Full text of DOI:10.1002/1520-6432(200102)84:2<54::AID-ECJB7>3.0.CO;2-Y

Electronics and Communications in Japan, Part 2, Vol. 84, No. 2, 2001
Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J83-C, No. 2, February 2000, pp. 118–127
Derivation of Uniform PO Diffraction Coefficients Based on
Field Equivalence Principle
Ken-ichi Sakina, Suomin Cui, and Makoto Ando
Department of Electrical and Electronic Engineering, Tokyo Institute of Technology, Tok yo, 152-8552 Japan
surface integrals [2, 3] leads us to line integral repre-
SUMMARY
sentations or closed-form expressions of fields, which
greatly contributes not only to reducing the computation
A novel approach for asymptotic reduction of physi-
time but also to mechanism extraction of PO [4]. In general,
cal optics (PO) integration is proposed for two-dimensional
the asymptotic reductions of PO surface integrals such as
line source diffraction from a half-sheet. The field equiva-
the geometrical theory of diffraction (GTD) become infi-
lence principle provides alternative integration surfaces not
nite at geometrical boundaries and caustics. To eliminate
these difficulties, several uniform expressions have been
proposed. In Ufimtsev’s physical theory of diffraction
(PTD) [5–7], PO currents are improved by adding another
component called fringe wave currents JFW. Many works
about the evaluation of surface integrals for diffraction from
a half-sheet are developed in the spectral domain [8–11].
Efforts to achieve surface to line integral reduction
have included both an exact approach based on the Helm-
holtz–Huygheng principle [12–15] and asymptotic ap-
proaches such as the high-frequency approximation
[16–21]. The asymptotic and local expressions are quite
different from the exact and global ones and sometimes
have the advantage that the former is applicable to a much
wider class of scatterers based on local features of the
diffraction phenomena.
on the original half-sheet but on the geometrical shadow
SB) and reflection (RB) boundaries, where analytical in-
(
tegration leads to the well-known Fresnel-type uniform PO
diffraction coefficient of UTD type. The superiority of the
uniform diffraction coefficient to those of other types is
explained in terms of the location of the integration surfaces
and is demonstrated numerically. © 2001 Scripta Technica,
Electron Comm Jpn Pt 2, 84(2): 54–62, 2001
Key words: PO; PO diffraction coefficient; field
equivalence principle.
1. Introduction
In two-dimensional (2D) problems of half-sheet dif-
fraction illuminated by a line source, the edge contribution
of the PO integral is asymptotically expressed in terms of
PO diffraction coefficients. PO diffraction coefficients of
the classical Keller type are nonuniform at geometrical
boundaries, such as shadow and reflection boundaries
(SB/RB) [16–18]. Two types of uniform expressions to
cope with these difficulties are available. The coefficients
of the first kind were derived by directly applying the
uniform asymptotic evaluationto integration on ahalf-sheet
Physical optics (PO) [1] is a high-frequency tech-
nique in which the total induced currents J are approxi-
mated in the sense of geometrical optics (GO). The PO
currents JPO thus defined are then integrated over the sur-
face to give finite fields everywhere, including geometrical
boundaries and caustics in focusing systems. PO has been
widely applied to the pattern analysis of reflector antennas.
In PO, the scattering fields are obtained by evaluating
the surface radiation integrals of JPO, which is performed
numerically in general. The asymptotic evaluation of these
©
2001 Scripta Technica
54

