12-4
Scattering from Chirally Coated Oblate Spheroids
Modifications
- A Study in RCS
R. Sharma and N. Balakrishnan
Department of Aerospace Engineering, Indian Institute of Science
Bangalore 560 012, INDIA
ABSTRACT
I n t h i s paper extended
boundary condition method has
been formulated to compute the
scattering from three dimensional dielectric scatterers
coated by a chiral material.
The applicability of this
method t o oblate spheroidal
shape is also discussed.
INTRODUCTION
T h e lack of geometric
symmetry between an object and
its mirror image is referred to
as chirality, and the mirror
image of such a chiral object
cannot be made to coincide with
the object itself by any operation involving rotations and/or
translations. In recent times
a lot of attention has been
focussed towards evaluation and
reduction of radar cross section (RCS) using chiral materials. Bohren [ l ] studied the
scattering from a chiral sphere
for the first time, using the
constitutive equations appropriate for an optically active
isotropic medium. The scattering and absorption from chiral
non-spherical objects was
studied by Lakhtakia et.al [ Z ]
U s l e n g h i [ 3 ] evaluated t h e
scaterring from the chirally
coated sphere. Scattering from
chirally coated planar surfaces
with an aim to reduce the RCS,
was analysed by Jaggard [ 4 ] .
chirally coated scatterers. The
analysis is carried out using
extended boundary condition
method (EBCM). This method was
first developed by Waterman [ 5 ]
using the Huygen's
Poincare
method to evaluate the scaterring from homogeneous nonspherical bodies. Barber and
Yeh [ 6 ] modified the analysis
using the Schelkunoff's field
equivalence principle, and
their formulation is convenient
to program on a computer and is
applicable t o dielectric
bodies. Barber and Wang [ 7 ]
applied this method to three
dimensional electromagnetic
scattering problems involving
multi-layered dielectric objects. In this paper, Barber
and Wang's 171 formulation has
been modified t o t a k e into
account the chiral nature of
the outer layer. The modified
formulation is used to evaluate
the scattered fields and the
radar cross section of chirally
coated non-spherical dielectric
bodies.
-
FORMULATION OF THE PROBLEM
The constitutive relations
in a chiral medium get modified
.
The present work is directed towards t h e study of
electromagnetic scattering from
three dimensional non-spherical
0-78034549-3/92
$3.00 43 1992 IEEE
251.
where f , . p aria /3 are the permittivity, permeability and
chirality parameter of t h e
medium.
Following Bohren [ l ] t h e EM
(2)
1992 Asia-Pacific Microwave Conference. Adelaide
aR
where qL and
are the LCP and
the RCP fields and satisfy the
conditions
along with the auxiliary conditions
zL
i
0 X
= kL gL
V*& = 0
V X QR = kR QR > V'QR = 0
assumed to be uniform throughout the entire space and equal
to po the permeability of the
free space. Applying the field
equivalence principle results
in the reduction of the two
layered scattering problem into
three sub problems as shown in
Fig. 2.
(3C)
(3d)
In equation (2) the matrix
is given as
A
[A] =
-
where kL = k/ (1 kp ) ;
aL = -jve/p ;
kR = k/(l + kj3 ) ;
a~ = - j / W / p ) ;
Thus, t h e electromagnetic
fields existing in the chiral
medium are given by
null lield
(b)
(C)
F i g . 2 A p p l i c a t i o n 01 e q u i v a l e n c e
pr i n c i p 1e;Or i g i n a1 scat ier i n g
problem i s r e p l a c e d by three
su b- p ro b l e m s
(4)
Consider a chirally coated
body, say an oblate spheroid as
shown in Fig. 1.
Modifying the Barber and Wang's
[7] formulation for the chirally coated body (Fig.1) we have
the following integral equations;
83(3
=
S(E) + v
X I (h2X
.):E
Sa
WE^)/(;^
%(k3z)ds'- V X V X
X
nj)-E
52
(k3E)ds'
outside S2
inside
=O
S
75)
z2&)
Fig.1 Chirally coated
oblate s p h e r o i d
It can be treated as a two
layered object which is characterised by the constitutive
parameters e l , p1 for the inner
layer and e 2 , p and p for
the chiral layer. +he permittivity of t h e surrounding
medium is e 3 which maybe free
space.
For the sake of simplicity the permeability is
252.
=
V X!(-G2
X E?)-z(k2R)ds1
32
E(k2E)ds1
+
E(k2i?)ds'-VXVX
e(k2E)dsp
= o
-G(kiE)
V
X
1 (hlXEz)e
s,
(1/jwe2)(
~ l m ~ ) ~
st
between S1 and
inside S1
is the
Green's
S2
(6)
dyadic
for that medium. E= Ir-r ' I and
and
are position vectors
from an interior origin
to
field and source points, respectively.
