ON THE SCATTERING OF SPHERICAL ELECTROMAGNETIC
WAVES BY A PENETRABLE CHIRAL OBSTACLE1
C. Athanasiadis(1) , P. A. Martin(2) , A. Spyropoulos(1)
and I. G. Stratis(1)
(1) University of Athens, Department of Mathematics, Panepistimiopolis,
GR 157 84 Athens, Greece (e–mail: istratis@math.uoa.gr)
(2) Colorado School of Mines, Department of Mathematical and Computer Sciences,
Golden CO 80401–1887, USA (e–mail: pamartin@mines.edu)
Abstract
The problem of scattering of spherical electromagnetic waves by a bounded chiral obstacle is considered. General scattering theorems, relating the far–field patterns due to
scattering of waves from a point source put in any two different locations (the reciprocity
principle, the optical theorem, etc), and mixed scattering relations (relating the scattered
fields due to a point source and a plane wave) are established. Further, in the case of a
spherical chiral scatterer, the exact Green’s function and the electric far–field pattern of
the problem are constructed.
1
Introduction
The interaction of an incident wavefield with a bounded 3-dimensional obstacle is a classic
problem in scattering theory. The vast majority of the literature is concerned with incident
plane waves. However, in some recent papers Dassios and his co–workers, as well as three
and all of the present authors, see [1], [2], [3], [5], have studied incident waves generated by
a point source in the vicinity of the scatterer. Point sources have been used by Potthast [8]
to solve standard inverse problems. For related work by other authors see the bibliographies
of all the previous references. In this work we consider the problem of scattering of spherical
electromagnetic waves by a bounded chiral obstacle. General scattering theorems, relating
the far–field patterns due to scattering of waves from a point source put in any two different
locations (the reciprocity principle, the optical theorem, etc), and mixed scattering relations
(relating the scattered fields due to a point source and a plane wave) are established. Further,
in the case of a spherical chiral scatterer, the exact Green’s function and the electric far–field
pattern of the problem are constructed, using spherical vector wave functions. These results
generalize related properties of the problem where the scatterer is achiral [1], [3].
2
Formulation
Let Ωc be a bounded three–dimensional obstacle with a smooth closed boundary S, the scatterer. The exterior Ω is an infinite homogeneous isotropic achiral medium with electric permittivity ε, magnetic permeability µ and conductivity σ = 0. The scatterer Ωc is filled with
a chiral homogeneous isotropic medium with corresponding electromagnetic parameters εc , µc
1
Advances in Scattering and Biomedical Engineering (ed. D.I. Fotiadis and C.V. Massalas) World Scientific,
New Jersey, 2004, 34–43.
and chirality measure βc . The parameters ε and µ are assumed to be real constants and εc , µc
and βc to be complex constants.
We consider an incident spherical electromagnetic wave due to a point source located at a
point with position vector a, with respect to the origin O. Suppressing the time dependence
e−iωt , ω being the angular frequency, the incident wave (E ia , H ia ) has the form [1], [3]
b)
E ia (r ; p
a e−ika
=
∇×
ik
!
eik|r−a|
b ×p
b ,
a
|r − a|
b) =
H ia (r ; p
1
b ),
∇ × E ia (r ; p
ikη
(2.1)
p
b is a constant unit vector with p
b · a = 0, η =
µ/ε is the intrinsic impedance,
where p
√
k = ω εµ > 0 is the free–space wave number, and a = |a|. Physically (E ia , H ia ) represents
b ×p
b ; see p.163 of [4], or p.23 of
the field generated by a magnetic dipole with dipole moment a
−ika
[5]. The coefficient a e
/(ik) in (2.1) assures that when the point source recedes to infinity
b and
the spherical wave reduces to a plane electric wave with direction of propagation −a
t
b . The total exterior electric field E a is given by
polarization p
b ) = E ia (r ; p
b ) + E sa (r ; p
b ),
E ta (r ; p
r ∈ Ω \ {a},
(2.2)
b ) is the scattered electric field, which is assumed to satisfy the Silver–Müller
where E sa (r ; p
radiation condition
b × ∇ × E sa + ikr E sa ) = 0,
lim (r
(2.3)
r→∞
b ∈ S 2 , where S 2 is the unit sphere.
uniformly in all directions r
s
The behaviour of E a in the radiation zone is given by
E sa (r ) = h0 (kr) g a (rb ) + O(r−2 ),
r → ∞,
(2.4)
where h0 (x) = eix /(ix) is the spherical Hankel function of the first kind and order zero, and
g a (rb ) is the electric far–field pattern.