[
3, 19, 22]. Those of the second kind have the symmetry
analogous to those based on uniform theory of diffraction
UTD) [8, 10, 23–26] that have been proposed in the
(
spectral domain [8, 10]. Though numerical comparison
indicated that the second kind is superior to the first kind,
comparison of the two in terms of accuracy or applicability
is insufficient since the direct derivation of the second kind
in the spatial domain has not been accomplished. For a full
understanding of the accuracy and the applicability of the
PO diffraction coefficient of UTD type, its spatial domain
derivation is indispensable.
This paper presents the mathematical derivation of
the uniform PO diffraction coefficient in the spatial domain
for the first time. In our derivation based on the field
equivalence principle [27, 28], the original PO integration
surface on the half-sheet is transferred into two surfaces
coinciding with GO-SB and GO-RB, on which the integrals
can be evaluated analytically [29–31]. We obtain some
important results by analyzing the problem in the spatial
domain as follows:
Fig. 1. The scattering problem of cylindrical waves by
the half-sheet C0.
where Z is the angular frequency, P is the permeability,
i
EꢀPꢁ is the incident field from the line source I at the
0
^
observation point P, n is the unit vector normal to the
x The diffraction coefficients are uniformly valid for
arbitrary combinations of the angular positions of
the source and the observer, provided that the
distance between the source and the edge is large.
x The distance between the source and the edge is
always larger than the parameter in the conven-
tional derivation on the original half-sheet, that
is, the distance between the source and the half-
sheet. Therefore, the superiority of the uniform
PO diffraction coefficient of UTD type is
clearly identified.
half-sheet C , and G is the 2D Green’s function with wave
0
number k for the far field:
(2)
i
H is the magnetic field incident on C and is given as
0
follows with the amplitude of the line current I assumed
0
to be unity:
(
3)
x The difference between the uniform PO diffrac-
tion coefficient of UTD type and the conventional
coefficient increases as the angular position of the
source moves to the infinite plane including the
half-sheet.
ꢀ
2ꢁ
where H is the first-order Hankel function of the second
^
1
kind, and Iis the unit vector in the direction of I. If the
distance dmin between the current and the half-sheet satisfies
the relation
The above results are confirmed numerically.
(4)
Equation (3) can be approximated as
2. Physical Optics and PO Diffraction
Coefficients
(5)
By substituting Eqs. (2) and (5) into Eq. (1), we get the PO
integral to be evaluated asymptotically,
We consider the 2D problem of cylindrical wave
diffraction by the half-infinite conducting sheet C shown
0
in Fig. 1. In the PO approach, the induced currents (PO
currents) on the surface of the sheet are given by
^
i
I
2n uH, M 0 and hence, the total field is
(6)
where Kdenotes the intrinsic impedance of free space and
^
z is the unit vector pointing into the positive direction along
(1)
55

the z-axis. Direct application of asymptotic theory [2] to the
integral leads us to the diffracted wave
(7)
i
where E is the z-component of the incident electric field at
the edge of half-sheet and D is the PO conventional diffrac-
tion coefficient, expressed as
z
(8)
This coefficient diverges at GO-SB(I = I + S) and GO-
i
RB(I= I – S), as the GTD diffraction coefficient of Keller
i
does. On the other hand, the use of the uniform asymptotic
method [22, 23] gives a slightly different uniform diffrac-
tion coefficient:
Fig. 2. The decomposition of PO currents.
(9)
decomposition in Fig. 2, we can rewrite the PO field in Eq.
(1) as
This coefficient gives finite fields even at GO-SB and
GO-RB [19, 22]. Fꢀxꢁis the modified Fresnel function
(10)
We denote the numerical integration in Eq. (6) by POap-
prox.inc. It is noted that in the uniform asymptotic manner
[3, 19, 22] we must evaluate the integration over the ex-
tended range ꢀꢂf, fꢁ. In this case we have the condition
(11)
kd !!1 instead of Eq. (4), df being the distance from
f
min
min
the source to the infinite plane containing the half-sheet C0.
This condition means that the accuracy of Eq. (9) degrades
Note that the original field in Eq. (1) remains unchanged
through these manipulations into Eq. (11).
when I o0° and 180°. It proves that the diffraction coef-
ficient of UTD type presented in this paper maintains high
accuracy for such locations.
In order to change the integration plane to the above
decomposed problem, we define the closed curves
C {C ‰C ‰C for <1>, which consist of a half-line C
i
0
1
f
0
onthehalf-sheet, C on GO-SB, and thecircular arc C with
1
f
3. Transformation of Integration Plane by
Field Equivalence Principle [30, 31, 33–36]
We first decompose the PO currents on the half-sheet
into two parts <1> and <2>, as shown in Fig. 2. The location
^
of the source as well as the definition n in <2> is then
i
i
changed to that of the image in Fig. 3, where Hcand Ec
denote the incident fields from this rotated current I ; these
g
o
fields are related to the original incidence as
n^ cuHic n uH and Ec E. Now the surface currents on
^
i
i
i
the two surfaces consist of the special combination of
^
I
n uH and M E un} and {I ncuHc and
i
i
i
^
^
i
{
^
M
Ecunc} for <1> and <2>, respectively, which appear
in the field equivalence principle [27–29, 35]. After the
Fig. 3. Closed curves for the field equivalence principle.
56

