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An Extended UTD Analysis for the Scattering and
Diffraction from Cubic Polynomial Strips
...... : :_:-- E.D. (_onstantmldes and R.J. Marhefka
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The Ohio State University
7-177- ElectroScienceLaboratoryDepartment of Electrical Engineering
Columbus, Ohio 43212
Final Report 725311-2
Grant No. NSG-1498
January 1993
National Aeronautics and Space Administration
.... 4 Langley Research Center
Hampton, VA 23665
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NOTICES
When Government drawings, specifications: or other data areused for any purpose other than in connection with a definitelyrelated Government procurement operation_ the United StatesGovernment thereby incurs no ffespoffsibility nor any obl_gationwhatsoever, and the fact that the Government may have formulated,furnished, or in any way supplied the said drawings_ specifications,or other data, is not to be regarded by implication or otherwise asin any manner licensing the holder or any other person or corporation,or conveying any rights or permission to manufacture, use, or sell
any patented invention that may in any way be related thereto.
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REPORT DOCUMENTATION 1. REPORT NO.
PAGE
4. Title and Subtitle
An Extended UTD Analysis for the Scattering and
Diffraction from Cubic Polynomial Strips
2. 3. Recipient's Accession No.
5. Report Date
January 1993
6.
7. Author(s) 8. Performing Org. Rept. No.
E.D. Constantinldes and R.J. Marhefka 725311-2
O. Performing Organizntlon Name and Address 10. Project/Task/Work Unit No.
The Ohio State University
ElectroScience Laboratory
1320 Kinnear Road
Columbus, OH 43212
12. Sponsoring Organization Name and Address
National Aeronautics and Space Administration
Langley Research Center
ttampton, VA 23665
11. Contract(C) or Grant(G) No.
(c)
(G) NSG-1498
13. Report Type/Period Covered
Final Report
14.
15. Supplementary Notes
16. Abstract (Limit: 200 words)
Spllne and polynomial type surfaces are commonly used in high frequency modeling of complex struc-
tures such as aircraft, ships, reflectors, etc. It is therefore of interest to develop an efficient and accurate
solution to describe the scattered fields from such surfaces. In this report an extended UTD solution for
the scattering and diffraction from perfectly conducting cubic polynomial strips is derived and involves
the incomplete Airy integrals as canonical functions. This new solution is universal in nature and can
be used to effectively describe the scattered fields from flat, strictly concave or convex, and concave-
convex boundaries containing edges. The classic UTD solution fails to describe the more complicated
field behavior associated with higher order phase catastrophes and therefore a new set of uniform re-
flection and first-order edge diffraction coefficients is derived. Also, an additional diffraction coefficient
associated with a zero-curvature (inflection) point is presented. Higher order effects such as double
edge diffraction, creeping waves, and whispering gallery modes are not examined in this work. Theextended UTD solution is independent of the scatterer size and also provides useful physical insight
into the various scattering and diffraction processes. Its accuracy is confirmed via comparison with
some reference moment method results.
17. Document Analysis a. Descriptors
:_t_yM[* l o']rlC
l)ll:Fl_ AC'FI( )N
SCNII I'_IlINC,
b. Identifiers/Open-Ended Terms
2-D
UTD (UNIFOIllXl TIIEORY OF DIFFRACTION)
c. COSATI Field/Group
18. Availability Statement
A. Approved for public release;Distribution is tmlimited.
19. Security Class (This Report}
Unclassified
20. Security Class (This Page)
Unclassified
21. No. of Pages
53
22. Price
See ANSI-Z$9.1S)See Instructions on Reverse OPTIONAL FORM 272 (4-77)
Department of Commerce
Contents
m
List of Figures
1 Introduction
2
iv
1
5
3
Uniform Asymptotic Analysis
2.1 Uniform Asymptotic Evaluation of e.,,h(6',8) .............. 8
2.2 Total Field Solution ........................... 12
Extended UTD Solution Formulation 14
3.1 Reflected Field Solution ......................... 20
3.2 Zero-Curvature Diffracted Field Solution ................ 22
3.3 First-Order Edge Diffracted Field Solution ...............
4 Numerical Results and Discussion
5 Summary and Conclusions
A_PPENDICES
A Ordinary Airy Functions
B Incomplete Airy Functions
I3ibliography
23
26
39
41
44
47
b
,,=
III
PRliCIEN'NQ PAGE" BLANK NOT FN..MED
PAGE I,_ cNI_LLY 6LANK
List of Figures
2.1
2.2
3.1
3.2
3.3
3.4
3.5
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Canonical geometry for the uniform asymptotic analysis ....... 6
Scattering mechanisms associated With a cubic polynomial boundary':
containing an edge ............................. 6
Complex plane topology and contour deformation for the incomplete
Airy integral when o" < 0. 15
Complex plane topology and contour deformation for the incomplete
A_rylntegralwhen_r>0. : . : ........... ::: ':::. ..... 17
Geometry for the reflected field from a cubic polynomial strip ..... 20
Geometry for the zero-curvature diffracted field from a cubic polyno-mial strip. : 22
Geometry for the edge diffracted field from a cubic polynomial strip.. 24
Scattering geometry and relevant parameters for the numerical results. 27
Scattered field contributions (TM polarization case) from a cubic
polynomial strip with a0 = 2.0A, al = 0.5, a2 = 0.1A -1, a3 = 0.1A -2
a -1.5A, b = 1.5A, and angle of incidence 0' ='45 o . : ....... 30
Bistatic echo width (TM polarization case) from a cubic polynomial
strip with a¢, = 2.0A, al = 0.5, a_ = 0.1A -_, a3 = 0.1A -_, a = -i.5A,
b = 1.5A, and angle of incidence 0' = -45 ° ................ 31
Bistatic echo width (TE polarization case) from a cubic polynomial
strip with at, = 2.0A, al = 0.5, a_ = 0.1A -l, a3 = 0.1A -2, a = -1.5A,
b = 1.5A, and angle of incidence 0' = -45 ° ................ 32
Monostatic echo width (TM polarization case) from a cubic poly-
nomial strip with a_ = 2.0A, al = 0.5, a2 = 0.1A -l, a3 = 0.1A -2
a = -1.5A, and b = 1.5A. 33
Monostatic echo width (TE polarization case) from a cubic polyno-
mial strip with ao = 2.0A, al = 0.5, a2 = 0.1A -I, a3 = 0.1A -2
a = -1.52, and b = 1.5A .......................... 34
Bistatic echo width (TM polarization case) from a cubic polynomial
strip with au = 2.0A, al = 0.5, a._ = O.1A -1, a3 = 0.12 -2, a = -0.33A,
b = 1.5A, and angle of incidence 0' = -45 ° ................ 35
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4.8
4.9
4.10
Monostatic echo width (TM polarization case) from a cubic poly-
nomial strip with au = 2.0A, al = 0.5, a2 = 0.IX -j, as = 0.1X -2,
a-- -0.33X, and b = 1.5A ......................... 36
Monostatic echo width (TM polarization case) from a cubic poly-
nomial strip with ao = 2.0X_ al = 0.5, a2 = 0.1X -l, aa = 0.0X -2
(parabolic strip), a = -0.33A, and b = 1.5A ............... 37
Bistatic echo width (TM polarization case) from a cubic polynomial
strip with ao = 2.0A, al = 0.5, a2 = 0.0A -l, as = 0.1)_ -2 (flat strip),
a = -0.33A, b = 1.5A, and angle of incidence 8 _ = -450 ......... 38
Contours of integration for the ordinary Airy Functions ........ 42
Contours of integration for the incomplete Airy Functions ....... 45
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Chapter 1
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Introduction
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Itlgh frequency electromagnetic analysis of problems involving complex geometry de-
scriptions of the scattering bodies has been of great interest in recent years. Spline
and polynomial type surfaces are commonly used in high frequency modeling of com-
plex structures such as aircraft, ships, reflectors, etc., and is therefore of interest to
develop efficient and accurate solutions for the purpose of RCS and antenna pattern
predictions. Numerical techniques for treating these problems such as Physical Op-
tics (PO) [1] and the Method of Moments (MM) [2], while simple in concept, still
require vast amount of computer resources for even intermediate sized targets. The
use of closed form ray optical solutions, whenever possible, is one way to increase
efficiency and also provide important physical insight into the various scattering and
diffraction processes.
