Electromagnetic wave scattering by many small particles and creating materials with a desired permeability

0.15 MB·English

by  A. G. Ramm

ProgressinElectromagneticResearch, M,(PIERM)14,(2010), 193-206.
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Electromagnetic wave scattering by many small
particles and creating materials with a desired
permeability
A.G. Ramm
Department of Mathematics
Kansas State University, Manhattan, KS 66506-2602,USA
[email protected]
Abstract
Scattering of electromagnetic (EM) waves by many small particles
(bodies), embedded in a homogeneous medium, is studied. Physical
propertiesoftheparticlesaredescribedbytheirboundaryimpedances.
The limiting equation is obtained for the effective EM field in the lim-
iting medium, in the limit a →0, where a is the characteristic size of
a particle and the number M(a) of the particles tends to infinity at a
suitablerate. Theproposedtheoryallowsonetocreateamediumwith
adesirablespatiallyinhomogeneouspermeability. Themainnewphys-
ical result is the explicit analytical formula for the permeability µ(x)
of the limiting medium. While the initial medium has a constant per-
meability µ0, the limiting medium, obtained as a result of embedding
manysmallparticleswithprescribedboundaryimpedances,hasanon-
homogeneous permeability which is expressed analytically in terms of
the density of the distribution of the small particles and their bound-
ary impedances. Therefore, a new physical phenomenon is predicted
theoretically,namely,appearanceofaspatiallyinhomogeneousperme-
abilityasaresultofembeddingofmanysmallparticleswhosephysical
properties are described by their boundary impedances.
PACS: 02.30.Rz; 02.30.Mv; 41.20.Jb
MSC: 35Q60;78A40; 78A45; 78A48;
Key words: electromagnetic waves; wave scattering by many small bodies;
smart materials.
2
1 Introduction
In this paper we outline a theory of electromagnetic (EM) wave scatter-
ing by many small particles (bodies) embedded in a homogeneous medium
which is described by the constant permittivity ǫ > 0, permeability µ > 0
0 0
and, possibly, constant conductivity σ ≥ 0. The small particles are em-
0
bedded in a finite domain Ω. The medium, created by the embedding of
the small particles, has new physical properties. In particular, it has a spa-
tially inhomogeneous magnetic permeability µ(x), which can be controlled
by the choice of the boundary impedances of the embedded small particles
and their distribution density. This is a new physical effect, as far as the
author knows. An analytic formula for the permeability of the new medium
is derived:
µ
0
µ(x) = ,
Ψ(x)
where
8π
Ψ(x) = 1+ iǫ ωh(x)N(x).
0
3
Here ω is the frequency of the EM field, ǫ is the constant dielectric pa-
0
rameter of the original medium, h(x) is a function describing boundary
impedances of the small embedded particles, and N(x) ≥ 0 is a function
describing the distribution of these particles. We assume that in any sub-
domain ∆, the number N(∆) of the embedded particles D is given by the
m
formula:
1
N(∆)= N(x)dx[1+o(1)], a → 0,
a2−κ Z
∆
where N(x) ≥ 0 is a continuous function, vanishing outside of the finite
domain Ω in which small particles (bodies) D are distributed, κ ∈ (0,1) is
m
a number one can choose at will, and the boundaryimpedances of the small
particles are defined by the formula
h(x )
m
ζ = , x ∈ D ,
m aκ m m
where x is a point inside m−th particle D , Re h(x) ≥ 0, and h(x) is a
m m
continuousfunctionvanishingoutsideΩ.Theimpedanceboundarycondition
on the surface S of the m−th particle D is Et = ζ [Ht,N], where Et
m m m
(Ht) is the tangential component of E (H) on S , and N is the unitnormal
m
to S , pointing out of D .
m m
Since one can choose the functions N(x) and h(x), one can create a
desired magnetic permeability in Ω. This is a novel idea, to the author’s
knowledge.
3
We also derive an analytic formula for the refraction coefficient of the
mediuminΩcreatedbytheembeddingofmanysmallparticles. Anequation
for the EM field in the limiting medium is derived. This medium is created
when the size a of small particles tends to zero while the total number
M = M(a) of the particles tends to infinity at a suitable rate.
The refraction coefficient in the limiting medium is spatially inhomoge-
neous.
