Simple approximation of diffraction surface currents on a conducting half plane, due to an incoming plane wave, is obtained with a line current (monofile) in complex space. When compared to an approximating current at the edge, the diffraction pattern is seen to improve by an order of magnitude for a minimal increase of computation effort. Thus, the inconvient Fresnel integral functions can be avoided for quick calculations of diffracted fields and the accuracy is good in other directions than along the half plane. The method can be applied to general problems involving planar metal edges

Lindell, I. V.

English

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https://ntrs.nasa.gov/search.jsp?R=19880014221 2020-03-20T06:18:19+00:00Z
PB88-171939
Complex Space Monofilar Approximation of
Diffraction Currents on a
Conducting Half Plane
I
I
	 Helsinki Univ. of Technology, Espoo (Finland)
Prepared for
National Aeronautics and Space Administration
Washington, DC
Jun 87
F	 WS. - bipartroent - of , Commerce
National Tiect*oal l intwfoation Service
BIBLIOGRAPHIC INFORMATION
PB88-171939
Complex Space Monofilar Approximation of Diffraction
Currents on a Conducting Half Plane,
Jun 87
by I. V. Lindell.
PERFORMER: Helsinki Univ. of Technology, Espoo (Finland).
Electromagnetics Lab.
ISBN-951-754-290-9 1 REPT-17
Contracts N00014-83-K-0528, DAAG29-85-K-0079
SPONSOR: National Aeronautics and Space Administration,
Washington, DC.
Sponsored in part by Grant NAG5-270. Sponsored by National
Aeronautics and Space Administration, Washington, DC.,
Office of Naval Research, Arlington, VA., and Army Research
Office, Research Triangle Park, NC.
Simple approximation of diffraction surface currents on a
conducting half plane, due to an incoming plaice wave, is
obtained with a line current (monofile) in complex space.
When compared to an approximating current at the edge, the
diffraction pattern is seen to improve by an order of
magnitude for a minimal increase of computation effort.
Thus, the inconvenient Fresnel integral functions can be
avoided for quick calculations of diffracted fields and the
accuracy is good in other directions than along the half
plane. The method can be applied to general problems
involving planar metal edges.
KEYWORDS: *Diffraction, *Foreign technology, *Diffraction
patterns, *External surface currents.
Available from the National Technical Information Service,
SPRINGFIELD, VA. 22161
PRICE CODE: PC E03/MF A01
i
VB88-171939
Complex Space Molnofilar
Approximation
of Diffraction Currents
on a Conducting 1-half Plan
I>mo V. Lindell
N^/^~ ^^O
°^	 fl	 a
r
HJOSyO^
REPRODUCED BYHelsinki University of Technology 	 U.S. DEPrIARTMENTOFCOMMERCE
Faculty of Electrical Engineering	 INFORMATION SERVICEE
	
