Wave scattering by small bodies and creating materials with a desired refraction coefficient

0.16 MB·English

by  A. G. Ramm

Progress in Electromagnetic Research, M (PIER M, 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
Scatteringofelectromagnetic(EM)wavesbymanysmallimedance
particles (bodies), embedded in a homogeneous medium, is studied.
Physical properties of the particles are described by their boundary
impedances. The limiting equation is obtained for the effective EM
fieldinthe limiting medium, inthe limita→0,wherea isthe charac-
teristic size of a particle and the number M(a) of the particles tends
to infinity at a suitable rate. The proposed theory allows one to cre-
ate a medium with a desirable spatially inhomogeneous permeability.
The main new physical result is the explicit analytical formula for the
permeability µ(x) of the limiting medium. While the initial medium
has a constant permeability µ0, the limiting medium, obtained as a
result of embedding many small particles with prescribed boundary
impedances, has a non-homogeneous permeability which is expressed
analytically in terms of the density of the distribution of the small
particles and their boundary impedances. Therefore, a new physical
phenomenon is predicted theoretically, namely, appearance of a spa-
tially inhomogeneous permeability as a result of embedding of many
smallparticleswhosephysicalpropertiesaredescribedbytheirbound-
ary 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.
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1 Introduction
In this paper we outline a theory of electromagnetic (EM) wave scattering
by many small impedance particles (bodies) embedded in a homogeneous
medium which is described by the constant permittivity ǫ > 0, permeabil-
0
ity µ > 0 and, possibly, constant conductivity σ ≥ 0. The small particles
0 0
are embedded in a finite domain Ω. The medium, created by the embed-
ding of the small particles, has new physical properties. In particular, it
has a spatially 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
8πi
Ψ(x) = 1+ h(x)N(x).
3ωµ
0
Here ω is the frequency of the EM field, µ is the constant permeability
0
parameter 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, see also [11].
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], [5], 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 techniques
are quite different from the usual methods. These ideas are similar to the
ideasdevelopedinpapers[7,8],wherescalarwavescatteringbysmallbodies
was studied, andin thepapers[9],[10]. However, thescattering of EMwaves
brought new technical difficulties which are resolved in this paper. The dif-
ficulties come from the vectorial nature of the boundary conditions. Our
approach is valid for small particles of arbitrary shapes, but for simplicity
we assume that the small bodies are balls of radius a.
We give a new numerical method for solving many-body wave-scattering
problems for small scatterers.
In Section 2 an outline of our theory is given and the basic results are
formulated and explained. In Section 3 the Conclusions are formulated. In
Section 4 short proofs of two lemmas are given. In Section 5, Appendix,
more difficult and lengthy proofs are given, and a numerical method for
solving many-body wave scattering problem in the case of small scatterers
is presented.
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
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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
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
5
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
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
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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
into impedance boundary condition (7), use the known formula (see, e.g.,
[6]):
σ (s)
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)
where the ± signs denote the limiting values of the left-hand side of (13) as
x → s from D (D ), and get the following equation:
m
σ (t) = A σ +f , 1≤ m ≤ M. (14)
m m m m
Here A is a linear Fredholm-type integral operator, defined by formula
m
A = −2[N ,B σ ], where the operator B is defined by formula (20),
m s m m m
and f is a continuously differentiable vector function, defined by formula
m
(15).
Let us find formulas for A and f . Equation (14) is derived in Ap-
m m
pendix and there the formulas for f and A are obtained.
m m
One has:
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
Boundary 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 known 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
7
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)
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 ].
m m s m
TheoperatorA islinearandcompactinthespaceC(S ), sothatequation
m m
(22) is of Fredholm type. Therefore, equation (22) is solvable for any f ∈
m
T if the homogeneous version of (22) has only the trivial solution σ = 0.
m m
Inthis case thesolution σ to equation (22) isof theorderof theright-hand
m
side f , that is, O(a−κ) as a → 0, see formula (15). Moreover, it follows
m
from equation (22) that the main term of the asymptotics of σ as a → 0
m
does not depend on s ∈ S .
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
∆
8
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
The central physical idea of the theory, developed in this paper, is simple:
one can neglect the second sum in (25) compared with the first sum.
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 one. This proof is
based on several estimates.
We assume in these estimates that a → 0, a → 0, and |x − x | ≥ d.
d m
Under these assumptions 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)
These esti(cid:12)ma(cid:12)tes show that one may neglect the second sum in (25), and
write
M
E (x) = E (x)+ [∇ g(x,x ),Q ] (30)
e 0 x m m
mX=1
with an error that tends to zero under our assumptions as a → 0, and when
|x−x | ∼ a then the term with m = j in the sum (30) should be dropped
j
according to the definition of the effective field. We will show that the limit
of theeffective field, as a → 0 does exist andsolves equation (39), see below.
9
Let us estimate Q asymptotically, as a → 0.
m
Integrate equation (22) 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
|Q | as a → 0. Thus,
m
Q = [f (s),N ]ds, a → 0. (32)
m e s
Z
Sm
Let us estimate the integral in the right-hand side of (32).
It follows from equation (15) that
ζ
m
[N ,f ]= [N ,E ]− [N ,[∇×E ,N ]]. (33)
s e s e s e s
iωµ
0
If E tends to a finite limit as a → 0, then formula (33) implies that
e
1
[N ,f ]= O(ζ )= O , a → 0. (34)
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, (35)
m
(cid:0) (cid:1)
because the integration over S adds factor O(a2). It will follow from our
m
arguments that Q does not vanish at almost all points, see formulas (37)-
m
(38).
As a → 0, the sum (30) converges to the integral
E(x) = E (x)+∇× g(x,y)N(y)Q(y)dy, (36)
0
Z
Ω
where Q(y) is the function uniquely defined by the formula
Q = Q(x )a2−κ. (37)
m m
The function Q(y) is defined uniquely, because,as a → 0 the set of points
{x }M becomes dense in Ω. The physical meaning of vector E(x) in
m m=1
equation (36) is clear: this vector is the limit of the effective field E (x) as
e
a → 0, and N(x) is the function from equation (23).
10