• In many instances, high-frequency scattering and diffraction mechanisms can be
described in terms of ray optical fields that behave locally like plane waves propagat-
ing along the trajectories of the Geometrical Optics (GO) [3] and the Geometrical
Theory of Diffraction (GTD) [4]. Although these rather simple ray optical techniques
can be used to solve many complex practical electromagnetic scattering/radiation
problems, they fail to describe the more complicated field behavior inside transition
regions where wave focusing due to higher Order phase catastrophes and or changes
from illuminated to shadow zones occur. Removing the errors within these regions
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would greatly enhance the Applicability of GTD as indeed has been shown for its uni-
form version, the so called UTD [5]. The UTD approach for dealing with transitional
field behavior involves the use of appropriate transition functions that eliminate the
errors of GO/GTD solutions, and also provide a smooth connection into the regions
where these classical ray optical formulations remain valid.
A common method of analysis for high-frequency scattering and diffraction prob-
lems involves the use of radiation integrals as well as plane wave integral represen-
tations for the fields, with the asymptotic approximations for the various scatter-
ing mechanisms found from the critical point contributions of the integrand. This
method of analysis when applied to a cubic polynomial surface containing an edge re-
sults in a stationary phase integral characterized by two stationary phase points that
are arbitrarily close to one another or to an integration endpoint. Uniform asymp-
totic evaluation of integrals with such analytical properties involves the incomplete
Airy function [6] which serves as a canonical integral for the description of transition
region phenomena associated with composite shadow boundaries. Transition regions
of such type result from the merging of the reflection shadow boundary associated
with an edge in the scattering surface, and the smooth caustic of reflected rays aris-
ing from the confluence of the two stationary phase points near a zero-curvature
(inflection) point. When the reflection shadow boundary is not in the immediate
vicinity of the smooth caustic the conventional UTD diffraction coefficient [5] which
involves the Fresnel integral as a canonical function can be used to effectively de-
scribe the edge diffracted fields. Furthermore, the ordinary Airy integrals and their
derivatives are the appropriate canonical functions for the description of the high-
frequency fields in the neighborhood of the smooth caustic [7]. However, when there
is a confluence of both reflected and caustic type shadow boundaries, neither the
Fresnel integral nor the ordinary Airy integrals adequately describe the transition
region phenomena, and they must be appropriately replaced by the incomplete Airy
functions. Although these functions are not so easily generated [8] as, for example,
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the Fresncl integral or the complete Airy functions, a rnetbod for their emcient and
accurate computation was recently developed [9] and allows for the formulation of
uniform asymptotic solutions that are useful for engineering purposes.
In this report, an extended UTD solution for the scattering and diffraction from
perfectly conducting cubic polynomial strips is derived and involves the incomplete
Airy integrals as canonical functions. The scattering mechanisms involved are de-
scribed by a new set of uniform reflection, first-order edge and zero-curvature diffrac-
tion coefficients that remain valid inside the transition regions and also provide
smooth connection into the regions where tile classic ray optical formulations still
apply. Although additional higher order mechanisms such as creeping waves, double
edge diffraction, and whispering gallery modes may also exist for the geometries con-
sidered, they will not be examined in this report. Also, this new solution is universal
in nature and can be used to effectively describe the scattered fields from flat, strictly
concave or convex, and concave-convex boundaries containing edges. Numerical re-
suits obtained using the extended UTD solution showed excellent agreement with
the method of moments for both polarizations, except for some limited regions in the
non-specular direction for the TE polarization where the higher order mechanisms
become significant.
The outline of this report is as follows: In Chapter 2, a uniform asymptotic
analysis for the plane wave scattering from a perfectly conducting cubic polynomial
boundary containing an edge is presented, and in Chapter 3 tile extended UTD
solution for the scattering and diffraction from cubic polynomial strips is formulated.
In Chapter 4, some indicative numerical results are presented and discussed with
their accuracy confirmed via comparison with reference moment method results. Ill
addition, results obtained using classic UTD are also shown to illustrate the need
for the new solution. Finally, some concluding remarks and the accomplishments of
this work are summarized in Chapter 5.
3
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It is assumed that all fields are tlme harmonic with t_me dependence e i_, wldeh
will be suppressed O_rougli0ut tiffs report. Also, the medium surrounding the scat-
terers is assumed to be free space.
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Chapter 2
Uniform Asymptotic Analysis
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In this chapter, a uniform asymptotic analysis for predicting the scattered fields
from perfectly conducting cubic polynomial strips is presented. The relevant canon-
ical geometry for the analysis that follows is a semi-infinite two-dimensional cubic
polynomial boundary illuminated by a plane wave as shown in Figure 2.1. The scat-
tering mechanisms to be examined, namely specular reflection, zero-curvature and
first-order edge diffraction are illustrated in Figure 2.2. As a plane parametric curve
the boundary C is given by:
_"(Q') : _z + _y(z) ; a _< x < oc (2.1)
where
y(_) = a0+ a,_ + a_ _+ _'_ • (2.2)
The zero-curvature point, zp, is a root of the second derivative of the surface, i.e.,
y"(zp) = 0 and is given by:
-_ (2.3)xp ± 3a3
The scattered fields at any point P away from the surface boundary can be
expressed in terms of the usual radiation integrals over the electric current f, induced
on the boundary by the incident plane wave as follows:
kgO fc[_ × _ × g:(q,)]H_2)(kr)dl, (TM case) (2.4)
kftZ(Q,)× _]g_)(k_)at,(WEcase) (2.5)#,(p) ___
5
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Q:y ' rI Y f,
II
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Figure 2.1: Canonical geometry for the uniform asymptotic analysis
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Figure 2.2: Scattering mechanisms associated with a cubic polynomial boundary
containing an edge.
6
where
=-
!
.m
= I_-PI-_ p- ¢(Q')" _ (far-zoneapproximation),
,_ = _cos0+_sin0,
£(Qr) = value of o_ at any point Q' on the surface,
H(_)(kr) = cylindrical Hankel function of the second kind of order zero,
Z0 = impedance of free space,
k = wavenumber of free space, and
dl' = llne integration element.