The I + '
and I - )
subscripts indicate
external
and internal fields respectively , evaluated at the surface.
The superscripts in E and H
denote the fields in the regions as shown in Fig.1.
An
equation for E1(R) could also
be written but is redundant.
Enforcing the boundary conditions that the tangential
electric and magnetic fields
are continuous across the surfaces S1 and S in (5) and ( 6 ) ,
gives a set OH integral equations,
E3(E)
= Ei
=
G(k3E?)ds'
(^n21&2_)
-
C = C
I(&, x E2,)-
+v x
-V
n
*
C
C
u=even m=O n-1
or odd
D is the normalization constant. The free space dyadic
Green's function is expanded as
in Morse and Feshbach [lo].
The superscripts 1 and 3 indicate Bessel and Hankel function
radial dependence respectively.
The index n is truncated at
some value N depending on the
accuracy r e q u i r e m e n t s .
To
e v a l u a t e t h e f i e l d s in t h e
chiral layer qL and GR are
expanded in t e r m s of vector
spherical harmonics [ll]
where
32
X V
(k3E) ds
=O
X J(l/jwc3)
s,
outside S2
inside S
(7f
52 (E)
c
=
v x j (oh2 x Z)
sa
-V
X
Using equation (10) along with
e q u a t i o n ( 4 ) t h e f i e l d s in
region ( 2 ) can be evaluated.
The dyadic Green's function in
chiral region is given as in
X J(l/jwe,)
s,
(fi, X ~ ~ ) o ~ ( k 2 ~ +) Vd sX n (f;,
G(k26)ds1
V
I
SI
X T3:)*3(k2E)$f1z
V-X V X
(l/jw~~)I(iiX H,). E(k2R)dst
5,
lfetween S1 and S
= o
b)
outside S2 and
inside S1
To s o l v e t h e integral equations, the fields are expanded
in terms of the vector spherical harmonics M and N [ 9 ]
and
I n s i d e t h e s u r f a c e S2, t h e
total field is zero a s in
e q u a t i o n (7). S u b s t i t u t i n g
(lo), (11) and (12) in equa-
c
G(k3E)dS'
253.
and absorption
Writing it in matrix form
characteristics
of lossy dielectric chiral non-
Similar expressions can be
written for other elements of
QI matrices.The scattered field
is given as
E?)5(k3E) v x v x { (l/jwe3)(n2 x G 2 ) d
Es(E) = 0 X(;,
r
X
32
A
*A
G (k3s)
ds
(16)
and can be evaluated once the
solution to eqn.(l5) is obtained. The RCS then can be
obtained directly from the back
scattered fields.
1953.
11. A .
CONCLUSION
Using the field equivalence principle, originally
applied to non chiral media,
the extended boundary condition
method has been shown t o be
applicable to the evaluation of
the scattered field and RCS of
chirally coated 3D-scatterers.
The formulation is versatile
enough to handle complex shapes
such a s t h e fuselage of an
aircraft
spherical objects. ( I , Applied
Optics, Vol. 24, No. 2 3 , pP
4146-4154 Dec. 1985.
3.P.L.E. Uslenghi *!Scattering
by an impedance sphere coated
with a chiral layer", EleCtrOmagnetics 10:201-211, 1990.
4.D.L.Jaggthd and N. Engheta
'*Chirosorb as an invisible
medium**, Electronics Letters
25, 173-174 ,1989.
S.P.C.Waterman, flSymmetry,
unitarity and geometry in
electromagnetic scattering,
Phys. Rev. D. 3 , 825,1971.
6.P. Barber and C. Yeh, "Scattering of electromagnetic waves
by arbitrarily shaped dielectric bodies" , Applied Optics
14 , 2864 , 1975.
7.D. S. Wang and P. W. Barber.
"Scattering by inhomogeneous
nonspherical objects" , Applied
Optics V01.18, No. 8 , pp 11901197 April 1979.
8.
J. A. Kong, Theory of
electromagnetic waves,Wiley,New
York,1975.
9.J. A. Stratton.Electromagnetic theory McGraw Hil1,New York
1941.
10.P. M. Morse and H.Feshbach,
Methods of theoretical physics
Vo1.2 McGraw Hill New York
.
REFERENCES
l.C.F.Bohren, "Light scattering
by an optically active sphere"
,Chem. Phys. Letters, 29, 3.1
458-462 Dec. 1974.
2.A. Lakhtakia, V. K. Varadan
and V. V. Varadan, *@Scattering
254.
Lakhtakia, V.K. Varadan,
V.V. Varadan, Time harmonic
electromagnetic fields in
chiral media, Springer Verlag
1989.