
Then J 0.
=
Proof. Denote A: g(x,t)J(t)dt andusetheformula
= S
R
A Bdx A Bdx N [A,B]ds A Bdx, (2.20)
∇× · = ·∇× − · = ·∇×
ZD′ ZD′ ZS ZD′
validforanyB C (D ). If A 0inD ,thenformula(2.20)yields
∈ 0∞ ′ ∇× = ′
A·∇×Bdx=0, ∀B ∈C0∞(D′). (2.21)
ZD′
Writethisformulaas
dtJ(t) g(x,t)F(x)dx 0, F : B. (2.22)
· = =∇×
ZS ZD′
Thesetofvector-fieldsF coincidewiththesetofdivergence-freefields F 0
∇· =
inD ,whereF C (D ).
′ ∈ 0∞ ′
Thesetofvector-fields
G(t) g(x,t)F(x)dx, F C (D ),
= ∀ ∈ 0∞ ′
ZD′
whereitisnotassumedthatthecondition F 0holds,isdenseinthesetL2(S)
∇· =
ofvectorfields.Indeed,ifthereexistsanh 0suchthat
6=
h(t) g(x,t)F(x)dxdt 0, F C (D ), (2.23)
= ∀ ∈ 0∞ ′
ZS ZD′
andw(x): g(x,t)h(t)dt,then
= S
R
w(x)F(x)dx 0, F C (D ).
= ∀ ∈ 0∞ ′
ZD′
Thus,
w(x) g(x,t)h(t)dt 0 inD . (2.24)
′
= =
ZS
7
Consequently,
( 2 k2)w 0 inD, w 0 on S. (2.25)
∇ + = =
Sincek2 isnotaDirichleteigenvalueoftheLaplacianinD,equation(2.25)im-
plies w 0 in D. Therefore, w 0 in D D . This implies h ∂w ∂w 0.
= = ∪ ′ = ∂N − ∂N =
Consequently,thesetG(t)isdenseinthesetL2(S)ofvectorfieldso+nS. −
Weclaimthatif F 0inD ,whereF C (D ),then G 0onS.
∇· = ′ ∈ 0∞ ′ ∇· =
Indeed,
g(x,t)F(x)dx g(x,t) F(x)dx g(x,t) F(x)dx 0. (2.26)
t x
∇ · =− ∇ · = ∇· =
ZD′ ZD′ ZD′
Conversely,if G 0onS,thenequations(2.26)showthat
∇· =
g(x,t) F(x)dx 0, t S.
∇· = ∀ ∈
ZD′
Let us use the local coordinate system with the axis x directed along the
3
outernormalN toS,andx (s),x (s)arecoordinatesalongtwoorthogonalaxes
s 1 2
tangentialtoS.Letusdenotebye (s)ande (s)theunitvectorsalongtheseaxes
1 2
atapoints S.
∈
Equation(2.22)canbewrittenas
J(t) G(t)dt 0 (2.27)
· =
ZS
forallsmoothG(t)suchthat G 0onS,G g(x,t)F(x)dx, F 0.
∇· = = D′ ∇· =
Let J(t) J (t)e (t) J (t)e (t) in the local coordinates. For an arbitrary
= 1 1 + 2 2 R
smallδ 0onecanchooseG (t)andG (t)suchthat
1 2
>
J G J G δ, (2.28)
1 1 L2(S) 2 2 L2(S)
|| − || +|| − || <
wheretheover-bardenotesthecomplexconjugate. WithG andG sochosen,
1 2
chooseG suchthat
3
G 0 on S, (2.29)
∇· =
whichisclearlypossible.Thenequation(2.27)yields
( J 2 J 2)dt O(δ). (2.30)
1 2
| | +| | =
ZS
Sinceδ 0isarbitrarysmall,relation(2.30)impliesJ J 0.Therefore, J 0.
1 2
> = = =✷
Lemma2.1isproved.