infinite radius as in Fig. 3. The closed curve C divides the
BB
whole space into subspaces V and V. In a similar way,
integrals over two semi-infinite lines on shadow boundaries
g
C and C for the PO diffracted field. The above transfor-
1
BB
1
g
g
Cc{C ‰C ‰C , Vc, and Vc are defined for <2>. Note
mation is exact and its significance is that the minimum
g
distance between the surface C1 (and C1) and the source I0
g
(and I0 ) is d and is always larger than dmin. This loosens the
0
1
f
^
i
^
i
g
1
that n uH and ncuHcvanish on C and C , respectively.
1
Now the relation between integrals on C and C is derived
0
1
based on the field equivalence principle so that the PO
restriction in Eq. (4) and enhances the accuracy of asymp-
totic reduction of integrals.
integration on the half-sheet C may be transformed into
0
g
integrations on GO-SB (C ) and GO-RB ꢀC ꢁ. In terms of
1
1
geometrical optics, there are three kinds of position of
4
. Uniform PO Diffraction Coefficient
observer P; in the shadow region (P  V), the reflection
BB BB
region (P Vc), and the rest ꢀP V ˆVcꢁ.
Applying the field equivalence principle to the closed
i
We focus on evaluation of the integral
ꢀE u
³
C
^
nꢁu’G dl in the far field region, referring to Fig. 4. Let the
incident electric field at Q on C be
curves C and Cc, we obtain the relation between integrals
^
i
1
as follows, under the conditions that n uH 0 on C and
ncuHc 0 on Cc:
^
i
(16)
(12)
where d and l are defined in Fig. 4. The following loose
condition is assumed:
i
^
where n ˜E 0 on C. In a similar way, we have
(17)
which is satisfied more easily than condition in Eq. (4).
Furthermore, since the integrals in Eq. (15) contain only the
diffracted component, an important contribution to the in-
tegral comes from small l near the edge. Thus, the distance
(13)
Ufrom Q to the observer and that U from the edge of the
0
sheet to the observer are related by
From Eqs. (12) and (13), the PO total field in Eq. (11) is
g
expressed in terms of integrals on C and C instead of that
1
1
on C as
0
(18)
(14)
i
where ꢂEcꢀPꢁis the incident field from the image source,
that is, the reflection field at P. Thus, the last term in this
equation indicatesthe geometrical opticscontribution in the
reflective region; we finally reach the following formula for
the original diffracted field from the half-sheet.
(15)
Thus, the integral over the original semi-infinite sheet C0
Fig. 4. The coordinate system for evaluating the
for the PO total field in Eq. (1) is now transformed into
integral on C1.
57

