696
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 5 , MAY 1988
Extinction Cross Section of a Dielectric Strip
SUBRATANANDA DOWERAH AND ALOKNATH CHAKRABARTI
Ahtruct-The problem of scattering of a plane electromagnetic wave
by a dielectric strip is formulated in terms of an uncoupled system of
three-part Wiener-Hopf equations by using a set of approximate
boundary conditions derived and u t i l i recently. The resulting WienerHopf problems are solved approximately for sufficiently large values of
the width of the strip by using Jones’ method. An analytical formula is
derived for the extinction cross section of the strip under consideration
from which numerical values are obtained in specific situations and the
results are presented graphically. The radar cross section of the strip is
also computed in special circumstances and these are presented separately.
I. INTRODUCTION
HE WIENER-HOPF technique has been utilized
extensively by various workers ([1]-[3], etc.) to attack
two-dimensional scattering problems associated with the
diffraction of plane electromagnetic/acousticwaves either by
perfectly conducting or metallic half-planes/strips wherein the
boundary conditions on the scatterer involve, at the most,
partial derivatives of the scattered field of first order. It is only
recently that Leppington [4] has derived a different type of
boundary conditions involving second-order partial derivatives of the scattered field that are to be satisfied on dielectric
scatterers and, as an application of these boundary conditions,
that Chakrabarti [5] has considered the solution of the problem
of diffraction by a dielectric half-plane. As a further application of these new types of boundary conditions which are
different from the ones used by Rawlins [6], we have
considered, in the present paper, the problem of diffraction of
a plane transverse electric wave by a dielectric strip. By
employing a standard procedure as explained in Jones’ book
[7], the diffraction problem under consideration is reduced to a
system of uncoupled three-part Wiener-Hopf equations and
these are solved approximately by assuming that the width b of
the strip is electrically very large. Utilizing these approximate
solutions and using the method of stationary phase, we have
ultimately derived a formula for the extinction cross section of
the strip under consideration for sufficiently general angles of
incidence (the only restriction on the angle of incidence is that
it must not be too close to zero). Derivation of the formulas for
extinction cross sections of strips of other types associated
with propagation of electromagnetic waves has been the
subject of several works existing in the literature (see [31, [71,
and [8]). We discover the structure of this important practical
quantity in the present problem as far as its dependence on the
various physical parameters of the dielectric strip is con-
T
cerned. The variation of the extinction cross section with the
angle of incidence for certain specific values of the physical
parameters is shown graphically. Results for the radar cross
section of the strip are also obtained and are presented
separately.
II. FORMULATION
AND REDUCTION
TO WIENER-HOPF
EQUATIONS
We consider a dielectric strip of electrically very small
thickness h and of dielectric constant el and magnetic
permeability p which occupies the region: 0 I x Ib, - 00 <
z < QO (x, y , z denote orthogonal Cartesian coordinates), and
is embedded in a medium of dielectric constant
and
magnetic permeability p, taken to be the same as that of the
strip, for simplicity. Throughout the analysis we shall assume
that the thickness of the strip is electrically very small and that
the terms of 0 (€3where
Eo=h(l-E),
E=Q/E,
(1)
can be neglected in our analysis. The width b of the strip will
be assumed to be at least O [ ( ~ E , ( ~ / ~(6
+ ~>) )0]; ke0 Q 1) so
that kb % 1. We shall confine our attention only to the case E
< 1, i.e., e l > €2. A transverse electric wave as given by
,!?;=Re (curl (0, 0, di)e-’@‘}
Hi=-Re
in Cartesian bases (x, y ,
( i w z ( 0 , 0, 4i)e-i@t}
(2)
z), with
qd=exp [ - i k ( x c o s Oo+y sin eo)]
(3)
strikes the dielectric strip as explained above. We shall drop
the factor e-”‘ and the symbol “Re” throughout the rest of
our analysis. It is required to determine the resultant total field
given by (2) with 4’ replaced by 4; + b“, where 4” denotes
the scattered potential. The mathematical problem of determining 4” turns out (with the help of Maxwell’s equations [7])
to be that of solving the reduced wave equation
Manuscript received April 16, 1987; revised March 4, 1988.