The total exterior electric field solves the equation
∇ × ∇ × E ta − k 2 E ta = 0 in Ω.
(2.5)
We note that the incident electric field satisfies the radiation condition (2.3), and hence
the total electric field also satisfies (2.3).
The incident electromagnetic waves are transmitted into the chiral scatterer. Let E ca be
the total electric field in the interior. Then E ca satisfies [6]
∇ × ∇ × E ca − 2βc γ 2 ∇ × E ca − γ 2 E ca = 0
in Ωc ,
(2.6)
where γ 2 = kc2 (1 − kc2 βc2 )−1 and kc2 = ω 2 εc µc . On the surface of the scatterer we have the
following transmission conditions:
b × E ta = n
b × E ca
n
b × ∇ × E ta = B1 n
b × ∇ × E ca + B2 n
b × E ca
n
where B1 = (µ/µc )kc2 /γ 2 and B2 = −(µ/µc )βc kc2 .
2
)
on S
(2.7)
3
The general scattering theorem
b ) due to a point source
In the sequel, for an incident time–harmonic spherical wave E ia (r ; p
located at a, we will denote the total field in Ω, the scattered field and the far–field pattern
b ), E sa (r ; p
b ) and g a (r
b; p
b ), respectively, indicating the dependence on the
by writing E ta (r ; p
b . Also, the total electric field in Ωc will
position a of the point source and the polarization p
c
b
be denoted by E a (r ; p).
We are interested in relations between these fields. We consider a point source at a with
b 1 and another point source at b with polarization p
b 2 . For a shorthand notation,
polarization p
we use
Z h
o
n
i
b × E ) · (∇ × E 0 ) − (n
b × E 0 ) · (∇ × E ) ds,
(n
E, E0 =
S
S
where the overbar denotes complex conjugation.
Let Sr denote a large sphere of radius r, surrounding the points a and b, and let Sa, =
{r ∈ R3 : |a − r | = } surrounding the point a. Then we have the following Lemma from [1].
b 1 ) be a point source at a. Let E ib (r ; p
b 2 ) be a point source at b, with
Lemma 1 Let E ia (r ; p
s
b; p
b 2 ). Then
b 2 ) and far–field pattern g b (r
corresponding scattered field E b (r ; p
n
o
b 1 ), E sb ( · ; p
b 2)
lim E ia ( · ; p
→0
where
Sa,
n
− lim
b 1 ), E sb ( · ; p
b 2)
E ia ( · ; p
r→∞
b 2 ) = eika a × ∇ × E sb (a; p
b 2) −
Gb (a; p
ik
2π
Z
S2
o
Sr
=
4πa
b · Gb (a; p
b 2 ),
p
ik 1
r ·a
b 2 ) eikb
b)
rb × g b (rb ; p
ds(r
(3.1)
is a spherical far–field pattern generator.