	
Report 7Electromagnetics Laboratory	 SPRINGFIELD, VA. 22161
	 P
Otakaari 5 A, SF-02150 Espoo, Finland 	 June 1987
COMPLEX SPACE MONOFII#AR APPROXIMATION Oil
DIFFRACTION CURRENTS ON A CONDUCTING HALF PLANE
Ierno V. Lindell
Report 17, June 1987
4
ABSTRACT
Simple approximation of diffraction surface currents on a conducting half plane, due
to an incoming plane wave, is obtained with a line current (monofile) in complex space.
When compared to an approximating current at the edge, the diffraction pattern is seen to
improve by an order of magnitude for a minimal increase of computation effort. Thus, the
inconvenient. Fresnel integral functions call be avoided for quick calculations of diffracted
fields and the accuracy is good in other directions Cian along tile half plane. The method
can be applied to general problems involving planar metal edges.
This work was funded by the Academy of Finland and, in part, by NASA grant NAG5-
270, ONR contracts NOOO14-83-K-0528, NOOO14-86-K-0533, and ARO contract DAAG-29-
85-K-0079.
ISBN 951-754-290-9
ISSN 0782-0607
L	 TKK OFFSET
r`
INTRODUCTION
The problem of wave diffraction by it conducting; half plane is a classical one, and its
first nom asymptotic solution was given by Sonmuerfeld in 1896 111. The exact result for the
surface current induced oil half plane involvs l^resnel integral functions, which although
straightforward to compute, are inconvenient for quick calculations. Hence, there have been
sitteuipts to use approximate expressions for the surface currents, Since ill(- surface current
hi ►s it singularity at the edge, it is most natural to try to use a line current at the edge as
an approximation for the diffraction current function (true current, minus the geometrical
optics current), The accuracyis, however, not too good, In fact, since a line current
radiates osnnidirectionally in tilt ()lane transverse to the line, the radiation pattern of
the diffraction current is only poorly approximated, The common technique in improving
the accuracy of the GTD is to use nonphysical, so-called equivalent edge currents, which
depend on the point, where the field is to be calculated, These techniques are compared in
the reference 12] but it is out of scope to delve into any comparisons in this short study,
However, the equivalent currents call
	 be expressed in terms of a mmltifslar series
expression (soon to be defined), the first term of which is the monofile, or line c` gent.
The problens studied here is to find all location for themonofile such that the next
tern) has the least: effect, to make the approximation through the monofile most! efficient.
The idea has its origin in the corresponding three-dimensional approximation of scatterers
by multipole series, (3], (4],
]
3
3
1
i
3
V LTIFiLAR, EXPANSION OF SOURCES
:
lit 	 (3], [4] the multipole expansion in complex space has been applied to
source approximation in three dimensions. The theory call modified to two dimensions
so that instead of pointlike mult;ipoles, threadlike multifslarelements are used. Tile problem
need not be truly two dimensional, since the multifilar currents need not be constant,
Let its consider a current source J(T) and its vector potential A(i) function
A(fl p	 G(D)J(r`)dl"	 (1)
1
Here, D is the distance function
D	 (,	 rid f (, : 'rf)	 (2)
and it, can be complex if rs is given complex values. G(D) is file Green function
...jk D
G(D) _ 47rD	 (3)
If the source is almost axial, i.e., of small extension in ti , direction transverse to the
axis, and we . m interested in approximations giving good accuracy for the far : field
calculation, the Green function can be approximated in two ^'.imensions (x, y) through the
Taylor expansion:
G(D) = G(p ) — P` ' VG(P) + 2P^Pi V VG(p) — .:.	 (4)
0
f
Here, G(p) is also a function of
	 ? , although not explicitly shown. When substituted
in (1), after partial integration, we have the expansion
A( 'r ) N	 106'(P) + I r VOW + X 12 VVC;(P) + ,.. I dz .	 (5)
x,	 !
This is equivalent with the following two-dimensional tnultipole or nucllifik expansion for
the current function:
t
	
J(r) =1owb(P) + It (;) - vb(P) + 212(x) : VV6(P)
 + --.	 (6)
The first (zeroth order) term corresponds to a monoflar current component, tile next
one (first order term) a biflar, t•he following one a guadrifilar component and so on. The
expansion (6) follows a similar pattern as given in reference 15) for three-dimensional sources
and it call formally obtained directly by expanding the function b(p — p) within the
integral J(r) = J(p, z) = fS b(p — p')J(p', z)dS'. In (5) and (6), the Ii's are polyadic
z-dependent current moments defined by
to	 r J(r)dS,	 (7)
s
t
x	 _
12 = PPi ( r )dS,	 (9)
i
	and the series (6) could be further extended [3], 14). If the axis of the multifile is moved	 [
{	 from the z axis to all 	 going through a point p = a, instead of (6), we can write,
suppressing the z dependence,
j(P) = f'o(a)b(P — a.) +)(a ) - vb(p — a) + 2i(a) : vvb(^ — a) + ...,	 (10)
with the new current, moments
oi
Io(a)	 1(r)dS =70 , (11)
n	 fS
r ,( a ) - J (p - a;)J(r)dS = 1 t — 610,	 (12)	
liS
fS12( a ) = (P — a)(P — 
a.)J(r)dS 12 — a, —	 kaci + a•a1o•	 (13)
	 *ti
In (13), we have applied the notation I t bici. For sources sufficiently small in the
transverse plane and of sufficiently slow variation, the series (5) for the radiation field
converges quickly and the first few terms will give a good approximation. To find a good
first- term approximation, we can try to move our multifilar current to a point a which
makes the first term most dominating. This amounts t.o finding a point a such that the
first moment in (12) is minimized. Following the method given in [3), [4), assuming Po # 0,
we find
r_
3
r
ii	 f,1Q	 (14)
!o Ip
III this paper, we Study currents that have constant polarization, Thus, the vector
1	 potential will silso hP of the sauce coust.mil
 polarization. Denoting this polarization by .1.
u))it vector rr, we ails write liar I t) ' 10 10,1 1 ». lau v , etc, "Thus, file rank of file currant
u ►om)e nt polyadics call be lowered by one, which makes the formulas appear simpler. For,
example, (14) call be written as
	
n	 J
.
	(140)
0'
Also, it is easy to see, that the choice (14a) not only minimizes the norm of (12), but
actually makes 1 1 (4) = 0, further, if the zeroth moment, is zero, the following value for a
will render tine first moment the most dominant one by making the second moment (13)
vanish:
	