The authors are with the Department of Applied Mathematics, Indian
Institute of Science, Bangalore-560 012, India.
IEEE Log Number 8819817.
0018-926X/88/0500-0696$01
.OO 0 1988 IEEE
(4)
697
DOWERAH AND CHAKRABARTI: EXTINCTION CROSS SECTION OF DIELECTRIC STRIP
and the radiation condition characterizing outgoing waves at
infinity. The boundary conditions (5) and (6) on the surfaces of
the scatterer have been derived by Leppington under the
following assumptions.
i) The tangential component of the total electric field E +
E is continuous across the faces y = -+h of the dielectric
strip.
ii) The tangential component of the total magnetic field i f i
+ if.is continuous across y = h.
iii) The thickness h of the strip is electrically so small that
kh 4 1, and that O ( ( k l ~ t
)e
~r)ms can be neglected.
For unique determination of the scattered fields we need, in
addition to the conditions mentioned above, appropriate edge
behavior of 4" at the two edges x = 0 and x = b of the strip.
We assume that 4", 4;, $&, and 4; possess at most integrable
singularities at x = 0 and x = b (these assumptions will be
verified subsequently). The physical requirements at the edges
x = 0 and x = b are that the energy in any region containing
these edges must be finite and the assumptions on integrable
singularities mentioned here satisfy this requirement (see [7]).
The boundary value problem posed above is next reduced to
that of solving two independent three-part Wiener-Hopf
problems by utilizing a standard procedure, known as Jones's
method (see Noble [l]). To this end we assume that the
wavenumber k has a small positive imaginary part k2, i.e., k
= k l + ik2 (k2 will be taken to be zero at the end) and define
the Fourier transform of the scattered potential 4" as given by
*
@"(a,y ) =
jm p ( x , y)eiax d ~ .
--m
and
j,"
U")(a)= [4;(x, 0+)-4;(x,
O-)]eiaxdx
(15)
+
where the subscripts " " and " - " appearing above indicate
that the corresponding functions are analytic in the upper and
the lower half-planes Im (a)> - k2 and Im (a)< k2 cos eo,
respectively, whereas the functions Uu)( j = 1 , 2) are
integral functions of a,and next using (3,(8), (lo), and (12)
we obtain the relation
P y ) ( a )+ L !)(a)+ [ A(a)- B(a)]f(')(a)
=F(')(a) (16)
with
f ( l ) ( a=
) 1
+(EO/E)Y(CY)
~ ) ( a
= (2EO/E)
)
=t
(17)
jb4 l ( x , O)eiax dx
1 -ei(or-kcoseo)b
a - k cos 00
A(1)
A ( ' ) ={ 2 ~ ~ ksin/ 0,.~ }
(18)
Also the relation
Py)(a)+ L !)(a)- y ( a ) [ A(a)+ B ( ~ x ) ] f ( ~=)f(lc2~) )( a )(19)
with
f"' (a)= 1- EoaZ/y(a)
(20)
(7)
Then the appropriate solution of the transformed partial
differential equation (4) satisfying the requirement that 4" is
outgoing at infinity may be written as
*"(a,y)=A(a)e-+)Y
=B(a)ev(a)Y
(y>O)
(y < 0)
(8)
where
~ ( a=)
(a2- k 2 )1'2
with
~ ( 0=)- ik.
(9)
Introducing the following unknown functions of a,as given by
P Y ) (=
~ (2EO/E)
)
L !'(a)= ( 2 4
j:
so
Sm 4;(x,
-m
O)ei*X
dx
(10)
obtained by integrating by parts. It is to be noted that while
deriving (16) and (19) we have made use of the analyticity of
the scattered field outside the obstacle. We obtain two more
relations from (14) and (15) as given by
U(I)(a)=A(a)-B(a)
(23)
U(')(a)= - ~ ( a ) [ A (+aB(a)].
)
(24)
and
Substitution of A(a) - B(a)and A(a) + B(a) as given by
(23) and (24) into (16) and (19), respectively, leads to the
following three-part Wiener-Hopf equations:
4;(x,
O)eiaxdx
U(')(a)= [+"(x, O + ) - @ ( x , O-)]eiaxdx
(13)
(14)
P':"(cu)+L'i"(a)+Uo"(a)fo"(a)=Fo)(a) ( j = 1, 2).