Now, the general scattering theorem, [1], for spherical electric waves scattered by a chiral
obstacle is formulated as follows.
b1
Theorem 2 For any two point–source locations in Ω, a and b, and for any polarizations, p
b 2 , we have
and p
b 1 · Gb (a; p
b 2) + p
b 2 · Ga (b; p
b 1) +
p
1
2π
Z
S2
b 2 ) · g a (r
b; p
b 1 ) ds(r
b ) = Ea,b (p
b 1; p
b 2)
g b (rb ; p
(3.2)
where
(
b 1; p
b 2) =
Ea,b (p
k
Im (B1 )
2π
Z
− Im (B1 γ 2 )
Ωc
b 1 )) · (∇ × E b (r ; p
b 2 )) dv
(∇ × E a (r ; p
Z
Ωc
b 1 ) · E b (r ; p
b 2 ) dv
E a (r ; p
+ i[βc γ 2 B1 + Re (B2 )]
Z
Ωc
−
i[βc γ 2 B1
Z
+ Re (B2 )]
Ωc
3
b 1 ) · (∇ × E b (r ; p
b 2 )) dv
E a (r ; p
)
b 2 ) · (∇ × E a (r ; p
b 1 )) dv .
E b (r ; p
(3.3)
Proof.
In view of the relations E tα = E iα + E sα , α = a, b, we have
{E ta , E tb } = {E ia , E ib } + {E ia , E sb } + {E sa , E ib } + {E sa , E sb }.
(3.4)
We use the transmission conditions (2.7) and apply the divergence theorem in Ωc ; this gives
{E ta , E tb } =
4πi
b 1; p
b 2 ).
Ea,b (p
k
(3.5)
Since E ia and E ib are entire solutions of (2.5), the vector Green’s second theorem gives
{E ia , E ib } = 0.
(3.6)
For the other terms in (3.4), we consider two small spheres, Sa,1 and Sb,2 , centred at a and
b with radii 1 and 2 , respectively, with Sa,1 ∩ Sb,2 = ∅, as well as a large sphere SR centred
at the origin, surrounding the whole system of the scatterer and the two small spheres. Since
E ia and E sb are solutions of (2.5) for r 6= a, b, the vector Green’s second theorem gives
{E ia , E sb } = {E ia , E sb }SR − {E ia , E sb }Sa,1 − {E ia , E sb }Sb,2 .
(3.7)
The last term in (3.7) is zero because E ia and E sb are regular solutions of (2.5) in the interior
of Sb,2 . Then letting R → ∞ and 1 → 0, using Lemma 1, we obtain
{E ia , E sb } =
4πi
b · Gb (a; p
b 2 ).
p
k 1
(3.8)
As {E sa , E ib } = −{E ia , E sb }, we easily deduce that
{E sa , E ib } =
4πi
b 1 ).
b · Ga (b; p
p
k 2
(3.9)
Finally, in view of the regularity of E sa and E sb in the region exterior to S, we have
{E sa , E sb } = {E sa , E sb }SR .
(3.10)
Then, letting R → ∞, we pass to the radiation zone and thus using (2.4) we get
{E sa , E sb } =
2i
k
Z
S2
b 1 ) · g b (r
b; p
b 2 ) ds(r
b ).
g a (rb ; p
(3.11)
Substituting (3.5), (3.6), (3.8), (3.9), and (3.11) in (3.7) gives (3.3), and the theorem is proved.
4
Reciprocity
In [5] a reciprocity relation for spherical waves scattered by an achiral obstacle has been proved.
The same relation also holds for a penetrable chiral scatterer.
b 1 and
Theorem 3 For any two point–source locations in Ω, a and b, for any polarizations, p
b
p2 , and for a penetrable chiral scatterer, we have
b×p
b; p
b 2 ) · ∇ × E sa (b
b 1 ) = h0 (kb) (a
b ×p
b 1 ) · (∇ × E sb (a
b; p
b 2 )) .
h0 (ka) (b
(4.1)
Proof.