a ^_ I?'f? —	 ­12 : I 1 1 1 , (1:5)
17, 1 2	 21111'
For a current depending on it single coordinate x i , (15) can be reduced to the simple
fornnula
12
u'2It,	 (150)'
If a planar surface current is replaced by the multifilar expansiori, it can be easily seen
that tine nnon)ofilar term gives thy, far field radiation correctly in the direction perpendicular
to the plane. By choosing the location of the monofile as given in (14a), it can further
be shown, that also the derivative o" the radiation pattern becomes correct in the Sallie
direction, To see this, we consider tine far field in tine direction of the unit vector u,., as
with the radiation function
I'
If h is the nunit vector normal to the plane, obviously n• • T = p when T is on the plane, and
the origin lies on tine plane, for a direction Ti. ?. differing from T. only a little, tine term in
the expo)ient in (1.6) is sivall and we may write
.f(wr) x	 (1 -f- jkzt,. - rn)J(T')dS)
is
Io -i- jk7t, - It - ( 1 -i- jkit,-- 010 dkOr•6 10	 (17)
F The last, expression is equal to the radiation function due to a nionofilar current at the
location j) = a, Thus, it gives a two-term approximation for the radiation pattern at the
direction 6, = h, or both the radiation and its angular derivative correctly. The nnonofile
at j) = 0 gives only the first term correctly,
e
(
1
r
1
1
a
l
i
?	 TllE IIALF-PLANE,  ('11111IBM' PROBLEM
Let us consider first the simple case of plane wove corning perpendicularly A -,- 0 and
Pd) --- 0...180c', Fig,l) to the edge with the field polarized so that either the electric or the
magnetic field) is ill file dircefion of the edge (: direction), These two cases are sometimes
called , respectively, the V wave and file If wive, We prefer calling them the TM wave
find the TE widve, where "1" denotes transverse to the dh -etion of the edge (^ coordinate),
For the induced surface current we can write
M x ) ^ J9-(X) -1 •Irf( X )+	 (18)
where the geometrical optics surface current is (n = fty)
Jyo(x) = 2,uy x ft,Cjk'xCOMO.	 (10)
According to the reference (G, 10511, the diffraction current ran be expressed for the TM
wave as
_ z
.ld M (x) _ -2J9 M() F- (v) Q2jU-^ ,	 (20)
and for the TE wave as
J E(x) = --'2.Igo (x) U jF (v).	 (21)
7r
Here, F-(v) is a Fresnel integral function and with
v = Vk' r ( 1 + cosO0),	 (22)
r
	and with positive branch for the square root in (22). The Fresnel integral has the properties
	
s
F_(0) =	 (24)	
i
oo ((2m,-, 1)t^mF (v)	 2^v	 1 f 2 , (m- 1)!(2v)2"-& 25 )
(25) converges quickly for large v values. It. is easy t,o see that, for large v, the function
(25), and hence (21), decay as 1/Vx-, whereas the function in brackets in (20) obeys the
law 1 /x f, Hence, the diffraction current. in the TM case is more concentrated near the
edge.
5
i'
i
i
Ii
LINT t't'fMEN'J' APPROXIMATIONS
For flit , approximation of the previous surface currents (20), (21), we have to eval-
mite the currt,nt rmnnents (7). (8), whieh for the 'f u wave rare of the form lohf
re=J" M1/ ,i 1	aey^•ie-!j',1f and fort)re 'I`l', wave of the form Iq 	 rr,•lp^',/t
10Ilalj j^ . '.I • he' uren lit-lit 1, earn I,e evaluated invoking the folluwhig integral expressions:
	
fo	
Vi da - 1 p ,	 (26)
3 p	 (n	 9pni
n X 3/2
`t hus, We crrn write for the rnono6Jar approximation of the surface diffraction rrarrents
Tj'^ r (x) _— EiJo Nth(x a il ),	 (211)
J, '(.r)	 Auy x It' ll Eb(x u^'F ) -	 (30)
:after a number of algehraie operations, fire normalized current moments earl he found
ill the following forr7r:
t
Tni	 N/ 7r 	 ii-17001./2j3/24, \/1 1 : cos-o(1	 v`7 `{_ cns'Zp)	 (31)J 
1
la	 24 i .	 i «^ ensr r '
	
(32)
17 1i
CON ,;, (
f,wre,n
II7;	
11
	 2 -+ v 1 1_'r_O4
$,
if)
1'hc quatitities JAY hard RTli can he solved from (:31)-(131) and N4a). wit It f 1 result asirr
tlne following e^xpre:isions:
rrY`t17
	
