Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
403
Electromagnetic Scattering from an Anisotropic
Uniaxial-coated Conducting Sphere
You-Lin Geng1,2 , Xin-Bao Wu3 , and Bo-Ran Guan2
1
Xidian University, China
Hangzhou Dianzi University, China
Shanghai Research Institute of Microwave Technology, China
2
3
Abstract
The scattering fields from an anisotropic uniaxial-coated conducting sphere by a plane wave are derived.
The electromagnetic fields in uniaxial anisotropic medium and free space can be expressed in terms of spherical
vector wave functions in uniaxial anisotropic media and isotropic medium. Applying the boundary condition
in the interface between the uniaxial anisotropic medium and free space, the surface of the conducting sphere,
the expansion coefficients of electromagnetic fields in uniaxial anisotropic medium are obtained, and then the
expansion coefficients of scattering fields and radar cross sections can be obtained. Numerical results between
this method and Mie theory are in good agreement as we expect. some numerical results are given in this paper.
Introduction
In recent years, there has been a growing interest in interaction between electromagnetic fields and anisotropic
media, mainly due to its many applications in the fields of antennas and microwave devices, etc. As this is an
interesting subject of many potential applications, there have naturally been some existing work, for instance, the
analysis of two-dimensional geometries [1,2] and three-dimensional geometries [3-8]. In this paper, on the basis of
electromagnetic fields in uniaxial anisotropic medium using spherical vector wave functions [6], electromagnetic
fields in anisotropic uniaxial-coated conducting sphere are formulated and numerically studied in this paper.
The present work in this paper serves as a further extension of the studies in [6], and the fields in free space can
be deduced from the present results and then expressed in terms of spherical vector wave functions in isotropic
medium [6,9]. Applying the boundary conditions of electromagnetic fields on the interface between uniaxial
anisotropic medium and free space and on the interface of conducting sphere, all the field expansion coefficients
in uniaxial anisotropic medium and free space are derived. Some numerical results are also obtained using the
formulas and presented herein. One special case is considered, where the results obtained using the present
method and the Mie theory [11] are compared to each other and a good agreement is observed.
Formulas
Let us consider an anisotropic uniaxial-coated conducting sphere illuminated by an incident plane wave. As illuminated in Fig.1, the coated sphere
with outer radius a1 and inner radius a2 is located at the coordinate origin. On the surface the inner conducting sphere, the uniaxial anisotropic
medium with permittivity tensor (ǫ) and permeability tensor (µ) is coated
with thickness d(= a1 − a2 ). It is assumed that the incident wave propagates
in the +b
z direction, the incident electric field has unity of amplitude, and
is polarized in the +b
x direction. In the following analysis, a time dependence of exp(−iωt) is assumed for the electromagnetic field quantities, but
is suppressed throughout the treatment.
Figure 1:
Geometry of a
The electric field vector wave equation in such a source-free uniaxial plane wave scattered by an
anisotropic medium can be written in the following form [2,5,6]:
anisotropic uniaxial-coated conducting sphere.
−1
∇ × µ · ∇ × E(r) − ω 2 ǫ · E(r) = 0.
(1)
where E denotes the electric field, while ǫ and µ represents the permittivity tensor and the permeability tensor
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
404
of uniaxial anisotropic medium, the expression are [6,8]


