A technique which is based on Fourier transformations is introduced for predicting scattering widths. For a strip it is shown that explicit determination of the linear current density is not necessary for bistatic or monostatic scattering width calculations. Comparisons of the predictions of the technique are made with the integral equation technique predictions, which do not require explicit evaluations of linear current densities

Beck, F. B.

Cockrell, C. R.

Harrah, S. D.

English

NASA Technical Reports Server

NASA Technical Memorandum 102729
/A/3_
p-!7
,'ill
.¢, Transverse Electric Scattering
Widths for Strips - Fourier
Transform Technique
C. R. Cockrell
S. D. Harrah
F. B. Beck
(NASA-TM-102729) TRANSVERSE ELECTRIC
SCATTFRING WIOTHS FOR STRTPS-FOURTER
TRANSFORM TECHNI_IJ_ (NASA) 19 p CSC1 20N
N91-27_01
Unclas
0028381
June 1991
National Aeronautics and
Space Administration
Langley Research Center
Hampton, Virginia 23665-5225
https://ntrs.nasa.gov/search.jsp?R=19910018087 2020-03-19T16:37:54+00:00Z

TRANSVERSE ELECTRIC SCATTERING WIDTHS FOR STRIPS - FOURIER
TRANSFORM TECHNIQUE
ABSTRACT
A technique which is based on Fourier transformations is
introduced for predicting scattering widths. For a strip it is
shown that explicit determination of the linear current density is
not necessary for both bistatic and monostatic scattering width
calculations. Comparisons of the predictions of the technique in
this paper are made with the integral equation (IE) technique
predictions, which do require explicit evaluations of linear
current densities.
_NTRQDUCTION
The most commonly used methods for predicting the scattering
widths (SW) of scattering objects are physical optics (PO),
integral equation (IE), and geometrical theory of diffraction
(GTD) [1-6]. The PO technique predicts the scattered fields quite
well for objects large in wavelengths, particularly in the
backscattered direction and for normally incident fields. The IE
technique requires the solution of an integral equation for the
induced current density on the scattering object. Numerical
techniques such as the method of moments (MOM) are used to solve
for the current density. The IE solution is usually referred to as
the exact solution. However, for bodies which are large in terms
of wavelengths, the solution can be quite time consuming, and
therefore, the IE technique is usually restricted to bodies a few
wavelengths in size. On the other hand, the GTD high frequency
technique is more applicable to scattering bodies which are large
in wavelengths. It is an extension of geometrical optics and
accounts for finite edge effects using diffraction theory.
In this paper a different approach for solving scattered
electromagnetic fields from flat strips is introduced. Even
though the strip model may be unrealistic, it provides a useful
geometry for verification of the technique since a wealth of
computational data is readily available for comparison [5]. The
technique is based on applications of the Fourier transform of the
tangential electric field in the plane containing the strip and
the current density on the strip. In general, once the current
density is determined, the far fields are found from the radiation
integral. The scattering width for a strip is then determined
from these scattered far fields. Also, only the case for which the
incident transverse electric field (TE to the infinite length of
the strip) illuminates the strip is investigated. The results
using this technique are then compared to results obtained by the
IE technique.
2
TE FOURIER_ _ _ FQRMULATION
WIT_____H_ _ _ ON A
A plane wave with wave _umber k, polarized in the x-y plane,
^ _sin8 ]e jk[xsinel+zc°sSi] is
and represented as E = [xcos8 l- i
assumed incident at an angle e I on a perfecting conducting strip