Using the transmission conditions (2.7) and applying the divergence theorem in
Ωc , we obtain {E ta , E tb }S = 0. Also, using some asymptotics we get {E sa , E sb }S = 0. Now
{E ia , E sb }S = 0, since the corresponding integral on the large sphere SR vanishes due to
b; p
b 1 ) = h0 (kr) g ia (r
b; p
b 1 ) + O(r −2 ) as r → ∞, where
the asymptotic form (2.4) and E ia (r
r ·a) (r
b 1 ) = ikae−ika(1+b
b × (a × p
b 1 )) . Combining the above we finally obtain (4.1).
g ia (rb ; p
4
5
The optical theorem
We define the scattering and absorption cross–sections due to a point source at a, [5], as
σas
1
= 2
k
Z
S2
b; p
b )|2 ds(r
b)
|g a (r
σaa
and
respectively, and the extinction cross–section by
in Theorem 2, we obtain
σae
1
= Im
k
=
Z
S
σas +σaa .
b · (E ta × ∇ × E ta ) ds,
n
b1 = p
b2 = p
b
If we put a = b and p
b · Ga (a; p
b )] + 2πk −2 Ea,a (p
b; p
b ).
σas = −4πk −2 Re [p
(5.1)
From the above definitions and (3.3) we have
b; p
b ).
σaa = −2πk −2 Ea,a (p
(5.2)
Hence, adding (5.1) and (5.2), the definition (3.1) gives
b · Ga (a; p
b )] .
σae = −4πk −2 Re [p
(5.3)
b; p
b ) is given in Theorem 2; it depends on the scatterer’s properties.
The value of Ea,a (p
6
Mixed scattering relations
b, p
b · r } be an incident time–harmonic plane electric wave, where the
b) = p
b exp{ik d
Let E i (r ; d
b
b gives the polarization.
unit vector d describes the direction of propagation and the unit vector p
We will indicate the dependence of the total field in Ω, the total field in Ωc , the scattered field
b and the polarization p
b by writing
and the electric far–field pattern on the incident direction d
t
−
s
b
b
b
b
b
b
b
b
b
E (r ; d, p), E (r ; d, p), E (r ; d, p) and g (r ; d, p), respectively.
Here, we consider mixed situations, and relate fields due to one spherical electric wave
b, p
b 1 ) and one plane electric wave E i (r ; −b
b 2 ); we do this by letting b → ∞ in our
E ia (r ; p
previous results.
b · a + O(r −1 ) and |r − a|−1 = r −1 + O(r −2 ),
Using the asymptotic forms |r − a| = r − r
b) =
we can easily show that for the spherical electric wave (2.1) we have limb→∞ E ib (r ; p
i
b
b ), that is the spherical electric wave, when the point source goes to infinity, reduces
E (r ; −b, p
b and polarization p
b . Similarly, we have
to a plane electric wave with direction of propagation −b
t
t
s
s
b
b
b, p
b ) → E (r ; −b, p
b ), E b (r ; p
b ) → E (r ; −b, p
b ) and g b (r
b; p
b ) → g (r
b ; −b
b ) as b → ∞.
E b (r ; p
Next, let b → ∞ in Lemma 1 to give the following result.
b, p
b 1 ) be an incident spherical electric wave and let E i (r ; −b
b 2 ) be an
Lemma 4 Let E ia (r ; p
incident plane electric wave. Then
n
o
b, p
b 1 ), E s ( · ; −b
b 2)
lim E ia ( · ; p
ε→0
n
o
b, p
b 1 ), E s ( · ; −b
b 2)
− lim E ia ( · ; p
Sa,ε r→∞
Sr
=
4πa
b, p
b · G(a; −b
b 2)
p
ik 1
where
b 2) =
G(a; −bb, p
b 2)
lim Gb (a; p
b→∞
h
b, p
b 2) −
= eika a × ∇ × E s (a; −b
is a plane far–field pattern generator .
5
ik
2π
Z
S2
i
r ·a
b 2 ) eikb
b)
rb × g (rb ; −bb, p
ds(r
b, p
b 2 ) and G(a; −b
b 2 ) we have the following limiting values [8].