In the PO approximation to (2.4) and (2.5), the induced current is assumed to be
given by GO as follows:
2h' ×/I'(Q'), on the fit portion of the boundary
0, on the shadowed portion of the boundary
(2.6)
in which ¢_' is the outward normal to the surface and/t'(Q') is the incident magnetic
field at Q' that under the plane wave incidence assumption is given by:
{g,(Q,) = Z?'(_ x _')d k(r''_')• .',t "i
_ejk(, ._ )
(TM case)(2.7)
(TE case)
where
_' = _ cos 0' + 9 sin 0'. (2.8)
Using the assumed current in (2.6) and the large argument form of H(2)(kr) given
by:f
H(2)(kr) .._ 21-_r e -jk_ , for kr >> 1 (2.9)
the far-zone scattered field assumes the form:
e-JkP (2.10)u:(p;o',o) __:F e,,_(O',O)--j
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where e,.i,(8', 8) is the angular field dependence and is given t)y the statlonnry phRse
integral
where
e_,h(O',0) = f_ f,,,h(z)e sk#'(x) ax (2.11)
dl' ,,,,,, { _' } (2.12)L,,,(,_) - _,_ t_ J" _ ,
¢(_) = _'(Q').(_' + _)= _c(o',o) + y(_)s(o',o), (2.13)
C(O',O) = cos0'+ cos0, (2.14)
S(0',0) = sin0' + sin0, (2.15)
dl'- h(x)= _/1 + [y'(x)] 2, and (2.16)
dx
_p'(x) _ (2.17)_'(Q') - h(_)
The next step in this procedure is the uniform asymptotic evaluation of e.,h(0',0)
given in Equation (2.11) ............
2.1 Uniform Asymptotic Evaluation of e_,h(O',O)
The integral in (2.11) can be transformed into a canonical form by first expanding
the phase function in a Taylor series around tile zero-curvature point x_,, i.e.,
¢(x) = ¢(xv) + (x - z.)¢'(xp) + (_ - xP)_ ¢"'(z,,) • (2.18)3
Notice that the above expansion is exact since ¢"(x.) = 0 and ¢(")(x) _ 0 for every
x with n > 3. Next we make the following linear transformation:
¢(z) = _(,) = _ + Z, + ,_/3. (2.19)
where
= _(0)=¢(_,). and1
= ¢'(_,,) ¢,,,_-_,,)
(2.20)
(2.21)
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The proper branch for _ depends on the sign of ¢'(z,,) and ¢'"(z,,), thus
f=
Lif ¢'"(_) _<o
, + for ¢'(x,,)_O (2.22)if ¢"'(zp) > 0
where L
I_l = I¢'(_)1 • (2.23)
Also, alternative expressions for _ and fl may be found directly from Equation (2.19),
i.e.,
2 (2.24)
2
where s,,2 = 4-(-/3) '/2 are the stationary phase points satisfying r'(s,,.z) = 0. Thus,
using Equation (2.19), the scattered field angular dependence, e,,h(O', 0), becomes:
e,,h(O', 8) = G.,,h(s)e jk'(') ds (2.26)a
where
!
d_
c,,_(_) = /.,,.(_)y,1
and
(2.27)
(2.2s)
(2.29)
a, depend on the sign of ¢'"(zp), thusThe proper branches for _,., and
_f¢,,,(_,,)< 0, -4- for a_zp, or
if ¢'"(zv) > 0
! 1
I= I¢'(,_)- ,/,'(_,,)1'/_ ¢,,,[_.) ,I_,,I= Ia - x,,t ¢"' ) 2
(2.30)
(2.31)
la,_ e-J_
d_f = J;
ds
_f¢'"(_,) < o
if ¢'"(zv) > 0
1
, and (2.32)
d__ = __ . (2.33)
Next we empioy the Chester et al' expansion [i0] for the amplitude function in
Equation (2.26), i.e.,
a_,h(_)=oo
tam ks "4-t3)" + b_,hs(s 2 -4-/3)"] (2.34)m=O
and since only the leading terms in the asymptotic expansion of (2.26) will be re-
tained, Equation (2.34) may be written as follows:
a.,,,(_)= a;;'_+ .E 'h+ (_ + #)g.,,,(.) (2.35)
where
a3,h 1= _[G,,h(8,) + a.,h(S2)], (2.36)
b_.h _ 1 [G.,h(s,)- G,,h(82)], and (2.37)2sl
oo
g,,_(_) Z' '" _ _)"-'= [a,,, ks + + b:;,hs(s 2 +/3)m-t]. (2.38)"=1
Now, using the far-zone observation approximation and the symmetry of the surface
near and around s = 0 or • = zv, Equations (2.36) and (2'37) simplify as follows:
where
dx I and (2.39)a_ 'h _-- G,,t,(s,,2)=f,,h(z,,2)_-s =,,2,i
bo'" -_ 0 (2.40)
d_ = i T"(,_,,_), I +2(-Z)'/_.=.,,_ ¢"(_,,_) ¢"(_,,_) ,and usingEquations(2.32),(2.33),(2.39)iand (2.41)wehave
_5,h= / I_o'"l_-J_if ¢'"(_.) < o[ la3'_l if ¢'"(_.) > 0
(2.41)
(2.42)
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where
I I±1"I<;'hl= IL,',(_',_)I_) = IL,_C_'a)lI_'"C_,')l
Then, using Equations (2.35) and (2.40) e.,,(0',0) may be given by the following
expression:
c.,,hIe',e)= ejkT_''_[a? ["_"L s_
e#(a'+''U3) ds +: f o"_"
1
(,2 + #)g,.h(,)¢_c_.,+.,.'l_)d,I .,I
(2.44)
The integrand of the second integral on the right is regular over the entire path of
integration, thus integration by parts yields
+
and using Equations (2.19) and (2.35)it simplifies further
e,,,h(8,O) e#'(")a_'h f ''_" [ a_'h, ~ o _J_c_.+.'l,,)d, + e_'_'c_'')jk(_ + (_)
(2.45)
where the terms of order higher than 1/k have been omitted. Next, we let s = k-1/:it,
= k-"/:'_ and G = k-l/3(= in (2.46), thus
e,,,,(O',o) ~ _J_o')k-,/:,_,."/.,,oo,, _ -_e j(kt+v'la) dt + e jk_(_) [ k-1/aa°'h
[JIn+ _o) c.h<<o}]jk,-'(¢o(2.47)
Finally, the field angular dependence is given by the following expressions depending
on the sign of 4Y"(Zv) and ¢'(zv):
1. 4/"(ae,,) < 0 and ¢'(azv)_O
_j_- '_ s,hIn this case, /7 = ,re ,, (o = _e_, a0
comes:
r._ll s,hi
e.,,(o',o)~ eS_C"")k-_la;','hl_T(-<,,¢o)+ _s,<<°)/_-"la"lLJ("-<_)
11
s,b _;__and
1"3_ ,t! be-= a o e -._ r_quauon ta.u,)
+j L,,,(a)k _'(a)
(2.48)
=--
M
2. ¢'"(,,.,.)-. o ,,,t ¢,(_.,,).>o
In this case,/3 = o', 4. = (,, a_'" = la;'hl and Equation (2.47) becomes:
[ k-_lai;'hl j f.,,h(a)"
e.,,do',e) _ dk4'('")k-_la;'hlN(_,¢o) + d k_'(o)Lj-(gT_-d)+ k ¢,(_)
= =
Both o- and if,, are real quantities with ¢r>0 if ¢'(_,,)>0 and ¢.>0 if a<>x,,.
the incomplete Airy integral defined by:
012
N(_,-r) _ d ('_+_/_1dz.