.1 (35)2k 1 •+ v 1 	 ca,4(fi^
TL,	 j 2 I ^1 t ros^pa	 _. __ _,.,	 _	 (36)2k 1 4 ^ 4- cam p``
It is seen tha4 the hest location far the monofile is on the negative x axis at a distance
dependent (ill: the angle 6, of the incoming wave.
6
r
d	 UIPPRAC'TION FlEI,DS
•	 To sec the effect of the shift of the 1nonofile from the origin tc ► the complex point, let
us consider the diffracted far field and the far fields arising from the monofile located loth
at the edge and in complex space, The far field arising from the exact diffraction current(20) call be evaluated in the form
	
3^7'p fJ sin	 ,	
37Ed
-}^ ens
The line current 11TAf at the edge gives rise to the following field:
)
GTAI	 j", d ^ jA,p_. 2'3_'	 gin `i	 3
which is seen to coincide with (37) at
	 -- zr/2, The line current at, file pol ► i• x s aTm
defined by (35), radiates the field
C —jkP 2Vi	 wind '	 1	 cosd^FTAf 
_ _E	 ex	 (3y)
► v^ T^N V'5	 . cas ^	 P 2f I + caa O
By expanding both (37) and (39) at ¢ = n + b, in power series in terms of b, which
is a small quantity, it can be seen that the first term is coincident with (38), and the
second terms in (37) and (38) are equal. This shows that (39) approximates ( 37) more
closely near the radiation direction 0 ;.= 90°, Examples of direction patterns for these fields
can be seen in figures 2, 3 for the incidence angles ¢p y 60°,120°.. It is seen that while
the current at the edge only gives omnidirectional radiation, the complex space monofile
gives quite good estimate for the diffraction patterns for angles about 45° -- 180 0 , wheras
the diffraction field in the direction of the half plane is poorly approximated. From the
previous we conclude, that the complex space line current gives the diffraction pattern and
its derivative correctly for the angle 90°. It is obvious that this is the best that call be
(toile with an ap^roximation by a single monofile. The value of J AI could be chosen also
to meet some ►alter rcquirenlent, For example. ^we might have wanted the correct. field in
the direction o - 0, for which the coefficient, of .cox;, within file exponent function of (39)
could have been accordingly chosen. In this case, however, it is obvious that, the radiation
in the direction t =. 18010 would be greatly deteriorated.
Corresponding expressions for the TF field can be written, respectively,
	
jA.p	 d
Ed E fijrlHi: 
c "	 2 cos	 {40)
J^P sin -+• cos
_jA,p
d	 d+l t-	
+coso '
_ A
	
T
fsin.
EdTE = fdoy i, $ ^ ^	 ^	 cap	 f }cas h- cas¢
	(42)din3- P I + cos 	2( 1 + cos	)
 (
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I., Ihr rurrews and held).. and with k e verYwhe rr in Iho' fors idas replaced by A• 4i it Off . 11.(llu^. file IIUr111ahled lisle' iulr #-Ills I 'I11, 132 1 ;11141 flit- I„tiltl , tlli' n( Ill y• 111(ow fllr^.
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p, ,I;ilizatstill Hits call Ili- done with the following expansion for flit Unit dva,ht .
k1.
	 IA	 11.114 . $I	 l	 (k	 IA ' u:)I(A , IA• • U.
licr, . lilt .
 til -1 t4-rnl d44-s s p ot alfe 1-1 who'll 11rtlltiplying t ll " fi4-lds, ,Incr A V	 Il and k• H
	 it
M lilt iplcifit , 	1 'tl Lt -1' flirts us Ihr de,onlpositilit 1.	 tl • F•TF. , l 111. multiplying I,v
11 (C4- ha.	 h, ,14- , oll,po ilio11 ll	 U	 1l	 11	 'I h11s, anti' platir wave not iucidrnl
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FIC'URE CAPTIONS
Figure l Basic geometry of plane wave incidence on a conducting half plane. The angle
of incidence q¢p is between 0 anti 180° and the angle 00 between 0 and 90".
Figm—e 2 TM polarized diffraction field patterns for 0 0 = 0 and 00 = 60". Solid line:
exact diffraction field. equation (37), dot line: nronofilar current at the edge, equation (38),
dashed line: monofilar current in complex space, equation (39). The quantity displayed is
arl:ually the bracketed term in the field expressions,
Figure 3 Same as Fig.2, but with 00 = 120°.
Figure 4 TE polarized diffraction field patterns for 00 0 and 00 = 60 0. Solid line:
exact diffraction field, equation (40), dot line: monofilar current at the edge, equation (41),
dashed line: monofilar current in complex space, equation (42). The quantity displayed is
actually the bracketed terry, in the field expressions.
Figure u Same as Fig.4, but. with 0 0 = 1200.
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Complex space monofilar approximation of diffraction currents on a conducting half plane

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