ǫt 0 0
ǫ =  0 ǫt 0  ,
0 0 ǫz

µt
µ= 0
0
0
µt
0

0
0 .
µz
(2)
Using Fourier transform [5,6], the expansion of plane wave factors in terms of spherical vector wave functions in isotropic medium [10], and the properties of spherical Bessel functions [9], the electromagnetic fields
(designated by the subscript 1 ) in the uniaxial anisotropic medium can be obtained as follows:
E1 =
2 X
2 X
X
(l)
Fmn′ q
l=1 q=1 mnn′
× Pnm′ (cos θk )kq2
H1 =
2 X
2
X
Z
π
0
i
h
e
(l)
e
(l)
(θ
)N
(r,
k
)
+
C
(θ
)L
(r,
k
)
Aemnq (θk )M(l)
(r,
k
)
+
B
q
q
q
mn
mnq k
mn
mn
mnq k
sin θk dθk ,
i
X (l) Z π h
h
(l)
h
(l)
Fmn′ q
Ahmnq (θk )M(l)
mn (r, kq ) + Bmnq (θk )Nmn (r, kq ) + Cmnq (θk )Lmn (r, kq )
l=1 q=1 mnn′
× Pnm′ (cos θk )kq2
(3a)
0
sin θk dθk .
(3b)
where n′ and n are summed up both from 0 to +∞ while m is summed up from −n to n, and r is pointing
(l)
in the (θ, φ)-direction in the spherical coordinates. The coefficients, Fmnq , are unknown, as in [6]. Apmnq (θk ),
p
p
Bmnq
(θk ), Cmnq
(θk ) (where p = e or h) and kq are functions of θk and they have been derived in [6]. The vector
(l)
(l)
(l)
wave functions, Mmn , Nmn , Lmn are spherical vector wave functions and they are also shown in [5,6,9,10]
Pm
dPm
n (cosθ) imφ b
n (cos θ) imφ b
(l)
M(l)
=z
(kr)
im
e
θ
−
e
φ
,
(4a)
mn
n
sin θ
dθ
(l)
(l)
zn (kr) m
1 d(rzn (kr)) dPm
Pm
n (cos θ) b
n (cos θ) b imφ
(l)
imφ
Nmn=n(n+1)
Pn (cos θ)e br+
θ+im
φe ,
(4b)
kr
kr
dr
dθ
sin θ
(l)
(l)
zn (kr) dPm
Pm
dzn (kr) m
n (cos θ) b
n (cos θ) b
imφ
imφ
.
(4c)
L(l)
=k
P
(cos
θ)e
b
r
+
θ
+
im
φ
e
mn
d(kr) n
kr
dθ
sin θ
(l)
(1)
where zn (where l = 1, 2, 3, and 4) denotes an appropriate kind of spherical Bessel functions, jn , yn , hn , and
(2)
hn , respectively.
The incident electromagnetic fields(designated by the superscript inc) can be expanded in an infinite series
in isotropic spherical vector wave functions [6,9,10]
i
Xh
x
(1)
Einc =
axmn M(1)
(r,
k
)
+
b
N
(r,
k
)
[δm,1 + δm,−1 ],
(5a)
0
0
mn
mn mn
mn
H
inc
i
k0 X h x
x
(1)
=
amn N(1)
mn (r, k0 ) + bmn Mmn (r, k0 ) [δm,1 + δm,−1 ].
iωµ0 mn
where the expansion coefficients are defined as:

2n + 1

 in+1
,
x
2n(n
+ 1)
amn =

 in+1 2n + 1 ,
2

2n + 1
n+1

 i
,
2n(n + 1)
bxmn =

 −in+1 2n + 1 ,
2
δs,l =
(5b)
m = 1,
(6a)
m = −1;
m = 1,
(6b)
m = −1;
1, s = l,
0, s =
6 l.
(6c)
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
405
According to the radiation condition of an outgoing wave (attenuating to zero at infinity) and the asymptotic
(1)
behavior of spherical Bessel functions, only hn should be retained in the radial functions, therefore the expansion
of scattered fields (designated by the superscript s) are
i
Xh
s
(3)
Es =
Asmn M(3)
(r,
k
)
+
B
N
(r,
k
)
,
(7a)
0
0
mn
mn mn
mn
i
k0 X h s
s
(3)
H =
Amn N(3)
mn (r, k0 ) + Bmn Mmn (r, k0 ) .
iωµ0 mn
s
(7b)
s
where the coefficients, Asmn and Bmn
(n varies from 0 to +∞ while m changes from −n to n), are unknowns
(l)
(l)
to be determined, Mmn (r, k0 ) and Nmn (r, k0 ) denote the spherical vector wave functions defined in Eqs.(4a) to
(4c), and k0 = ω(ǫ0 µ0 )1/2 identifies the wave number of free space, respectively.
Applying the boundary conditions at the surface of uniaxial anisotropic medium, for example, when r = a2 ,
the expansion coefficients of electromagnetic fields in uniaxial anisotropic medium can be obtained by the
following equations:
2 X
2 X
∞
X
(l)
Fmn′ q
l=1 q=1 n′ =0
(l)
Fmn′ q
Z π
0
e
Bmnq
π
Aemnq zn(l) (kq a2 )Pnm′ (cos θk )kq2 sin θk dθk = 0,
(8a)
(l)
zn (kq r)
1 d (l)
e
rzn (kq r) + Cmnq
Pnm′ (cos θk )kq2 sin θk dθk = 0.
kq r dr
r
r=a2
(8b)
l=1 q=1 n′ =0
2 X
2 X
∞
X
Z
0
and r = a1 it can be obtained the following expression
2 X
2 X
∞
X
l=1 q=1 n′ =0
2 X
2 X
∞
X
l=1 q=1 n′ =0
(l)
Fmn′ q
Z
(l)
Fmn′ q
π
0
Z
0
m
2
x
Q(l)
mnq Pn′ (cos θk )kq sin θk dθk = [δm,1 + δm,−1 ] amn
i
,
(k0 a1 )2
(9a)
i
.
(k0 a1 )2
(9b)
π
(l)
Rmnq
Pnm′ (cos θk )kq2 sin θk dθk = [δm,1 + δm,−1 ] bxmn
(l)
(l)
where expansion coefficients axmn and bxmn can be expressed in Eqs.(6a) and (6b). Qmnq and Rmnq have the
following expression
1 d (1)
iωµ0 h
1 d (l)
(l)
e
(l)
Qmnq =
Amnq
rhn (k0 r) zn (kq r) −
Bmnq
rzn (kq r)
k0 r dr
k0
kq r dr
(l)
zn (kq r)
h
+Cmnq
· h(1)
,
(10a)
n (k0 r)
r
r=a1
iωµ0 h
1 d (1)
1 d (l)
(l)
(l)
e
Rmnq =
Amnq
rhn (k0 r) zn (kq r) − Bmnq
rzn (kq r)
k0
k0 r dr
kq r dr
(l)
zn (kq r)
e
+Cmnq
· h(1)
.
(10b)
n (k0 r)
r
r=a1
From the Eqs.(8a) to (9b), it shows that
• firstly, the unknown coefficients of electromagnetic fields in the uniaxial anisotropic medium can be obtained;
• secondly, the coefficients of scattered fields in region 0 are calculated; and
• lastly, the far scattering field of electromagnetic fields from an anisotropic uniaxial-coated conducting
sphere by a plane wave, and the radar cross section are thus obtained.
406
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
Numerical Results and Discussion
In the last section, we have presented the necessary theoretical
formulation of the electromagnetic fields of a plane wave scattered
by an anisotropic uniaxial-coated conducting sphere. To gain more
physics insight into the problem, we will provide in this section some
numerical solutions to the problem of electromagnetic scattering by
an anisotropic uniaxial-coated conducting sphere.
Numerical computations have been performed by applying the theoretical formulae derived earlier in the previous sections. In order to
check the accuracy of the newly obtained numerical results, we performed one trial, that is, we calculated the radar cross sections using
the present method and the Mie theory in reference [11]. The results
are shown in Fig.2, where electric dimensions of outer and inner spherical surfaces are k0 a1 = 2.1π and k0 a2 = 2π, while the permittivity and
permeability tensor elements are ǫt = ǫz = 2.5ǫ0 , µt = µz = 1.6µ0 , respectively, (where and subsequently, ǫ0 and µ0 stand for the free space
permittivity and permeability, respectively).It must be noted that the
incidence wave propagates in the negative z-direction in this figure.
Figure 3: Radar cross sections (RCSs) versus scattering angle θ (in degrees) in the E-plane (solid curve)
and in the H-plane (short dashed curve).
Figure 2: Radar cross sections (RCSs)
versus scattering angle θ (in degrees):
Results of this paper (solid curve) and
of Mie theory(block square)(Fig.7 in
Reference [11]).
Figure 4: Radar cross sections (RCSs) versus scattering angle θ (in degrees) in the E-plane (solid curve)
and in the H-plane (short dashed curve).
From Fig.2, it is seen apparently that the radar cross sections calculated by using the two methods (i.e., the
present method in this paper and Mie theory) are in very good agreement in both the E- and H-planes, where
the maximum number of n′ used in Eqs.(8a) to (9b) is only 10 to achieve the convergence. It partially verifies
the correctness and applicability of our theory as well as the program codes.
After this, we obtain some new results unavailable elsewhere in literature. Two examples are considered
herein, and their radar cross sections are plotted in Figures 3 and 4.
Fig.3 represents radar cross sections of an anisotropic uniaxial-coated conducting sphere of more general
uniaxial medium, where the permittivity and permeability tensor elements are characterized by ǫt = 2ǫ0 ,
ǫz = 4ǫ0 , and µt = µz = µ0 , the electric size of the uniaxial anisotropic spherical shell is chosen as k0 a1 = 3π
and k0 a2 = 2.5π. The maximum number n′ in Eqs.(8a) to (9b) to achieve a good convergence is found to be 16.
To illustrate further applicability of the scattering solution for an electrically large sized anisotropic uniaxialcoated conducting sphere(for example, in its resonance region), the radar cross sections of a relatively large
uniaxial anisotropic sphere with k0 a1 = 5π and k0 a2 = 4π, under the illumination by an incident plane wave,
are obtained and depicted in both the E-plane and the H-plane in Fig.4. The permittivity and permeability
tensor parameters used for this case are: ǫt = (2 + 0.2i)ǫ0 , ǫz = (4 + 0.4i)ǫ0, and µt = µz = µ0 . As the electric
dimension of the sphere is increased, the maximum number of n′ used in Eqs.(8a) to (9b) must be significantly
increased to 24 to achieve the convergence.
Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26
407
Conclusion
The spherical vector wave function expansion solution to the plane wave scattering by an anisotropic uniaxialcoated conducting sphere is obtained analytically in this paper. The solution has only one-dimensional integral
which can be calculated easily. Numerical results are obtained using the present method and compared with
Mie theory and a fairly good agreement is observed. It is shown that the obtained solution is stable even for
almost isotropic scatterers, since the proposed solution is an analytical one of the uniaxial anisotropic media,
and the result of the Mie theory is a special case of the present method. The general numerical results, including
the lossy anisotropic uniaxial-coated conducting sphere and resonance region, are given and are found reducible
to those of spacial cases. The present analysis are believed to be useful in antenna and satellite communication
system designs.
*This work is partially supported through No. Y104539 by the Natural Science of Zhejiang Province of China and a
Research Grant No: 60071025 by the National Natural Science Foundation of China (NSFC).
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