2.2 Total Field Solution
(2.49)
_i(_, _) is
(2.50)
Using Equations (2.10), (2.48) and (2.49), the total scattered field is given by:
1. ¢'"(_,,) 5 0
u:(p)
with o-X0 if ¢'(x_)X0.
._ 1
+ejk_{o) k_,lai;'hle_ e-J7 f.,j,(a)] /jv/_(a + _:) _ ¢'(a) J J
e-JkP(2.r,1)
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2. ¢'"(_,,) > 0
( j-_ __
) ejk¢(_:v ) e " k.
u;(p) ~ T [ -_-Ia;'"l_(_,¢,)
r .-,, ,.,,, ,-- " L,_,(_)I ] e-j_"+¢,,_(o) / _01_o_ ± _-_ • (2.52)
with a<>0 if ¢'(z,,)_0.
The q: in front of the expressions in (2.51) and (2.52) corresponds to the soft or
hard polarization cases. Notice that the last two terms in Equations (2.51) and (2.52)
12
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_r_ pllr,'ly e,tge diffract inn ¢o,_r_h,,tion_ wl,rr_ t_ _r._t _erm in (2.51) _,¢1 (2.._2)
appears to be a contribution from the zero-curvature point. In fact, three different
scattering contrlbutions_ namely reflection, edge and zero-curvature diffraction are
implicitIy contained in the incomplete Airy integral. It is advantageous to separate
the scattering mechanisms_ extract the appropriate reflection and diffraction coeffi-
cients_ and cast the total solution in a UTD format. In this form_ the solution would
be applicable to more general problems and also provide important physical insight
into the individual scattering processes. The detai]s of the separation procedure are
presented in the next chapter.
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Chapter 3
Extended UTD Solution
Formulation
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The total scattered field from the semi-infinite perfectly conducting cubic polynomial
boundary of Figure 2.2 can be expressed as follows:
Ui'(p) -_ U;(p) + U_(p) + U_(p) (3.1)
where U_, U_ and U d are the reflected, zero-curvature and edge diffracted field
components. Although additional higher-order mechanisms such as surface waves,
double diffraction and whispering gallery modes may also exist, they will not be
considered in this analysis since the PO approximation to the induced surface cur-
rents does not account for such effects. Before proceeding with the expressions for
each field component, the various field contributions that are implicitly contained
in the incomplete Airy integral are extracted by deforming the originM contour of
integration into appropriate steepest descent paths through the critical points of the
integrand.
First, let's consider tile complex platte topology of the incomplete Airy integral
when the saddle points are real ((r < 0), as depicted in Figure 3.1. There exist three
cases depending on the location of the endpoint 7 relative to the saddle points zl
and z2, and Ai(_r, 7 ) may be written as follows:
A-_(_r, 7) = f_¢ ex,,(j_,2)ej(. _+,., I3) dz + ft.23 e7("_+:'_ I:_)dz
14
M
I
m
m
Ell °
j
z
M
m
w _
M _
/L,,',
/
Ii _ X
orll|nalcontour
(e) 7 <_.-(-o')_'2
/ ;• t
/ i
/ \t •
(b)-(-u)tZt<7 < (-,)vt
g....;
w
u
(c) 7 > (_f)Vt
m
w
= =
m
Figure 3.1: Complex plane topology and contour deformation for the incomplete
Airy integral when tr < 0.
mu
+ f. e"''+''/:'_ d_- ir-r _<-(-_)'/_, (3.2)
¢_¢ 31
if _(__)./2 < _ < (__),/?, and (3.3)
F °°_'_p(j¢') ei("'+""l:_)dz if 7 > (-_)_/_ (3.4)
Using the definitions for the ordinary and incomplete Airy functions found in Ap-
pendices A and B the integrals in (3.2)-(3.4) may be expressed in terms of these:i z_ 7
functions as follows:
c_"l'_'_eJ(_z+'"/3_ez= g,,(_,_), (3.5)
f e j(_'+''_/:')dz = 7r[Ai(a) - jBi(o')], (3.6),72
e"_''_'+'_"/_)dz = g:,(_,_), (3.7)
f e j('=+''/:'O dz = _[Ai(cr) + jBi(_)], and (3.8),31
v'eexI'(JWl)ej(erz+z'a/3)dz = g,(a, 7). (3.9)
Thus, using Equations (3.2)--(3.9), the total scattered field solution in (2.51) and
(2.52) when o" < 0 (lit side of the caustic) is given by:
1. ¢'"(x,,) < 0
e- 5'- ' [AV(a) - jBV(a)]ul(-tr)'f' - (.:,]j_t ¢y'l 3 [ 2
-_ ._/_ .+ eJ_*¢_) e'_¢-'l [a,(¢)+ jBi(¢)]_[-(-¢) '/_- _°]}
+ e;k_¢"l{ ej'k"_ I_'" I [J-_+_g+ e_¢"_"+_/:')X;(°,]¢°)
-e-J'f"'h(a)]}) e-j_v (3.10)¢'(_) J v_
2. ¢'"(_.) > o
u;(p) _ + \v-_el_ _-,_1,_.._1,{eS_C._,)e_(-,,W_[Ai(o.) + jBi(o.)]u[(_o.),/_ _ (.]
16
m
U
iI
I
I
mmI
i
I
[]
U
J
w
I
r_mu
w
m
(a)
Y
contour
(3) '
_0
Y
conLour
\
(b) _' > 0
I
m
• ;_..
w
Figure 3.2: Complex plane topology and contour deformation for the incomplete
Airy integral when o" > 0.
+
+
.2 --or _/? ,,,e./k4'O'') e-'_ ( ) [A1 (o'1- jBi'(o')lu[-(-a) '/_ ¢_1}
{e)'k_'[-Je-J(_¢o+¢]/3)I,]eJko(°) ---_-_l_;'hl ,,'T_g + (_'¢°)
.e -j' f.,,h(a) })e -jk°_'(_)(3.11)
where
It(er, _) = g3(o', (:_) (3.12)
g,(_,C.)
and u(x) is the Heaviside unit step function. The first two terms in Equations (3.10)
and (3.11) represent uniform reflected field contributions where the last two terms
represent edge diffraction contributions.