of width a located in the z_ = 0 plane (see fig.l). The incident
electric field induces _a linear electric current density J×,
which in turn radiates the scattered fields. Once Jx is known,
the scattered field is readily found from the radiation integral.
Since only the x component of the electric field induces a
linear current density on the strip, the x component of the
magnetic vector potential is written as
y ejkRAx(x,z) = _-_ Jx (x'' ) R dx'dy' (i)
strip
where R = / (x-x') 2+ (y_y,)2+ z2 and _ is the permeability of free
space. Performing the y'integration (assuming J×is independent of
y'),
a
2
,z ) o k/(x-x' ) Z 2
a
m
2
(2)
where H (2)0 () is the zero-th order Hankel function of the second
kind. From one of Maxwell's curl equations, the y component of
the H field can be written as
2
H (x,z) = - j Jx (x') 0z 0
Y
m __
2
Also,since H satisfies the wave equation
Y
a2H a2H
Y + Y + k2H = 0
@x 2 az 2 Y
it can be expressed in terms of spectrum solutions as
H+(x,z) = 1 [ G,(y kx)
- I I ®Hy(X,Z) = _-_ G-(kx)
-JkxX -Jkzz
e e dk
x t Z > 0
-JkxX JkzZ
e e dk , z < 0
x
where the G's are the spectrum functions and
(4)
(5)
I /k 2 - k2 k2 k2 1
x ' x <
k - (6)
Z
- J /k 2 _ k 2 k2 > k 2
X f X
In equation (5), the superscripts + and - denote the regions z > 0
and z < 0, respectively.
Under the assumption that the strip is now replaced with a
linear current density J , the boundary conditions on the
x
tangential field components in the z = 0 plane are applied. The
tangential H field is discontinuous by the amount of linear
current density flowing on the region which was occupied by the
strip; that is,
J = - H + + H- (7)
x y y
Substituting equa{i0n (5) into equation (7) for z = O, the linear
cuTrent density becomes
- 1x _-_ - G÷(kx ) + G-(k ) e × dkx x
The tangential electric field at z = 0 is continuous.
from Maxwell's curl equation
1
E = _V X H
(8)
Therefore,
(9)
the continuity of E yields
x
i jkx i- x 1 (kx) (jkz) e x dk x_2_e (kx)(Jkz)e dk = _ G÷ -
-- DO -- DO
(I0)
where e is the permittivity of free space. The solution of
equation (i0) is
G_(kx) _G+(k×) (11)
Substituting equation (11) into equation (8), the linear current
density becomes
i° - ;_ 2 (kx) e k X (12)Jx (x' ) = _-_ G + Jkxx
Q0
Its inverse Fourier transform is
O0
- 2 G+(kx) = I Jx(X')
-- DO
jk x"
e x dx' (13)
with the understanding that Jx(X') is zero outside the strip
width.
For the approach in this study, it is necessary to express
the magnetic vector potential component in terms of a Fourier
spectrum representation; therefore, let
Jx6(X') = I 6(x'-x") (14)
where I is a current and 6(x'-x') is a delta function located at a
general point x" on the strip [i]. The subscript _ is used to
denote the solution for an impulsive linear current density. From
equation (3)
Hy 6 = - j I _-{ o
and from equation (13)
JkxX"
- 2 G6(kx) = I e (16)
Using equation (16), the first equation of equation (5) can be
written as
Do jk x"
x kzZdkx (17)1 ] e-Jk× x e -jHy6 = _-_ [ - Ie 2
-- CO
5
Equating equations (15) and (17)
I_ "-- " X X u___ H(2) [k/(x _ x,,)2+ z2 } 1 -].x_- ) -jkzaz o = _ e e z dk x (18)
The solution of equation (18) is found by integrating over z;
therefore,
1 -Jkx( ) z[ _( __ =-- eH (2) k x x")2+ z2 e dk (19)0 _ z x
-- 00
The magnetic vector potential component from equation (2) becomes
a
2 i_ -jkI 1 e-Jkx (x-x") zAx(X'Z) = - J_ Jx(X') _z e z dkxd x, (20)
x'= a k =
2 x
where the double prime on x has been changed to a single prime.
The tangential electric field in terms of A x is [5]
Ex(X,Z ) = -j_Ax(x,z ) - J
1 02 A (x,z) (21)
_e aX 2 x
With equation (20) substituted into equation (21), the tangential
electric field becomes
a
m
I I- z1 x l dk dx'Ex(X,Z ) _ _ _ jx ( ) kz e e x
x,- a k= -_
2 x
Interchanging the order of integration, equation (22) becomes