For the generators Gb (a; p
b 1 ) and E ib (r ; p
b 2 ), we have
Theorem 5 For two incident point–source electric waves, E ia (r ; p
b 2 ) = g b (−a
b; p
b 2)
lim Gb (a; p
(6.1)
b, p
b, p
b 2 ) = g (−a
b ; −b
b 2 ).
lim G(a; −b
(6.2)
a→∞
and
a→∞
We can now let b → ∞ in the general scattering theorem, Theorem 1. The proof of the
following result is similar to that of Theorem 2 and is omitted for the sake of brevity.
b, p
b 1 ) be an incident spherical electric wave and let E i (r ; −b
b 2 ) be an
Theorem 6 Let E ia (r ; p
incident plane electric wave. Then
1
b, p
b; p
b 1 · G(a; −b
b 2) + p
b 2 · g a (b
b 1) +
p
Z
2π
S2
b; p
b 2 ),
b; p
b 1 ) ds(r
b ) = Ma (−b
b 1, p
b 2 ) · g a (r
g (rb ; −bb, p
where
b; p
b 1, p
b 2 ) = lim Ea,b (p
b 1; p
b 2)
Ma (−b
b→∞
(
k
=
Im (B1 )
2π
Z
− Im (γ 2 B1 )
Ωc
b, p
b 1 )) · (∇ × E (r ; −b
b 2 )) dv
(∇ × E a (r ; p
Z
Ωc
b, p
b 1 ) · E (r ; −b
b 2 ) dv
E a (r ; p
Z
2
b, p
b 1 ) · (∇ × E (r ; −b
b 2 )) dv
E a (r ; p
+ i[βc γ + Re (B2 )]
Ωc
Z
− i[βc γ 2 B1 + Re (B2 )]
)
b, p
b 1 )) · E (r ; −b
b 2 ) dv .
(∇ × E a (r ; p
(6.3)
Ωc
To conclude, we note that we also have
b, p
b 2 ) = lim lim Gb (a; p
b 2 ) = g (−a
b ; −b
b 2 ).
lim lim Gb (a; p
a→∞ b→∞
b→∞ a→∞
(6.4)
This can be used to verify that the known scattering relations for plane–wave incidence [4], [5]
are recovered when a → ∞ and b → ∞. Furthermore, (6.2) and the reciprocity principle for
plane waves [5] give the following limiting property:
b, p
b 1 · G(a; −b
b 2 ) = lim p
b 2 · G(−b; a
b, p
b 1 ).
lim p
a→∞
7
b→∞
(6.5)
Exact Green’s function for a chiral dielectric sphere
Consider a spherical scatterer of radius a. Take spherical polar coordinates (r, θ, φ) with the
origin at the centre of the sphere, so that the point source is at r = r0 , θ = 0, and so that the
b = x
b is in the x–direction. Thus, r 0 = r0 z
b and b
b , where x
b and z
b are
polarization vector p
unit vectors in the x and z directions, respectively.
6
Using spherical vector wave functions, and in particular (13.3.68), (13.3.69), (13.3.70) of
[7] we obtain the following expansion for the incident field, see [3]
b) =
E inc
r 0 (r ; x
∞
o
X
i
2n + 1 n
e n (kr0 ) M 1 (r )
hn (kr0 ) N 1e1n (r ) − h
o1n
h0 (kr0 ) n=1 n(n + 1)
(7.1)
(1)
e n (x) = x−1 hn (x) + h0 (x) =
for r < r0 , where hn ≡ hn is a spherical Hankel function, h
n
1
1
x−1 [xhn (x)]0 , and M σ1n and N σ1n , for n = 1, 2, . . . and σ = e or o are the spherical vector
wave functions of the first kind.