Next, let's consider the complex plane topology of _-_(_r, 3') when the saddle points
are complex (cr > 0), as depicted in Figure 3.2. There exist two cases depending on......... :7 ........
the endpoint location relative to the zero-curvature point (z = 0)' and the incom-
if ¢o<_-(-o') '/_
if-(-_)'/_ < ¢o_<(-_)'/_
if G > (__)1/2
17
z
plete Airy integral may be expressed in terms of the complete and incornplele Airy
functions in a fashion similar to the previous case, i.e.,
]_](o-,3,) = g2(cr,3,) + 27rAi(o') if 3' < 0, and (3.13)
= g,(_,3') if3' > 0. (3.14)
Thenl using Equations (3113) and (3.14), the tot_iscattered field of (2.51) and (2.52)
when _r > 0 (dark side of the caustic) is given by:
1. ,'"(_,,) _<o
u;(p)
2. _'"(_,,) > o
"; T ( v/_ne k_ta;'hlAi'(a)u(-(o)
+
(3.15)
u;(p) eJk4_(xp) 21re.i_ 'T ( v/_ k_'la_;"lAi(o')u(-(o)
_+¢_-- + e--i("_°+_/a)Id(o. , (,,)]
(3.16)
where
i_(_,¢) = / g"(_'¢°) if ¢o_<0 (3.17)
( gi (_r, (,,) otherwise
In this case the first term in Equations (3.15) and Equation (3.16) represent
zero-curvature diffraction contribution where the last two terms in (3.15) and (3.16)
represent edge diffraction contributions. Notice that the zero-curvature diffraction
contribution is a uniform version of the complex ray (evanescent) field interpretation
of Ikuno and Felsen [11, 12], and it appropriately reduces to their expressions when
iI
I
I
E
[]
I
m
I
i
I
I
I
W
I
I
18
tr >> 1. The main advantage of our formulation is of course uniformity, and also
the need for a complex extension of the reflecting boundary and the evaluation of
surface parameters in complex space is avoided.
Another observation concerning the edge diffracted field component of the to-
tal field is that the last term in Equations (3.10), (3.11), (3.15) and (3.16) can
be recognized as the PO half-plane diffraction coefficient, and using the following
expressions:
f,,h(a) (a) (Q,) = h(a) (3.18)._ sin _,
= h(a)t (Q_) (h i + = h(a)C(_p',_)q_'(a) " ' • 6) (3.19)
w
L _
w
w
w
it may be written as follows:
DPO, , , e-J_ (sin_,'; sin_) (3.20).,,h t_, _) = + vrf_ (COS _' + COS_)
where _,' and _, are the angles of incidence and observation, respectively, measured
from the edge-fixed coordinate system. It is well known that the PO diffraction
coefficient does not satisfy reciprocity nor does it satisfy the local boundary condition
on the surface for the TM polarization case. These shortcomings produce errors in
the total scattered field for observation points away from the optical boundaries.
A rigorous method of providing the necessary corrections to the PO approximation
involves the introduction of an additional non-uniform induced current near the edge.
A rather heuristic but simpler approach for improving the PO half-plane diffraction
coefficient is to introduce a pair of edge correction multiplication terms derived by
James [13], i.e.,
(7 ' = , (3.21)sin( /2) sec( o'/2) and
= cos(¢/2) csc( /2) (3.22)
which correct the PO diffraction coefficient so that it yields the exact diffraction
coefficient outside the optical boundaries. These correction factors reduce to zero
19
=
u
Q;
Q "_ik.
Q" i RS.B.
w
F
: 52
trlbutlon consists of two specular components that only exist in the lit side of the
caustic. It is given by the following expression:
v;(e) ~ v;'(e)_,(b- _,) + vT(p)_,(xT_- a) (3.23)
where U_ ''2 (p) are tile two specular contributions given by
i_ ,h,'(O' )._iu:'._(o)~ dk_'('_'_'.')_'n,_(O' ) n,(O'_.7, ,-._,_,s _, rl, 2 ) ,
eikr'(Q'_,2)'_ e-ikP
v9(3.24)
R,(Q',.,) is the principal radius of curvature at the reflection point given by
R ' h3(x_').(Qt,) : y"(x_,)
(3.25)
and x,._.2 are the specular points found by
-a_ + _/a_- 4a.,a,Xrl,_ : 253
(3.26)
where
_, = _.(_i+,_)+a,_l.(_i+_)=C(O',O)+a,S(O',O), (3.27)
a, = 2a2_I. (._i + .i")= 2a2S(0',0), and (3.28)
a:) = 3a.s_. (h i + k")= 3a.)S(0',8). (3.29)
The quantity s,h(Qrl) is the uniform acoustic soft or hard reflection coefficient that
!
remains valid as Q',., --* Qp and is given by
' I T_(o',.) if R.q(Q'_,) > 07"t,,h(Q,,) = Rs,h -- -- (3.30)
T; (cry) if Rg(Q'_,) < 0
where
R,,h = _T:I:;' for a perfectly conducting boundary, (3.31)
•7 ._17
T,.(o,.) = x/_er_l'e-J_e -')s"" [Ai(-er,.)- jBi(-o'_)], and (3.32)7
or k[e'(Q'_,)- (Q,,)] .(._'+
21
I
Q;
Q Y
sV"S
.:!: I
;T;
dark side ::///IHside.", :"
caustic
% I
I
xp 0 b x
mm
I
I
m
I
I
R
==
Figure 3.4: Geometry for the zero-curvature diffracted field from a cubic polynomial
strip.
Ill
The quantity T, (err) is the appropriate transition function that cancels the GO field
singularity at the caustic. Also, when the specular point is far removed from the zero-
curvature point (observation in the deep lit side of the caustic), o'r >> 1 and using
the large negative argument forms of the ordinary Airy functions (see Appendix A)
we have that T,(a_ >> 1) ,-- 1 and Ti.,,h -_, ]:i.,,h so that the extended UTD reflected
field solution of Equation (3.24) reduces to the usual GO expression for tile reflected
field.
3.2 Zero-Curvature Diffracted Field Solution
U
i
m
I
u
g
The geometry for the zero curvature diffracted field is shown in Figure 3.4. The
zero-curvature diffracted field contribution exists only in the dark side of the caustic
lira
22
r_
r --
Im'
W
and compensates the reflected field discontinuity. It is given by
u;(p) ~ _Jk_'l¢_).._'v_.rO,_oV(¢,l..,,_-J_o,,(_,,- _),,(b- _,,) (3.34)
where 79'_,,h_wvJrr3'_ is the scalar zero-curvature diffraction coefficient and is given by
where
y'"(z,,-----_ L_(_,', _)Ai(a_)
Icos( )p
(3.35)
(3.36)
_ y'(x,,) 1
with c, h(zv ) and c2- h(rv )
is the diffraction coefficient angular dependance, and
Ic(¢,_,)1_.,I_,c(_', _) - _s(_,, _)l,/Z" (3.37)
Notice that the zero-curvature diffraction contribution is significant near the caustic
and decays exponentially away from the caustic as Ai(crc >> 1) _ 0.