a
E(x,z) _ [ I _ eJkxX' ] -JkxX -JkzZdkx1 I kz= _ _ Jx(X') dx' e e
a
x 2
After rewriting equation (23) as
a
[ jk.]jk.1 I 1 x' e xEx(X,Z ) = _-_ _ _ kz J×( )e dx'
a
- 00
2
the term in square brackets is defined as Fx(k x).
(22)
(23)
-jk z
Z
e dk (24)
X
Therefore,
6
w"
equation (24) becomes
JkzZdk
i [ -jkx -Ex(X,Z ) = _-_ Fx(k×) e e x (25)
J
-- 0D
At this point the problem is formulated as a scattering one;
that is, the total tangential electric field is written as
ET(x,z) = EZx(X,Z) + E:(X,Z) (26)
X
where EZ(x,z) is the incident field and E:(x,z) is the scattered
X
field which is assumed to be representable as equation (25).
Therefore,
ET(X,Z) = EIx(x,z) + _ Fx(kx)
X
-
-Jk x -jk z
X Z
e e dk (27)
X
where
a
Fx(kx) = - _ Jx (x')
a
2
jk x'
X
e dx' (28)
Since the unknown function Jx(X') appears only under the integral
sign, equation (28) is an integral equation of the second kind.
Such equations are often solved approximately by the method of
moments where the unknown function is expressed as a sum of basis
functions with unknown constants. The constants are then found
through a point matching technique which requires a matrix
inversion.
An approximate solution for Jx(x') is determined by a Fourier
transformation inversion. At z = 0 equation (27) becomes
Jkx'sinel 1 [ -jk x'
ET(x,,0) = cosBie + _-_ ] Fx(k ) e × dk X (29)X
If the ET(x ',0) were known over the complete z = 0 plane, the true
X
Fx(kx) can be found via the inverse Fourier transform. However,
the field ET(x',0) is known only over the strip (E: = 0) and not
X
outside. As an approximation the total ET(x''0)x field outside the
strip is assumed to be the tangential incident field. Written in
mathematical terms, the tangential electric field in the z = 0
plane is
7
ET x' jkx'sine
×a ( ,0) m [I - s(x')] cosele i
where the subscript a on ET
x(X',0) denotes an approximation and
(30)
1 , ix'Is ±s(x') - 2
0 , otherwise
Substitution of equation (30) into equation (29) gives
i ['F -JkxX' Jkx' sine
_-_ xa(kx)e dk x = - s(x')cose e '
J
-
The inverse Fourier transform yields
(31)
(32)
FxG(kx) = - a
san + k sine
+ k sine i
cose i (33)
An approximate solution for the linear current density in equation
(28) is found by a Fourier transformation of an approximate linear
current density Jxa(X'); that is
I jkx'
x 2&>e
Jxa(x')e dx' = - -_z Fxa(kx) (34)
The inverse Fourier transform yields
° /i / 1Jxa(X') = _-_ _ a cose e x dk (35)a i x
z k + k sinelJ _
- O0
This inverse transformation produces linear current density inside
and outside the strip width. Since the linear current density
Jx(X') given in equation (24) flows only on the strip, it is set
equal to Jxa(X') only over the strip width; that is,
Jxa (x') Ix'l s
Jx(x' ) = ' 2 (36)[ 0 , otherwise
For normal incidence (el= 0), equation (35) is evaluated
numerically for a strip two-wavelengths wide. In figure 2, these
computations are compared with the linear current density obtained
by the integral equation technique which uses the method of
moments. Even though the two solutions do not agree in magnitude
over the complete strip, their shapes are very similar.
Techniques for improving the agreement between the solutions will
be investigated in future work.
TE _WIDTH OFASTRIP
The definition of the scattering width of a strip (object) is
given as [5]
p -_ eo
lim
O"
2--D
lim [ 2_p IEs---!2]IEII2
(37)
[ 2.p IHS-_J.12 ]is I 12
where p is the distance from the strip to the observation point,
ES,H s are the scattered fields, and E* H*, are the incident fields.
In this paper the second definition is used where the scattered H
field component is given by equation (3). Explicitly performing
the differentiation in equation (3), the scattered y-component H
field becomes
&
2
H (x,z) = j J(x')_x -, (38)
a
2
(2)