Our theory may be viewed as a ”homogenization theory”, but it dif-
fers from the usual homogenization theory (see, e.g., [1], [2], and references
therein) in several respects: we do not assume any periodic structure in the
distribution of small bodies, our operators are non-selfadjoint, the spectrum
of these operators is not discrete, etc. Our ideas, methods, and techiques
are quite different from the usual methods. These ideas are similar to the
ideas developed in papers [4, 5], where scalar wave scattering by small bod-
ies was studied, and in the papers [6],[7]. However, the scattering of EM
waves brought new technical difficulties which are resolved in this paper.
The difficulties come from the vectorial nature of the boundary conditions.
Our arguments are valid for small particles of arbitrary shapes.
We also give a new numerical method for solving many-body wave-
scattering problems for small scatterers, see Section 5.2.
2 EM wave scattering by many small particles
We assume that many small bodies D , 1 ≤ m ≤ M, are embedded in a
m
homogeneous medium with constant parameters ǫ , µ . Let k2 = ω2ǫ µ ,
0 0 0 0
where ω is the frequency. Our arguments remain valid if one assumes that
the medium has a constant conductivity σ > 0. In this case ǫ is replaced
0 0
by ǫ +iσ0. Denote by [E,H] = E ×H the cross product of two vectors,
0 ω
and by (E,H) = E ·H the dot product of two vectors.
Electromagnetic (EM) wave scattering problem consists of finding vec-
tors E and H satisfying the Maxwell equations:
∇×E = iωµ H, ∇×H = −iωǫ E in D := R3\∪M D , (1)
0 0 m=1 m
the impedance boundary conditions:
[N,[E,N]] = ζ [H,N] on S , 1≤ m ≤ M, (2)
m m
and the radiation conditions:
E = E +v , H = H +v , (3)
0 E 0 H
4
where ζ is the impedance, N is the unit normal to S pointing out of
m m
D , E ,H are the incident fields satisfying equations (1) in all of R3. One
m 0 0
often assumes that the incident wave is a plane wave, i.e., E = Eeikα·x, E
0
is a constant vector, α ∈ S2 is a unit vector, S2 is the unit sphere in R3,
α·E = 0, v and v satisfy the radiation condition: r(∂v −ikv) = o(1) as
E H ∂r
r := |x| → ∞.
By impedance ζ we assume in this paper either a constant, Re ζ ≥ 0,
m m
or a matrix function 2×2 acting on the tangential to S vector fields, such
m
that
Re(ζ Et,Et)≥ 0 ∀Et ∈ T , (4)
m m
where T is the set of all tangential to S continuous vector fields such
m m
that DivEt = 0, where Div is the surface divergence, and Et is the tan-
gential component of E. Smallness of D means that ka ≪ 1, where
m
a = 0.5max diamD . By the tangential to S component Et of
1≤m≤M m m
a vector field E the following is understood in this paper:
Et = E −N(E,N) = [N,[E,N]]. (5)
Thisdefinitiondiffersfromtheoneusedoftenintheliterature, namely, from
thedefinitionEt =[N,E]. Ourdefinition(5)correspondstothegeometrical
meaning of the tangential component of E and, therefore, should be used.
The impedance boundary condition is written usually as
Et = ζ[Ht,N],
where the impedance ζ is a number. If one uses definition (5), then this
condition reduces to (2), because [[N,[H,N]],N] = [H,N].
Lemma 1. Problem (1)-(4) has at most one solution.
Lemma 1 is proved in Section 2.
Let us note that problem (1)-(4) is equivalent to the problems (6), (7), (3),
(4), where
∇×E
∇×∇×E = k2E in D, H = , (6)
iωµ
0
ζ
m
[N,[E,N]] = [∇×E,N] on S , 1 ≤m ≤ M. (7)
m
iωµ
0
Thus, we have reduced our problem to finding one vector E(x). If E(x) is
found, then H = ∇×E.
iωµ0
5
Let us look for E of the form
M eik|x−y|
E = E + ∇× g(x,t)σ (t)dt, g(x,y) = , (8)
0 m
Z 4π|x−y|
mX=1 Sm
where t ∈ S and dt is an element of the area of S , σ (t) ∈ T . This E
m m m m
for any continuous σ (t) solves equation (6) in D because E solves (6) and
m 0
∇×∇×∇× g(x,t)σ (t)dt = ∇∇·∇× g(x,t)σ (t)dt
m m
Z Z
Sm Sm
−∇2∇× g(x,t)σ (t)dt
m
Z
Sm
= k2∇× g(x,t)σ (t)dt, x ∈ D.
m
Z
Sm
(9)
Here we have used the known identity divcurlE = 0, valid for any smooth
vector field E, and the known formula
−∇2g(x,y) = k2g(x,y)+δ(x−y). (10)
The integral g(x,t)σ (t)dt satisfies the radiation condition. Thus, for-
Sm m
mula (8) solveRs problem (6), (7), (3), (4), if σm(t) are chosen so that bound-
ary conditions (7) are satisfied.