3.3 First-Order Edge Diffracted Field Solution
The geometry for the edge diffracted field is illustrated in Figure 3.5. The edge
diffracted field contribution consists of two components (one from each of the two
edges) and is given by
Va_(p) = U]=(p) + U]"(p) (3.38)
where
U]o._(p) ,_ e_ " tvo _j.. _ rc_, _o_k_'(o" _)._"e-Sk,' s,ht_Ola,b) _ , v_ (3.39)
and 7):_,h(Q'_,_)is the scalar uniform edge diffraction coefficient that remains valid as
Q,',,s ---* QI," It is comprised Of two parts, i.e.,
,. ! i,')ttp Q! !Vs,h(Qa,b) A"s,h( _,b) + D:,h(Q.,b) (3.40)
23
Figure 3.5: Geometry for the edgediffracted field from a cubic polynomial strip.
nh_,/,q, _ is the half-plane diffraction coemcient given bywhere r,s, h L_ ,l,l, ]
,,,h_Wa,b/- 2_ sec 2 2
D" to, _ is a term that provides curvature correction and is given byand .,,h_W.,bS
D"..j,(q'_,b) = T
(3.41)
d,,,t,k_eJ¼ 2
y"(a,b) {5 tL.(_o.,_,vo,_) T.(_o,_,(.,_)T2(_o,_,_,_,)
if u'"(_,b)9. (_' + _") < 0
if y"'(a,b)f_. (._ + _") > o(3.42)
where
!L.(_,_,_,_,o,,,) =
O'a _b
ld,C(_, s,_.,s) _ d2S(so, ,,p_,b)l.,/3cos
j ej(._l_.l+l¢l../:,){ g;(,_,l¢l)+ I¢1----_ + g;(_, I_1)
= -T_(_, I¢1),
= hi
sgn
(3.43)
if ,_+ Iffl_<__o, (3.44)
if _r + I(I 2 > 0
2 I._ Id:,C(so',b,_o_,_) + d4S(So'.,_,Soa,b)ly'"(a,b) ld, C(_o'.,_,_o.,_) - d',S(so'.s,,S°.,_,)I '/:;
• # ' z " _ #)]
(_'+ )+u( ,,)_( +¢,,(,_,b)_. (_, + _')
(3.45)
(3.46)
24
[]
g
m
m
i
m
m
[]
D
I
I
m
m
B
I
I
I
w
_a,b
1
d t t
d_,t, = -4-1,
+y'(a, b)d, = -h(a,b) '
1
d,, = _(a,b)'+ y'(_,b)y'(_,,,)
d3 = =t=h(a,b)
y'(a,b)- y'(_,,)dl = h(a,b)
, and
a_ bX xp _ (3.47)
(3.4s)
(3.49)
(3.50)
(3.51)
(3.52)
Notice that the curvature correction part of the uniform edge diffraction coefficient in
Equation (3.40) provides significant contribution near the reflection shadow bound-
ary (RSB) where o + (2 ._ 0. In fact both the half-plane and curvature components
in (3.40) become singular near the RSB, however these singularities cancel each
other out and the total diffraction coefficient remains finite. Away from the RSB
(a + _2 >> 0) the curvature correction component reduces to zero as the diffraction
transition function Ta(cr,_) _ 0, and thus the edge diffraction solution reduces to
the half-plane solution as it should.
_m
L
25
Chapter 4
Numerical Results and Discussion
In this chapter, some numerical results for the scattered fields from a general cubic
polynomial strip are presented. The scattering geometry and the relevant parameters
are illustrated in Figure 4.1. The accuracy of the extended UTD solution is verified
via comparison with method of moments results. Also, some results obtained using
classic UTD are shown and illustrate the need for the new solution.
The first example considered is a cubic polynomial strip with a0 = 2.0_, al = 0.5,
a2 -- 0.1)_ -1, a:_ = 0.1)_ -2, a = -1.5_, and b = 1.5_. In this case both edges
are far removed from the zero-curvature point and thus the two reflection shadow
boundaries and the caustic of the reflected rays are clearly distinct. Figure 4.2 shows
a plot of the magnitude of the various field contributions to the total field for the
TM polarization case and an angle of incidence 0 _ = -45 °. Notice that the reflected
field component exhibits a total of three discontinuities. The first discontinuity
occurs across the caustic of tile reflected rays at 0 ._ -82 ° and is compensated
by the zero-curvature diffracted field. The second discontinuity occurs at the RSB
associated with tile edge Q, at 0 _ -50 ° and is compensated by the edge diffracted
field from Q_. Similarly the third discontinuity occurs at the RSB associated with
the edge Qt, at 0 _ -23 ° and is compensated by the edge diffracted field from Qt,.
Also notice that the two edge diffraction terms become singular near the incidence
shadow boundary (ISB) at 0 = 135°; however, they combine to give a finite result.
U
[]
iD
m
!
g
I
I
mi
I
B
I
m
26
g
Qb
dark side
caustic
Y
b x
Ix2+a x 3 • a< x<b IY(x) = ao+a 1 x +a 2 a , - -
I
Figure 4.1: Scattering geometry and relevant parameters for the numerical results.
27
Figure 4.3 shnws a plo! for the total.qcatleredfieldin terms of lfisfallcecho wMth
for the TM polarization. Tlle extended UTD result shows excellent agreement with
the method of moments where classic UTD gives an erroneous result near the caustic
at O _ -82 °, and in both the lit and dark sides. This is expected since tile classic
UTD formulation does not contain zero-curvature diffraction information in the dark
side and also uses the non-uniform GO expression for the reflected field contribution
in the lit side. Figure 4.4 shows results for the bistatic echo width for the TE
polarization. Again the extendedUT_D result_shows good agreement with method of
moments. The discrepancies for observation directions near grazing are attributed
to higher order mechanisms missing from the total field. These higher order effects
such as edge exited surface rays and whispering gallery modes are stronger for this
polarization since the grazing fields do not vanish on the boundary as is the case for
the TM polarization. The failure of the classic UTD near the caustic is again clearly
illustrated. Figures 4.5 and 4.6 show plots of the mouostatic echo width for the
TM and TE polarizations, respectively. The extended UTD solution gives accurate
results for the monostatic case also, except for the regions where the higher order
mechanisms become significant. Contrary to tl'le classic UTD result, the extended
UTD solution remains finite and continuous across the caustics.
For the second example we consider a cubic polynomial strip with a, = 2.0A,
a, = 0.5, a2 = 0.1A-', a 3 =O.1A -2, a = -0.33A, and b = 1.5A. In this case, the edge
Q, coincides with the zero-curvature point Qv and thus the RSB and the caustic of
the reflected rays coalesce to form a composite shadow boundary. Figure 4.7 shows
a plot Of the bistatic echo width for the TM polarization and an angle of incidence
6' = -45 °. The classic UTD result exhibits a singularity at the RSB associated with
edge Q,, where the extended UTD result remains finite and is in excellent agreement
with the reference solution. Figure 4.8 shows a plot of the monostatic echo width for
the same geometry. Again the non-uniformity of the classic UTD solution is clearly
i
I
m
im
mm
/
m
g
I
mi
I
28Ju
r _
F.
J
evldent where the extends'el UTD solntion rem_in,_ valid across the R,qFt and reduces
to the classic UTD result away from the transition region.
The remaining two examples illustrate the universal nature of the extended UTD
solution. For the third example we consider a cubic polynomial strip with a_, --=2.0)_,
al = 0.5, a2 = 0.1)_ -1, aa -- 0.0)_-:, a = -0.33A, and b = 1.5)_. This of course
corresponds to a parabolic screen and is well known that classic UTD gives accurate
results. In this case the zero-curvature point theoretically moves to negative infinity.
Figure 4.9 shows a plot of the monostatlc echo width for the TM polarization and
the extended UTD solution remains valid and shows excellent agreement with the
reference solution.