where HI, () is the first order Hankel function of the second
kind. Since scattering widths are based on far-field expressions,
the square root term in equation (38) can be approximated as
\
p - x'sine s , for phase terms
V_(x - x')2+ z 2 _ (39)
p , for amplitude terms
where e is the scattered angle off the normal to the strip (see
fig.l). For k(p - x'sine ) large, the Hankel function can be
written as
-jk(p - x'sinSs) j_
He2)
, () _ J _--_p e e (40)
-jkp
.s e
Substituting equations (39) and (40) into equation (38) with
z = pcosO , the total scattered H far field becomes
S
a
w
j_ 12 jkx'sinOe cosO s Jx (x')e s dx' (41)
2
Instead of explicitly solving for the linear current density, the
integral representation given by equation (35) is substituted
directly into equation (41). The order of integration is then
interchanged, permitting the analytical integration over x'. This
procedure greatly reduces the computational time since only a
single numerical integration is now required. The total H far
field is now written as
H s _ _ _ _e _ e -jkp J_
--_ _p e cose cose i
[_ [ sinlkx+ k sin0111 J[ sin kx- k sin8)_
• a [ s
for bistatic scattering and
_e _ e -jkp J_HS _ - _ _-- e cos2e
s' Isinlk'ksinex l• az k + k sine
x l
-- CO
A
(k x- k sine I
dk (42)
X
dk (43)
X
for monostatic scattering.
H field
are given as
_2[bistatic) -
-[
The scattering widths for the incident
/_ jkxsineHI = _ _ C e !
jkzcose i
e (44)
k 3
cos2e cos2e
4T[2 s !
I I.sin kx+ k sine i a
kx+ k sine t _
sin[k - k sine la
x s 2
a ,
k x- k sine B
dk×
2
(45)
and
k 3
_2(_onostatic) - cos4e
• 4r[ 2 ]
] f _ (t sinel Ii
sin[k×+ k
• a
z kx+ k sine i
-- G0
a sinlkksineI kxsine2dkxl (46)
Equations (45) and (46) are normalized with respect to
wavelength. These normalized equations are then evaluated
numerically and compared with MOM solutions. The bistatic
solutions for a two wavelength strip are shown in figures 3, 4,
and 5 for incidence angles of 00,-30 ° , and -45 ° , respectively.
Figures 6 and 7 compare the monostatic solutions for two and three
wavelength strips, respectively. As evident from these figures,
the Fourier transform technique accurately predicts the null and
i0
peak locations in the scattering width patterns. The predicted
levels agree quite well with the MOM predictions in the specular
regions but tend to be lower than the MOM predictions for other
angles.
ii
CONCLUDING REMARKS
This paper introduces a different approach for predicting the
scattering widths for strips. The technique is based on Fourier
transformations for determining the linear current density flowing
on the strip. However, it has been shown that there is no need to
solve for the linear current density explicitly, thus reducing the
computational time required to obtain scattering width patterns.
On the other hand, the MOM technique requires explicit solutions
for the linear current densities which can be time consuming,
particularly for monostatic scattering widths.
The technique presented in this paper predicts quite well the
shapes of the scattering widths but not their absolute levels for
all angles. Methods of improving the accuracy of this technique
will be investigated in future work.
12
i. Harrington, R.F.: Time-Harmonic
McGraw-Hill Book Company, 1961.
Electromagnetic Fields,
2. Balanis, C.A.: Antenna Theory: Analysis and Design, Wiley,
1982.
3. Keller, J.B.:" Geometrical theory of diffraction," J. Opt. Soc.
Amer., vol. 52, no. 2, February 1962.
4. Kouyoumjian, R.G. and Pathak, P.H.: " A uniform geometrical
theory of diffraction for an edge in a perfectly conducting
surface," Proc. IEEE, vol. 62, no. ii, November 1974.
5. Balanis, C.A.: Advanced Engineering Electromagnetics, Wiley,
1989.
6. Stutzman, W.L. and Thiele, G.A.: Antenna Theory and Design,
Wiley, 1981.
13