Define the effective field E (x) = Em(x) = E(m)(x,a), acting on the
e e e
m−th body D :
m
E (x):= E(x)−∇× g(x,t)σ (t)dt := E(m)(x), (11)
e m e
Z
Sm
where we assume that x is in a neigborhood of S , but E (x) is defined
m e
for all x ∈ R3. Let x ∈ D be a point inside D , and d = d(a) be the
m m m
distance between two neighboring small bodies. We assume that
a
lim = 0, limd(a) = 0. (12)
a→0d(a) a→0
We will prove later that E (x,a) tends to a limit E (x) as a → 0, and
e e
E (x) is a twice continuously differentiable function. To derive an integral
e
equation for σ = σ (t), substitute
m m
E =E +∇× g(x,t)σ (t)dt
e m
Z
Sm
6
into (7), use the formula
σ (t)
m
[N,∇× g(x,t)σ (t)dt] = [N ,[∇ g(x,t)| ,σ (t)]]dt± ,
m ∓ s x x=s m
Z Z 2
Sm Sm
(13)
(see, e.g., [3]), the -(+) signs denote the limiting values of the left-hand
side of (13) as x → s from D (D ), and get the following equation (see
m
Appendix):
σ (t) = A σ +f , 1≤ m ≤ M. (14)
m m m m
Here A is a linear Fredholm-type integral operator, and f is a contin-
m m
uously differentiable function. Let us specify A and f . One has (see
m m
Appendix):
ζ
m
f = 2[f (s),N ], f (s) := [N ,[E (s),N ]]− [∇×E ,N ]. (15)
m e s e s e s e s
iωµ
0
Condition (7) and formula (13) yield
1
f (s)+ [σ (s),N ]+[ [N ,[∇ g(s,t),σ (t)]]dt,N ]
e m s s s m s
2 Z
Sm (16)
ζ
m
− [∇×∇× g(x,t)σ (t)dt,N ]| = 0
m s x→s
iωµ Z
0 Sm
Using the formula ∇×∇× = graddiv−∇2, the relation
∇ ∇ · g(x,t)σ (t)dt = ∇ (−∇ g(x,t),σ (t))dt
x x m x t m
Z Z
Sm Sm (17)
= ∇ g(x,t)Divσ (t)dt = 0,
x m
Z
Sm
where Div is the surface divergence, and the formula
−∇2 g(x,t)σ (t)dt = k2 g(x,t)σ (t)dt, x ∈D, (18)
x m m
Z Z
Sm Sm
where equation (10) was used, one gets from (16) the following equation
−[N ,σ (s)]+2f (s)+2Bσ = 0. (19)
s m e m
Here
Bσ := [ [N ,[∇ g(s,t),σ (t)]]dt,N ]+ζ iωǫ [ g(s,t)σ (t)dt,N ].
m s s m s m 0 m s
Z Z
Sm Sm
(20)
7
TakecrossproductofN withtheleft-handsideof (19)andusetheformulas
s
N ·σ (s) = 0, f := f (s) := 2[f (s),N ], and
s m m m e s
[N ,[N ,σ (s)]] = −σ (s), (21)
s s m m
to get from (19) equation (14):
σ (s) = 2[f (s),N ]−2[N ,Bσ ]:= A σ +f , (22)
m e s s m m m m
where A σ = −2[N ,Bσ ]. The operator A is linear and compact in
m m s m m
the space C(S ), so that equation (22) is of Fredholm type. Therefore,
m
equation (22) is solvable for any f ∈ T if the homogeneous version of
m m
(22) has only the trivial solution σ = 0. In this case the solution σ to
m m
equation (22) is of the order of the right-hand side f , that is, O(a−κ) as
m
a → 0, see formula (15). Moreover, it follows from equation (22) that the
main term of the asymptotics of σ as a → 0 does not depend on s ∈ S .
m m
Lemma 2. Assume that σ ∈ T , σ ∈C(S ), and σ (s) = A σ . Then
m m m m m m m
σ = 0.
m
Lemma 2 is proved in Section 2.