(25)
The basic unknowns A and B appearing in (8) can be
698
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36. NO. 5, MAY 1988
recovered in terms of the Wiener-Hopf unknowns U?)( j =
1, 2) by using (23) and (24).In the next section we shall
describe a procedure of solving the Wiener-Hopf equations
(25) approximately for electrically large values of the width b
by reducing these equations to two uncoupled systems of
integral equations as explained in Jones’ book [7].
and
H y ) ( a )= L !)( - a)-p’J”(a)e-iab
( j = 1, 2) (29)
we obtain the following two systems of independent integral
equations valid for Im (a)> - el:
where
withj = 1, 2.
III. APPROXIMATE SOLUTION
OF THE WIENER-HOPF
EQUATIONS In the Appendix we will present approximate expressions
(25) FOR LARGEbk
for the Wiener-Hopf factors fy) ( j = 1, 2) valid for small
values of E , by directly utilizing an analysis due to Leppington
[4].The integral equations (30) and (31)can be cast into more
convenient forms as given by
Utilizing the Wiener-Hopf factorizations of the functions
in the form f G ) ( c y ) = fY)(cy)f”‘(a)with f y ) ( a ) = f 0’)
fy)
+
eizbdz
(33)
(34)
( - a)(j = 1, 2), and following Jones’s procedure we deduce
(leaving aside the details) the following systems of integral
equations for the unknowns L?) and PY)( j = 1, 2):
where
?:)(a) = Zy)(a)
L(i’)(a)
1
-=fy)(a) 2?ri
+m+k~
{ py)(z)-F(j)(z))
dz
(Im ( a ) < e l )
E;(;)(a)=Hy’(a)+
(26)
+
1
- k COS e,)
1
e - ikb cos00
1
( a + k c o s Bo)-(a-kcos 0,)
1
+
(a+ k cos e,)
e-ik6cosOo
and
with ( j
=
(38)
1, 2), where
Equations (33) and (34) have similar structure as thOSe
obtained in Noble’s book [ l , p. 1991 and therefore Jones’
~~~Y
Changing to - in (26) and z to - z in (271,taking e2 = method described by Noble 111 can be S U C C ~ S S ~employed
to obtain an approximate solution for large b. Leaving aside
-el and defining
the details we present the functions 1:) and H y ) (j = 1, 2) in
zY(a)= L ‘i“(- a) P y)(a)e-iab
( j = 1, 2) (28) the following approximate forms (for large M =
-k2<el, e2<k2 cos e,.
+
699
DOWERAH AND CHAKRABARTI:EXTINCTION CROSS SECTION OF DIELECTRIC STRIP
O((k€o)-(2/3+6)); (k€o)a 1):
with
F(u) =
Im
eiu2
du
U
+ f (:'(a>[{
WO[- w a + k)l
and H(x) is the Heaviside unit function defined by
N ( x )= 0,
1
for X C O
- ~ , [ - 2 i k b { e ( e , ) ) 2 ] }(a- EO>
=1/2,
forx=O
+ iwO[-2ikb{s"(e0))2i
= 1,
for x> 0.
- WO[- ib(a+ k ) ] }
e(a+ 60)
ikob
1
[
+-2ak-I
~
a + k(O)
a
( k + k(O))
(av!)
I::
(- j 2) f(:)(a)x
with j = 1, 2, where
+2H(j-2)
1
At this stage we note that while obtaining the above solutions
of the systems of integral equations (33) and (34) we have not
utilized the approximate expressions of the Wiener-Hopf
factorsfii) as presented in the Appendix (only the relations*)
(a)= fo)(a)andf-j) (a)=
(- a),the analyticity
properties of the factors f;)(a),and the asymptotic results
(a)= O(a)a -+ 03 in appropriate half-planes have been
(40)
made use of). Using (39) and (40) along with (28), (29), and
(25) we arrive at the following results:
A(j)a(j)
- ijU)k-' w,(- ib[a+ k])
(45)
e)
fi)
700
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 5, MAY 1988
&(ab+%)
+-
2f !'(a)
[bo)
1
I
Wo[-ib(a+k)
2A
+-k(O)
+a H ( 2 - j )
1
[ ~ $ + x $ i ] k - ' . (46)
The scattered potential #Is is finally obtained by using (S),
(23), and (24) in the'form
with - kz < c < k2 cos 0,.