8
As was stated above, it follows from Lemma 2.1 and from the Fredholm
alternativethat equation (2.13) is uniquely solvable for any right-hand side if
k2 σ(∆ ), that is, if k2 is not a Dirichlet eigenvalue of the Laplacian in D. If
D
6∈
D is sufficientlysmall, which we assumesince a 0, then a fixed numberk2
→
cannot be a Dirichlet eigenvalue of the Laplacian in D because the smallest
DirichleteigenvalueoftheLaplacianinD isO( 1 ) k2ifa 0.
a2 > →
Remark2.1. The assumption k2 σ(∆ ) can be discarded if g(x,t) is replaced
D
6∈
byg (x,t),theGreenfunctionoftheDirichletHelmholtzoperatorintheexterior
ǫ
ofaballB : {x: x ǫ},whereǫ 0ischosensothatk2 σ(∆ ). Thischoice
ǫ = | |≤ > 6∈ D\Bǫ
ofǫ 0isalwayspossible(see[8],p. 29).
>
Let us denote by V the operator that gives the tangential to S component
v oftheuniquesolutionv tothescatteringproblem(2.1)–(2.3),(2.6):
Eτ E
E E v , v V( [N,E ]). (2.31)
0 E Eτ 0
= + = −
If the tangential component v is known, then v is uniquely defined in D .
Eτ E ′
Thisisaknownfact,see,forexample,[15].TheoperatorV islinearandbounded
inC(S). It mapsC(S) ontoC(S) and v has the same smoothness as the data
E
[N,E ]. Forexample,ifS Cℓ,thenv Cℓ(D ),whereℓ 0.
0 E ′
∈ ∈ >
Define
Q: J(t)dt. (2.32)
=
ZS
Fromformulas(2.7),(2.9)and(2.32)itfollowsthat
ik
A(β,α,k) [β,Q]. (2.33)
= 4π
ForbodyD onehas
[N,E ]ds E dx E D E c a3, (2.34)
0 0 0 0 D
= ∇× =∇× | |=∇×
ZS ZD
where D isthevolumeofD andc 0isaconstantdependingontheshape
D
| | >
ofD. Forexample,ifD isaballofradiusa,thenc 4π.
D = 3
Onehastheformula(see[15],p.8):
∂g(s,t) 1
ds o(1), a 0. (2.35)
− ∂N = 2+ →
ZS S
9
SinceN J(s) 0andSisC2 smooth,itfollowsthat N J(t) c s t J(t).
s s
· = − | · |≤ | − || |
Therefore
1
I : ds dt g(s,t)N J(t) c ds dt J(t), (2.36)
s s
= ∇ · ≤ s t | |
¯ZS ZS ¯ ZS ZS | − |
¯ ¯
¯ ¯
and I O(a¯) J(t)dt. If I wouldsatis¯fytheestimateI o(Q),asa 0,then
≤ S| | = →
the theorywould simplify considerably and one would haveQ E D
R =−∇× | |=
Ec a3. Unfortunately,estimateI o(Q)isnotvalid,andonehastogive
D
−∇× =
anewestimatefortheintegralI : ds dt g(s,t)N J(t). Todothis,inte-
1 = S S ∇s s·
grateequation(2.13)overS,useequations(2.15)and(2.35),andget
R R
Q I c a3 E . (2.37)
1 D 0
+ =− ∇×
LetuswriteI as
1
I e Γ (t)J (t)dt, (2.38)
1 p pq q
=
ZS
where{e }3 isanorthonormalbasisofR3,
p p 1
=
∂g(s,t)
Γ (t): N (s)ds, (2.39)
pq q
= ∂s
ZS p
and the integral in formula (2.39) is understood as a singular integral. Thus,
equation(2.37)takestheform
(I Γ)Q c a3 E . (2.40)
D 0
+ =− ∇×
HeretheconstantmatrixΓisdeterminedfromtherelation
ΓQ e Γ (t)J (t)dt, (2.41)
p pq q
=
ZS
thesummationisunderstoodovertherepeatedindicesp,q,soΓisthematrix
which sends a constant vector Q onto the constant vector I defined by the
1
equation(2.38).
OnecanprovethattheconstantmatrixΓexistsandcanbedeterminedby
equation(2.41),andthematrixI Γisnon-singular.
+
To prove that a constant matrix Γ exists assume that for every p 1,2,3,
=
thesetoffunctions{Γ (t)}3 islinearlyindependentinL2(S), Γ2 (t)dt 0
pq q 1 S pq 6=
and Q J(t)dt 0. Here=J(t) 3 e J (t). For a fixed p let M be the
= S 6= = q 1 q q R p
set in L2(S) orthogonalto thelinearspa=n of Γ (t). Then every function J (t)
R P pq q
10