The scattered field due to a chiral sphere does not have the same φ–dependence as the
incident wave, so it has the following general form, [6], p. 394,
b )=
E sr0 (r ; x
∞
o
X
2n + 1 n
i
an N 3e1n (r ) − bn M 3o1n (r ) + cn M 3e1n (r ) − dn N 3o1n (r ) ,
h0 (kr0 ) n=1 n(n + 1)
for r > a. In order to evaluate the electric field in the interior of the chiral sphere, we consider
the Bohren decomposition, [6]
E c (r ) = QL (r ) −
q
µc /εc QR (r )
q
and H c (r ) = −i εc /µc QL (r ) + QR (r ),
r ∈ Ωc ,
where QL (r ) and QR (r ) are Beltrami fields, satisfying ∇ × QL = γL QL and ∇ × QR =
−γR QR . We employ the following expansions for the Beltrami fields, [6], p.395
QL (r ) =
∞
i
h
io
h
X
2n + 1 n
i
An M 1o1n (r ) + N 1o1n (r ) + Bn M 1e1n (r ) + N 1e1n (r )
h0 (kr0 ) n=1 n(n + 1)
QR (r ) =
∞
h
i
h
io
X
i
2n + 1 n
Cn M 1o1n (r ) − N 1o1n (r ) + Dn M 1e1n (r ) − N 1e1n (r )
h0 (kr0 ) n=1 n(n + 1)
Using the transmission conditions on r = a, we obtain
an =
SnL WnR + SnR WnL
,
VnL WnR + VnR WnL
cn = −
bn =
TnL VnR + TnR VnL
VnL WnR + VnR WnL
e n (kr0 )
e n (kr0 ) SnL VnR − SnR VnL
h
h
dn =
hn (kr0 )
hn (kr0 ) VnL WnR + VnR WnL
where, for A = L, R,
e n (ka)jn (γA a) − (η/ηc ) hn (ka)e
WnA = h
jn (γA a)
e n (ka)jn (γA a)
VnA = hn (ka)ejn (γA a) − (η/ηc ) h
h
i
h
i
SnA = hn (kr0 ) jn (ka)ejn (γA a) − (η/ηc ) ejn (ka)jn (γA a)
e n (kr0 ) (η/ηc ) jn (ka)e
TnA = h
jn (γA a) − ejn (ka)jn (γA a) ,
e
jn (x) = x−1 jn (x) + jn0 (x) = x−1 [xjn (x)]0 ,
ηc
1
e n (kr0 )jn (ka)
An =
−hn (ka)
cn + bn − h
2jn (γL a)
η
7
(7.2)
1
ηc
ηc
Bn =
hn (ka)
an + cn + hn (kr0 )jn (ka)
2jn (γL a)
η
η
ηc
1
e n (kr0 )jn (ka)
hn (ka) − dn + bn + h
Cn =
2iηc jn (γR a)
η
1
ηc
ηc
Dn =
hn (ka)
an − cn + hn (kr0 )jn (ka)
2iηc jn (γR a)
η
η
(7.3)
(7.4)
(7.5)
p
and ηc = µc /εc is the interior intrinsic impedance.
Let us calculate the electric far–field pattern. Since hn (x) ∼ (−i)n h0 (x) and h0n (x) ∼
(−i)n−1 h0 (x) as x → ∞, and using (13.3.68) and (13.3.69) of [7] we find that
M 3σ1n (r ) =
q
b)
n(n + 1) hn (kr) C σ1n (r
∼
q
b)
n(n + 1) (−i)n h0 (kr) C σ1n (r
(7.6)
and
N 3σ1n (r ) = n(n + 1) (kr)−1 hn (kr) P σe1n (rb ) +
∼
q
q
e n (kr) B σ1n (r
b)
n(n + 1) h
b)
n(n + 1) (−i)n−1 h0 (kr) B σ1n (r
(7.7)
b ), P σ1n (r
b ) and B σ1n (r
b ) can be found e.g. in the Appendix of [3].
as kr → ∞, where C σ1n (r
Therefore for the electric far–field pattern, we have
∞
X
(2n + 1)(−i)n
1
b) = −
p
g r0 (r ; x
h0 (kr0 ) n=1
n(n + 1)
(
b ) + ibn C o1n (r
b) +
an B e1n (r
)
b ) + dn B o1n (r
b) .
icn C e1n (r
(7.8)
References
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43 (2002) 5683–5697.
[2] Athanasiadis, C.; Martin, P. A.; Stratis, I. G.: IMA J. Appl. Math. 66 (2001)
539–549.
[3] Athanasiadis, A.; Martin P. A.; Stratis I. G.: Z. Angew. Math. Mech. 83 (2003)
129–136.
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