The actual behavior of the scattered potential #Is near the
two edges (0, 0) and (b, 0) can be derived by using Abelian
theorem to the result (47) (see Williams [9]) and our previous
assumptions about the integrability of #Is, #,;I #IL,and 4; near
x = 0 and x = b can be verified easily by observhg that #Is =
O ( ( - ~ ) ~ " ) a s x + 0- a n d @ = O((x - b ) 3 / 2 ) a s x - +b + .
The scattered fields?
!, and are now completely specified
by (2) with #Ii replaced by #Is.
+2
H ( m - 2)A
(k(O)+R(#I))
1
X
{2f
+2
( m = 1, 2)
?)(b))}
(52)
H ( m - 2)A
(k(O)-
R(+))
*
IV. THEFARFIELDAND THE EXTINCTION
CROSS
SECTION
Evaluating the integral (47) approximately for large kr by
the application of the steepest descent method after writing x
= r cos #I and Iyl = r sin #I (0 < #I < T), we obtain the
following expression for the scattered far field after some
lengthy but straightforward manipulations (see Noble [l] and
Jones [7])
ei(kr- r/4)
-X
where
1/2+
(27rkr)
O[(kr)-3/21 (48)
We note that WO(- iy) defined in terms of the complex Fresnel
function by (44) for real positive y, has the asymptotic
development
as y + 00 and this will be utilized for later calculations. The
extinction cross section U, of the dielectric strip can now be
70 1
DOWERAH AND CHAKRABARTI: EXTINCTION CROSS SECTION OF DIELECTRIC STRIP
derived by using the formula [7]
ue=lim Im r-m
(bk)
[ $ [t$ig+4sk]
an
an
CIS] (56)
(COS
where c is a large circle of radius r with its center at the origin,
and stars denote complex conjugates. Substituting the expression for +s as given by (48) and using Kelvin's method of
stationary phase we deduce that
'1
1
-I-
[2kb{ ?(Bo)} + 3 ~ / 4 ] )
[wO)i4
1
cos2 eo
sin [2kb{ e(eo)} + 3n/4]
[am][
(57)
0;i
where
(a - 00) stands for the limiting values of the
functions U;] (4) as 4 tends to a - Bo ( j = 1, 2, 3, 4 , 5 ; rn
= 1, 2).
Assuming that 8, is not near zero and using (48), (50)-(53),
and the result (55) in (57) we obtain, after neglecting terms of
O(P3) (note that bk is assumed to be at least of O(0-(2/3+6));
0
4 l), the following expression for the extinction cross
section:
E
O(0) =
8
cos4 Bo + sin4 0,
sin Bo
]
1
- a- -
(r)
(59)
sin (kb + 3a/4)]
{ (2akb)1/20)
702
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 5, MAY 1988
*
A(i)= 2P2 sin { kb(I + 2/32))
+
- cos (kb+ 37r/4)+ ( 1 + 2(2kb)'I2[sin (2kb)F(,
(
[(2(2kb)'l2[sin (2kb)17(~,{(2kb)'/~)
+ cos (2kb)F(,){ (2kb)'/2)])
*
{(2kb)'/2}-cos (2kb)F(j){(2kb)'"}])
- sin (kb+ 37r/4)]
(66)
The functions F(r)and F(i)occuring on the right-hand sides of (65)and (66)are defined in terms of the real Fresnel functions
by the equations
F ( i ) ( X )= (7r/2)'/2
sin ((7r/2)t2)dt)
The final expression €or the extinction cross section U, appears
The backscatter radar cross section (T of the strip can be
to be rather lengthy, but it is not difficult to see the dependence computed by using the result (see [lo])
of U, on the parameter 0 = 1/2 kh (1 - e), especially
through the terms 0 In 0 and p2 In 0 (see also Faulkner [3]).