Let us assume that in any subdomain ∆, the number N(∆) of the em-
bedded bodies D is given by the formula:
m
1
N(∆)= N(x)dx[1+o(1)], a → 0, (23)
a2−κ Z
∆
whereN(x) ≥ 0isacontinuousfunction,vanishingoutsideofafinitedomain
Ω in which small bodies D are distributed, κ ∈ (0,1) is a number one can
m
choose at will. We also assume that
h(x )
m
ζ = , x ∈ D , (24)
m aκ m m
where Re h(x) ≥ 0, and h(x) is a continuous function vanishing outside Ω.
Let us write (8) as
M M
E(x) = E (x)+ [∇ g(x,x ),Q ]+ ∇× (g(x,t)−g(x,x ))σ (t)dt,
0 x m m m m
Z
mX=1 mX=1 Sm
(25)
where
Q := σ (t)dt. (26)
m m
Z
Sm
8
Since σ = O(a−κ), one has Q = O(a2−κ). We want to prove that the
m m
second sum in (25) is negligible compared with the first sum. One has
1 k
j := |[∇ g(x,x ),Q ]|≤ O max , O(a2−κ), (27)
1 x m m (cid:18) (cid:26)d2 d(cid:27)(cid:19)
1 k2
j := |∇× (g(x,t)−g(x,x ))σ (t)dt| ≤ aO max , O(a2−κ),
2 Z m m (cid:18) (cid:26)d3 d (cid:27)(cid:19)
Sm
(28)
and
j a a
2
= O max ,ka → 0, = o(1), a → 0. (29)
(cid:12)j (cid:12) d d
(cid:12) 1(cid:12) (cid:16) n o(cid:17)
(cid:12) (cid:12)
Thus, one(cid:12)ma(cid:12)y neglect the second sum in (25), and write
M
E(x) = E (x)+ [∇ g(x,x ),Q ] (30)
0 x m m
mX=1
with an error that tends to zero as a → 0.
Let us estimate Q asymptotically, as a → 0. Integrate equation (22)
m
over S to get
m
Q = 2 [f (s),N ]ds−2 [N ,Bσ ]ds. (31)
m e s s m
Z Z
Sm Sm
We will show in the Appendix that the second term in the right-hand side
of the above equation is equal to −Q plus terms negligible compared with
m
the first one as a → 0. Thus,
Q = [f (s),N ]ds.
m e s
Z
Sm
Let us estimate the first term. It follows from equation (15) that
ζ
m
[N ,f ]= [N ,E ]− [N ,[∇×E ,N ]]. (32)
s e s e s e s
iωµ
0
If E tends to a finite limit as a → 0, then formula (32) implies that
e
1
[N ,f ]= O(ζ )= O , a → 0. (33)
s e m (cid:18)aκ(cid:19)
By Lemma 2 the operator (I −A )−1 is bounded, so σ = O 1 , and
m m aκ
(cid:0) (cid:1)
Q = O a2−κ , a → 0, (34)
m
(cid:0) (cid:1)
9
because integration over S adds factor O(a2). As a → 0, the sum (30)
m
converges to the integral
E = E +∇× g(x,y)N(y)Q(y)dy, (35)
0
Z
Ω
where Q(y) is the function such that
Q = Q(x )a2−κ. (36)
m m
The function Q(y) can be expressed in terms of E:
8π
Q(y) = − h(y)iωǫ (∇×E)(y), (37)
0
3
see Appendix. Here the factor 8π appears if D are balls. Otherwise a
3 m
tensorial factor c , depending on the shape of S , should be used in place
m m
of 8π.
3
Thus, equation (35) takes the form
8π
E(x) = E (x)− iωǫ ∇× g(x,y)∇×E(y)h(y)N(y)dy. (38)
0 0
3 Z
Ω
Let us derive physical conclusions from equation (38). Taking ∇×∇× of
(38) yields
∇×∇×E = k2E (x)
0
8π
− iωǫ ∇×(grad div−∇2) g(x,y)∇×E(y)h(y)N(y)dy
0
3 Z
Ω
8π
= k2E −k2 iωǫ ∇× g(x,y)∇×E(y)h(y)N(y)dy
0 0
3 Z
Ω
8π
− iωǫ ∇×(∇×E(x)h(x)N(x))
0
3
8π
= k2E(x)− iωǫ h(x)N(x)∇×∇×E
0
3
8π
− iωǫ [∇(h(x)N(x)),∇×E(x)].
0
3
(39)
Here we have used the known formula ∇×grad = 0, the known equation
(10), and assumed for simplicity that h(x) is a scalar function. It follows
from (39) that
8πiωǫ
∇×∇×E = K2(x)E− 3 0 [∇(h(x)N(x)),∇×E(x)], (40)
1+ 8πiωǫ h(x)N(x)
3 0
10