For the final example we consider a fiat strip by letting both the quadratic and
cubic coefficients go to zero. Figure 4.10 shows a plot of the bistatic echo width for
the TM polarization and an angle of incidence 8' = -45 °. Tlle extended UTD solu-
tion remains valid for this special case also, and clearly demonstrates its flexibilily
for treating general surfaces that are highly curved, slightly curved, or completely
fiat.
w
w
m
29
Ill
i
If
4.0
3,2--
24-C
1,6-_J
U-
0,8--
Ref_cted
......... Zero-curvature diffracted
....... [dge cl;ffrocled (0o)
EdQe diffracted (lOb)
/I f
I
I
f i
iI i , /
/ _I "j
-135 -90
II '1
t
/L/
9QJ t' i
! i
I I! I
I I
a o,b x
-t,x'_, /t
\//\
II
Jf
lt!
It.
0,0- [
-180 -45
Angle of Observation
\\ /)i / _\.
\... ///' \\_\
1 l I l
0 45 90 135 180
8 (degrees)
Figure 4.2: Scattered field contributions (TM polarization case) from a cubic polyno-
mial strip with a_ = 2.0A, al = 0.5, a2 = 0.1A -I , a_ = 0.1A -2, a = -1.5)_, b = 1.5_,
and angle of incidence 0'= -45 °.
In
mM
i
i
mIm
i
i
m
i
mIll
m
i
mm
i
3OmII
i
;,4
w
b
t<
t-
Ot-
t,tA
.W
re_
25 I Qb
,, /',,'IG- ' ,
I \n F T :
I ! --
7- a ' e,b _-
,\@-11 ,
'1
A
I Method of Moments
........ Ck3$sl¢ UTD
Extended
-20 , , , , , , ,-180 -135 -90 -45 0 45 90 135
Angle of Observation e (degrees)
180
Figure 4.3: Bistatic echo width (TM polarization case) from a cubic polynomial stripwith a, = 2.0_, al = 0.5, a2 = 0.i$ -1, az = 0.1A -2, a = -1.5_, b = 1.5_, and angle
of incidence _' = -45 °.
m
L31
m
J
m
25
I
180
Figure 4.4: Bistatic echo width (TE polarization case) from a cubic polynomial strip
with a,_ = 2.0A, al = 0.5, a2 = 0.1A -1 , a_ = 0.1A -2, a = -1.5A, b = 1.5A, and angle
of incidence #' = -45 °.
mm
roll
m
m
m
I
mmm
n
I
m
m
mm
I
I
32
U
m
m
mm
50
tb"Ov
e-
wm
O¢-t9
ix3
t9.n
t_
¢/5
Ot-O
20
10
0
Method of Moments
........ Clossic UTD
Extended UTDqb
^ I / ;q , I ,.
I
-30 , , _ , , , ,-1 80 -1 35 -90 -45 0 45 90 135 180
Angle 0 (degrees)
Figure 4.5: Monostatic echo width (TM polarization case) from a cubic polynomial
strip with au = 2.05, al = 0.5, a2 = 0.1)_ -l , a3 = 0.1)t -2, a = --1.5)q and b = 1.5_.
33
mI
z!
3O
¢(
&
e-
"o°J
O¢-t9
taJ
_9
0C0
15-
O-/_,,,
-1 5-
-30-
-45 -
-60 --1 80
Q_
K
Method of Moments I
L......... CIo_ s;c UTO
Extended UTD
I I I I I I I
-1 35 -90 -45 0 45 90 135 180
Angle # (degrees)
Figure 4.6: Monostatic echo width (TE polarization case) from a cubic polynomial
strip with at, = 2.0A, aL = 0.5, a: = 0.1A -l, a3 = 0.1A -2, a = -1.5A, and b = 1.5A.
34
k
r--F
= =
I
I
2O
.-. 15]
,'510-
5
=o
t3
,i
re_
i !,I
i Qb
, /II
I1
' I ,_x
I [ I I
0 45 90 135
Observation 0 (degrees)
180
-t0 I I Method or Moments
......... Closslc UTD
Extended UTD
-15 i , I-180 -135 -90 -45
Angle of
Figure 4.7: Bistatic echo width (TM polarization case) from a cubic polynomial strip
with a,, = 2.05, al = 0.5, a2 = 0.1)_,1, a:_ = 0.1,_ -2, a = -0.333_, .... b = 1.5)_, and angleof incidence 8' = -45 °.
w
r_
35
I
i
i
I
I
24
0-
"_-8-
@e"o -16 -
-24
I Method of MornenI9 I
......... Clo_Js;c UTD
Exlended U'rD
I I I I I i I
-180 -135 -90 -45 0 45 90
Angle P (degrees)
135 180
Figure 4.8: Monostatic echo width (TM polarization case) from a cubic polynomial
strip with ao = 2.0)_, al = 0.5, a2 = 0.1A -j, aa = 0.1,_ -2, a = -0.33_, and b = 1.5A.
I
I
mI
I
=i
I
i
m
i
Ii
i
36=
I
i
=
kJ
2O
Parabolic StripI
Method of moments J
I......... Extended UTD
x
-20 , , , l , , ,
-180 -135 -90 -45 0 45 go 135
Angle 8 (degrees)
180
Figure 4.9: Monostatic echo width (TM polarization case) from a cubic polynomial
strip with au = 2.0A, al 0:5, a2_= '(}.1_-'i, a3 = O:OA:2 (parabolic strip), a =
-0.33A, and b = 1.5A.
t_a
37
U
u
m
2O
.-. 151
lo-
5
0
k__ 5
I ......... Method of Moments I
y
-10 - I I t _ _ t ,-180 -1`35 -90 -45 0 45 90 1.35 180
Angle of Observation 8 (degrees)
Figure 4.10: Bistatic echo width (TM polarization case) from a cubic polynomial
strip with al, = 2.05, aj = 0.5, a2 = 0.0,_ -j, a,3 = 0'1_ -_ (flat strip)_ a = -0.33_,
b = 1.55, and angle of incidence 8'= -45 °.
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Chapter 5
Summary and Conclusions
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We have presented an extended UTD solution for the scattering and diffraction
from cubic polynomial boundaries containing edges. This new solution involves the
incomplete Airy integrals as canonical functions and effectively describes the tran-
sitional field behavior associated with composite shadow boundaries and caustics of
the reflected rays. The total solution is presented in a ray optical format by deriving
the appropriate uniform reflection, zero-curvature and edge diffraction coefficients
that remain valid inside the transition regions, and also provide smooth connection
into tile regions where the classic ray optical formulations remain valid.
It was shown by comparison with a reference method of moments solution that
the extended UTD TM polarization solution yields excellent results. The TE polar-
ization solution also showed good agreement with the reference solution although it
would benefit from the inclusion of higher order effects such as edge excited creep-
ing waves, double edge diffraction, and whispering gallery modes in certain regions
where grazing fields exist. The failure of classic UTD solution to describe the scat-
tered fields near caustics of the reflected rays and composite shadow boundaries was
also clearly illustrated. The universal nature of the extended UTD solution was
demonstrated by considering examples of a parabolic screen and a fiat strip. In both
special cases, the new solution gave excellent results. Therefore, the extended UTD
39
II
._1111ion ran he _i._edtn eff_,cllvelyde._cril_ethe ._cal.tere(llleldsl*rnm general r11rved
surfaces ranging from strictly concave or convex, concave-convex, or completely fiat.