For normal incidence (eo = 712) most of the expressions
given above simplify considerably and we obtain the following
remarkably simple expression in this case:
where Siis the incident power density and S, is the scattered
power density at a distance r from the origin in the direction
Ue=2(kh)2{1/€- 1 ) 2 { 1 - l/bk}.
(87)
back in the direction of incidence. Using the result (48) we
We have shown in Figs. 1 the variation of the extinction find that
cross section U, with the angle of incidence for three different
sets of values of the parameters (hk, bk), and for the value of e
= 0.166767which is a fairly good representative of a mica or
a porcelain strip (see Jones [7, p. 61).
704
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 5, MAY 1988
35
35
0
LO
.
45
1
50
55
3
60
3
65
70
]
75
80 ' 8 5
90
,
1
0
50
45
40
55
60
65
70
75
80
85
90
- 0.005
3
- 0.12
- 0.11
- 0001
- 0.10
- 009
- 008
-
- 0003
007
-006
- 0002
- 0.05
- 0.04
35
LO
45
'
0,0013-
0.0011
50
'
55
'
65
60
70
75
80
I
I
85
90
I
I
h k ='0.005
bk = 1200
E = 0.17
00012
- o.oon
-00012
-
-
00011
-0~Omo
0~0010
-
0.0009-
0.0009
0.0008
-04007
-
00006
- 0.0005
- 0.0004
0.0003
35
io
5:
o:
5;
$5
$0
50
Bofdeg)
-A
CO
L
bo
Ja~oOo'
(c)
Fig. 1. Variation of the extinction cross section (ue) with the angle of incidence (e") for three different sets of values of thickness
(hk) and width (bk) of the dielectric strip for a particular choice of E .
(here A = 27r/k is the wavelength in the medium of dielectric
constant E~ of the incident wave).
Taking four different values of E, i.e., E = 0.125, 0.17,
0.25, and 0.5 we have presented in Figs. 2 the variation of u/A
with the angle of incidence eo, for a particular choice of the
parameters hk and bk given by hk = 0.05 and bk = 1200.
The variation of u/A with E for normal incidence is also
depicted separately (Fig. 3).
Numerical values of both the important physical quantities
ue and U for other choices of the various parameters can be
obtained from the final results obtained in our work, i.e., (58)
and (89).
DOWERAH AND CHAKRABARTI: EXTINCTION CROSS SECTION OF DIELECTRIC STRIP
I
I
I
I
I
I
I
705
I
-
b k=l200
hk=0.05
E=0.125
0-
-
5
0
15
25
35
I
I
I
45
I
55
65
75
85
I
I
I
I
bk=1200
h k i 0.05
0-
-0
- -10
--lo
m
-40-501
5
1
I
I
I
15
25
35
45
I
55
(dag)
eo
1
I
I
65
75
85
- -20
5-20-
-40
- -30
- -30
- -40
- -40
1-50
-50
5
I
I
I
I
15
25
35
45
1
e,
(a)
55
(dcg)
I
I
1
65
75
85
1-50
(b)
15
25
35
45
55
65
75
E5
I
1
I
I
I
I
I
I
bk=1200
hk= 0.05
C L 0.5
bk-1200
hk= 0.05
ci0.25
0
Fig. 2. Variation of the radar cross section ( o h ) with the angle of incidence (e") €or a particular choice of the thickness (hk)and the
width (bk)of the dielectric strip for four different values of E .
APPENDIX
0.1
I
50
40
0.2
I
03
0.1
1
I
0.6
0.5
0'1
0.8
0.9
I
-
- 50
-
-LO
30-
- 30
20
-
- 20
10
-
-
0
Approximate Wiener-Hopf factors of the functions fu) (j
= 1, 2) defined by (17) and (20) can be obtained by directly
utilizingan analysis of Leppington [4] (also see [5]). When a
is not near f k we find that
1
0.1
1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
09
p" =f':"(a)f"(a)
with
fy(- a)=f"(a>
10
0
(90)
( j = 1 , 2),
(91)
+-hw]
(92)
where
E-
Fig. 3. Variation of the radar cross section ( d X ) with the ratio of the
dielectric constants (e = e2/eI) for normal incidence (e = 90").
l
i
i
2 7 r T
~~
a)
2
706
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36, NO. 5 , MAY 1988
and
[7] D. S. Jones, The Theory of Electromagnetism. London, U K :
Pergamon, 1964.