XPerfectly conducting
strip
a
Y
Figure 1. Strip geometry
0 I I I I I I I I I I I I .I. I ! I I | .I I I f I ! I I I I i 1 I I
-tOO -.75 -.50 -25 .00 .25 .50 .75 tOO
X (WAVELENGTHS)
Figure 2. _ of Ineor current densities txkJoed
on a two m_lmgth _trip at normal inc_dence
14
(/1
20
lO
0
-10
-2O
-30
-4O
20
10
o
-10
-20
-3O
-4o
m
- j
- h
- p
- p
o P
.P
.r
.[
.r
/
p
f_
/ \
i
I I
_J_ JJll IIIIJ Ill Illl I11 1111 i!11 Ilia
-90.-60 -70--60-50-40 -30-20 -10 0 10
LLLL
...... w
m
pL ...................li
u.u.l ,LU.__LUJ LLU_
20 30 40 50 60
%
e,
LLLL _.I-
70 80 90
BISTATIC ANGLE (DEGREES)
Flguro_3. Comparlemnof "rEblafofb Icoflerlng widthsa two wavelmgth efrlp err normol Inoldmlce
- --- FT -_ MOld
!/.
:/ee_ o_ e
i
,.......__
'i
,,\ - -_ ,
t
tq
\
\
\
"1111 IIII III IIII IIII III IIII] III illil IIII IIII IIII II1 III1: III Illl 1111 I111
-00_-70 .._ .-_-_-30-20-_0 0 10 20 _ 4o _ 60 70 80 oo
BISTATICAHGU:"(DEORET:S)
Flgure 4. Comparl_)n of "rE:blltatle leafleting wldttl
of a two wav,lm_h u_p ot -30 d_'eu Inold,r_
15
20
0
m
"-" -10
-20
-3O
-40
2O
0
-20
-3O
-4O
.Jill JJJ. LJ.JJ_ .ILl ILL LII
I
\
\
+
LJ_U_ +.J_LL LIJJ_
-90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 80 90
mTAT_ ANaLE(DEC_ES)
Flgure 5. Comparh_n of "rE:blstatle _aflerlng wlclttm
of a two wavelenglh etrip at -45 clegreee lnalck_e
/
/
p ;
p
1
I
i
[ , ]
I .
f ,,
I *
I *
i ,
I
l i I |
i i
.LLI..LLLJ _J.J.
ff-i.......FT -- MO_
| _ = _ ........
' T
/"+ \
- . .
I
I
!
- _
_[ '_ + * * I
o_ • /
, a
"' i t I.
I ')ill
!
!
!
II 1 I 1 Irl Jill I LLJ-J JLI_ ill
m,
!,
.......... .....
i
I
_i .I_LLI IA_ I J.l_
|
_---.'+ *_4 -- -
i I
..... i ....
I
I
i
+:-""-'I,_ ........
I 'k{ ,
.LdJ .u_L u,-
-90-80-70--60-_)-,,40-30-20-10 0 10 20 30 40 50 60 70 80 90
_JOO4CCANm._(D£C,R_S)
Flgure 6. Comporlson of TE rnonostatlc ecaflerlng widths
for o two woveler_ strip
16
20
o
_ w e e
_20 _ " " "'°e '
w
" J ||
!llll Ill rllll fill II|! I1|1 Ill ,Jill II :ll
-go -8G -70 --60 -50 -40-Z,O -20 -10 0 10
L"" FT --
; •. ",,,
e_ b
_T
q
p --
i
IJlllllJ] Jill IllJ Jilt llll[ li
20 _0 40 50 60 70 80 gO
Flgure 7. Cemporhce of TE monoet_cic m:otterbg wk_che
for a three wav_ _a-ip
17
Report Documentation Page
I. Report No.
NASA TM-I02729
4. Title and Subtitle
Transverse Electric Scattering
Fourier Transform Technique
2. Government Accession No.
7. Author(s)
C. R. Cockrel]
S. D. Harrah
F. B. Beck
9. Performing Organization Name aM Address
NASA Langley Research Center
Hampton, Virginia 23665-5225
12. Sponsoring Agency Name and Address
National Aeronautics and Space Administration
Washington, DC 20546-0001
15. Supplementary Notes
3. Recipient's Catalog No.
5. Report Date
June 1991
6. Performing Organization Code
8. Performing Organization Report No.
t0. Work Unit No.
505-64-70-01
11. Contract or Grant No.
13. Type of Report and Period Covered
Technical Memorandum
Widths for Strips -
14. Sponsoring Agency Code
16. Abstract
A technique which is based on Fourier transformations is introduced for predicting scattering
widths. For a strip it is shown that explicit determination of the linear current density is not
necessary for both blstatic and monostatic scattering width calculations. Comparisons of the
predictions of the technique in this paper are made with the integral equation (IE) technique
predictions, which do require explicit evaluations of linear current densities.
17. Key Words (Suggestod by Author(s))
strip
Fourier transform
scattering width
i 18. Distribution Statement
unclassified - unlimited
subject category 32
19. SecuriW Classif. (of this report)
unclassified
NASA FORM 1626 OCT 86
_. SecuriW Cla_if. (of this page)
unclassified
21, No. of pa_s _. Price
18 A03


Transverse electric scattering widths for strips-Fourier transform technique

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