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Appendix A
Ordinary Airy Functions
Tile ordinary Airy functions satisfy the following differential equation:
(A.1)
which has two independent solutions, Ai(_) and Bi(cr). In integral form they are
given by
1
Ai(cr) _ 2_- _f._, eJ("_+_/:')dz (A.2)
Bi(o') = j f. e j(''+_"/3) dz (A.3)271" _._+ I.+._
where the contours of integration L21, L23 and L3w are shown in Figure A.1. For
small arguments they can be computed using their ascending series form [14], i.e.,
m
where
Ai(a) = c,f(a) - c,_g(tr) (A.4)
Bi(_) = VS[_,f(_) + _9(_)] (A.5)
1.4 6 1 "4 " 7o.9f(_) = a + _,_'_ + --g-o" + 9 +""
2al 2i5aT 2.5.8o.1.g(a) = a+-_ +-- + 1------0-- +""
c, = Ai(0): Bi(0)/v_ = s-_/z/r(2/3)
c2 = Ai'(0) = Bi'(0)/v_ : 3-'/3/F(1/3).
41
(A.e)
(A.7)
(A.S)
(A.9)
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Figure A.I: Contours of integration for the ordinary Airy Functions.
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Thclr large argument forms ar_ given by
-- " I 2 .*I/_
/'2 :lMAi(-<,-) -.. ,,-'/_<,--'/"sint,_<_
Bl(o') _ 7r-i/20"-I/ie -_','=/_
[2 "_i' 4)ni(-_) ~ .-'/_-'/'cost_ +
+ 7 (I,,g <,1< 2_-13)
(I,<rg<_1< ,_13)
(l_g<,l < 2_/3).
Complete asymptotic expansions may be found in [14].
(A.10)
(A.11)
(A.12)
(A.13)
i
43
I
Appendix B= : _£; :
Incomplete Airy Functions
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The incomplete Airy functions satisfy the parabolic differential equation applied by
Fock [15] and others to the study of fields near the surface of convex diffracting
bodies, i.e.,
0 0- _r- j_--_ y(a,7) = 0 (B.1)
which has three independent solutions, g,(_r, 7), i = 1, 2, 3. In integral form they are
given by
/ocexp{j¢_) ej{_+z._13 ) dz ;g,(_r,7 ) = i= 1,2,3 (B.2)
where the contours of integration are shown in Figure B.1. The functions g2 and g:_
can be obtained from gl and the ordinary Airy functions as follows:
g2(tr, 7) = g,(tr, 7 ) - 2_rAi(tr), and (B.3)
g:,(a, 7) = g,(tr,-y)- _r[Ai(a) + jBi(a)]. (B.4)
For small values of tr, gl can be computed using its ascending series form [9], i.e.,
g,(_,_) = _ a,,(_)a" (B.5)tl------O
where
a,,(3') = g,(0,3')= eJ"/_3-2/:'F(1/3,-JT:'/3), (B.6)
a,('_) - ff g,(O,7)=-e-J'_/63-'/3F(2/3,-JY_/3), (B.7)¢Ja
44
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im
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=-
Ill
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D
\
Y
(2) (1)
2X
Figure B.I: Contours of integration for the incomplete Airy Functions.
45
J !
a,,(7) = a,,-3(7) + jaJ,,_._(7) (B.9)n(n - 1) , n>3
and F(x, y) is the incomplete Gamma function. The asymptotic forms of gj are given
by
g, (0., 7) "_ J (B.10)o-+7 2
e-j](-.)"/_g,(_,7)
(_0.)1/,J
i 2
2j 0"+7 2
_¢ c_+_'_/_) ('r >>I_1'/_)
j_.. F'(r/2) e¢,,2]v/-_e , u(-r/) 2jr/
1 ej(_+._.,/.3)
,1(-,_ ) '1'(0. <<-1) (B.11)
where
2 _0.)3/2] 1127/ = + 0.7 + 73/3 + _( ; 7_(--0.) '/2 (B.12)
and F(z) is the Kouyoumjian-Pathak transition function [5]. Complete asymptotic
expansions for gl along with some representative plots may be found in [9].
46
Bibliography
w
E
[1] Crispin, J. W., and Siegel, K. M., Methods of Radar Cross-Section Analysis,
New York, Academic Press, 1968.
[2] IIarrington, R. F., Field Computation by Moment Methods, Robert E. Krieger
Publishing Company, 1968.
[3] K]ine, M., and Kay, I. W., Electromagnetic Theory and Geometrical Optics,
Interscience Publishers, Inc., 1965.
[4] Keller, J. B., "Geometrical Theory of Diffraction," 3". Opt. Soc. of America,
Vol. 52, No. 2, pp. 116-130, Feb. 1962.
[5] Kouyoumjian, R. G. and Pathak, P. H. "A Uniform Geometrical Theory of
Diffraction for an Edge in a Perfectly Conducting Surface," Proc. of IEEE, Vol.
62, pp. 1448-1461, Nov. 1974.
[6] Levey, L. and Felsen, L. B., "On Incomplete Airy Functions and their Applica-
tion to Diffraction Problems," Radio Science, Vol. 4, No. 10, pp. 959-969, Oct.
1969.
[7] Pathak, P. H. and Liang, M. C., "On a Uniform Asymptotic Solution Valid
Across Smooth Caustics of Rays Reflected by Smoothly Indented Boundaries,"
IEEE Trans. Antennas Propagat., Vol. 38, No. 8, pp. 1192-1203, Aug. 1990.
[8] James, G. L., Geometrical Theory of Diffraction for Electromagnetic Waves, p.
37, Peter Peregrinus Ltd., London, UK, 1976.
[9] Constantinides, E. D. and Marhei'ka, R. J., "Efficient and Accurate Computa-
tion of the Incomplete Airy Functions," to appear in Radio Science.
[10] Chester, C., Friedman, B. and Ursell, F., "An Extension of the Method of
Steepest Descent," Proc. Cambridge Phil. Soc., Vol. 53, pp. 599-611, 1957.
[11] Ikuno, H. and Felsen, L. B., "Real and Complex Rays for Scattering from a
Target with Inflection Points," Radio Science, Vol. 22, No. 6, pp. 952-958, Nov.
1987.
[12] Ikuno, H. and Felsen, L. B., "Complex Ray Interpretation of Reflection from
Concave-Convex Surfaces," IEEE Trans. Antennas Propagat., Vol. 36, pp. 1260-
1271, Sept. 1988.
47
[1.q],lames,G. L., Geometrical Theory of Diffraction for Electromagnetic Wave.q, p.
150, Peter Peregrinus Ltd., London, UK, 1976.
[14] Abramowitz, M. and Stegun, I. A., editors, Handbook of Mathematical Func-
tions, Dover Publications, Inc., pp. 446-452, New York, 1972.
[15] Fock, V. A., Electromagnetic Diffraction and Propagation Problems, Pergamon,
Oxford, England, 1965.
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