[8] A. Chakrabarti and S . Dowerah, “Diffraction by a periodically
corrugated strip,” J. Tech. Phys. Polish Acud. Sci. (Warszawa), vol.
25, no. 1, pp. 113-126, 1984.
[9] W. E. Williams, Partial Differential Equations. Oxford, UK:
Clarendon Press, 1980.
[lo] I . Appel-Hansen, “Accurate determination of gain and radiation
patterns by radar cross-section measurements,” IEEE Trans. Antennas Propagat., vol. AP-27, no. 5, pp. 640-646, Sept. 1979.
f‘Z’(a)= 1- -cos - 1 ( a / k )
7r-Y (a)
((+:):
-t€0
-:(i
In
Eok)] a. (93)
For a near k we have
fY’(a)=1
f!)(a)=1 + ( ~ 0 / ~ ) ( 2 k ) ’ / ~ ( a - k(94)
)’/~
fy)<a)
=I
f(?)(a)
= 1 - ~ ~ k 3 / *1/2/(a
2 - - k)1/2.
and
(95)
For a near - k the approximate factors can be obtained from
(94) and (95) by replacing Q by -a.
ACKNOWLEDGMENT
Subratananda Dowerah was born in Dibrugarh,
Assam, India, in 1958. He received the B.Sc.
(maths., Hons.) degree in 1978 and the M.Sc.
(maths.) degree in 1982, both from Gauhati University, India.
Since August 1982 he has been with the Department of Applied Mathematics, Indian Institute of
Science, Bangalore, India, where he has completed
his Ph.D. work. A thesis entitled “The WienerHopf technique and allied methods in a class of
scattering problems” has just been submitted to the
IISc., Bangalore, for the Ph.D. degree.
The authors are grateful to the referee for his many valuable
suggestions that improved the presentation of the work.
REFERENCES
B. Noble, The Wiener-Hopf Technique. London, UK. Pergamon,
1958.
T. B. A. Senior, “Diffraction by a semi-infinite metallic sheet,” Proc.
ROY. SOC.,vol. A 213, pp. 436-458, 1952.
T. R. Faulkner, “Diffraction of an electromagnetic plane wave by a
metallic strip,” J. Inst. Math. Appf. vol. 1, pp. 149-163, 1965.
F. G. Leppington, “Travelling waves in a dielectric slab with an abrupt
change in thickness,” Proc. Roy. Soc. London, vol. A 386, pp. 443460, 1983.
A. Chakrabarti, “Diffraction by a dielectric half-plane,” IEEE Trans.
Antennas Propugat., vol. AP-34, no. 6, pp. 830-833, June 1986.
A. D. Rawlins, “Diffraction by an acoustically penetrable on an
electromagnetically dielectric half-plane,” Int. J. Eng. Sci., vol. 15,
pp. 569-578, 1977.
Aloknath Chakrabarti was born in the village of
Dongalon, Bankura, West Bengal, India, in 1943.
He recieved the B.Sc. (maths., Hons.) degree in
1963, the M.Sc. (applied mathematics) degree in
1965, and the D. Phil (science) degree in 1970, all
from the University of Calcutta, India.
He was a Lecturer of Mathematics at the B.E.
College, Howrah, India during 1967-1968. He was
a Commonwealth Academic Staff Fellow at the
University of Dundee, UK, during 1974-1975, and
a Leverhuime Visiting Fellow at the University of
Surrey, UK, during 1979-1980. He was appointed as a National Lecturer for
the year 1979-1980, by the University Grants Commission, India. He is an
Associate Professor of Applied Mathematics at the Indian Institute of Science,
Bangalore, India, where he has been teaching and carrying out research in
theoretical elasticity, diffraction theory, as well as integral equations and their
applications in boundary valve problems of mathematical physics, since 1968.