It follows that
5. Numerical Discussion
The superiority of the derivation is now demon-
strated numerically. The necessary conditions for the
new and conventional derivation in approximating the
PO surface integration are (17) and (4), respectively. We
compare various expressions for PO diffraction coeffi-
cients with the original PO integration (1) with exact
incidence (3). They are “Uniform” defined by this paper
in Eq. (23), “Uniform (Direct)” in Eq. (9), “POap-
prox.inc.” in Eq. (1) with approximation (5), and
(19)
Then the integral for C become
1
“
Nonuniform” in Eq. (8). We predict the general accu-
racy of various expressions as in Table 1. The key pa-
rameters used in the comparison are illustrated in Fig. 1.
First, the half-sheet problems are discussed for decreas-
(20)
ing values of d with I = 45° in Figs. 5(a), 5(b), and 5(c).
i
Figures 6(a) and 6(b), 7(a), and 7(b) compare the total
field for d 1O, 3O, and 0.2O, respectively. From these
figures,uniformexpression(23)derivedfor thealterna-
tive surfaces on GO-SB is the most accurate and gives
resultsalmost identicaltoPO for smallvaluesofddown
to 0.2O.Ontheother hand,theuniformexpression“Uni-
form(Direct)”derivedfor theoriginalhalf-sheetsuffers
where U /U|1 is used implicitly. The integral in Eq. (20)
isidentical to thedefinition of themodified Fresnel function
Fꢀxꢁin Eq. (10). We then get
0
from error in Fig. 5(b) for I = 45° and d = 1O (d
|
i
min
0.7O).“POapprox.inc.” haslargeerrors,while“Nonuni-
form” hassingularitiesat GO-SBand GO-RB. Thus, the
prediction inTable1isconfirmed.
(21)
6. Conclusion
g
The integral on C can be obtained by simply replacing I
1
i
A new derivation for the uniform diffraction coeffi-
cient for the half-sheet is provided, based on the field
equivalence principle. The PO integral on the original half-
sheet is transformed into semi-infinite integrals on the
geometrical shadow boundaries, expressed by using the
with 2SꢂI in Eq. (21), taking account of the direction of
i
^
nc, as
(22)
Table 1. Prediction of accuracy (!, high accuracy; u,
low accuracy) for various integral evaluations for
different source angles
We finally obtain the uniform PO diffraction coefficient of
U
UTD type D ꢀd, I, Iꢁas
i
(23)
This gives finite fields even at GO-SB and GO-RB, and is
similar to the uniform expression in UTD [24].
58

Fresnel function. The coefficient is valid when the distance
between the source and the edge of the half-sheet is larger
than about 0.2 wavelength. The superiority of the uniform
diffraction coefficient to other typesisexplained for thefirst
time in terms of the location of the integration surfaces and
is demonstrated numerically.
Fig. 6. Degradation of conventional uniform diffraction
Fig. 5. Accuracy of diffraction coefficients as functions
coefficients for small distance dmin |0.2O. (a) I = 5°, d =
i
of distance d to the edge of the half-sheet (I = 45°).
1O; (b) I = 175°, d = 1O.
i
i
59

Fig. 8. Accuracy of diffraction coefficients for I =
i
135°. (a) d = 3O; (b) d = 0.2O.
Fig. 7. Accuracy of diffraction coefficients for I = 90°.
i
(a) a = 3O; (b) d = 0.2O.
2
3
4
. Copson ET (editor). Asymptotic expansion. Cam-
bridge University Press; 1965. p 27–35.
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AUTHORS (from left to right)
Ken-ichi Sakina received his B.S. degree from Kanto Gakuin University in 1972 and M.S. degree from Chiba University
in 1975, in electrical engineering. He studied general relativity and spinor in space-time as a research student from 1976 to 1980
at Tohoku University. Since 1998, he has been pursuing a D.Eng. degree at the Tokyo Institute of Technology. His research
interests include the scattering of electromagnetic waves and electromagnetic theory.
Suomin Cui received his B.S. degree in physics from Shaanxi Normal University, China, in 1989, and M.S. and Ph.D.
degrees in electrical engineering from Xidian University, China, in 1992 and 1995, respectively. He joined the Department of
Electrical Engineering, Nanjing University of Science and Technology, as a lecturer in 1995 and was appointed an associate
professor there in 1997. He was a research fellow at the Tokyo Institute of Technology from 1997 to 2000. His primary fields
of interest include high-frequency diffraction analysis, computational electromagnetics, and RCS computation of complex
objects. He was awarded postdoctoral fellowships from China’s Postdoctoral Council in 1995 and from the Japan Society for
the Promotion of Science in 1997, and a research fellowship from the International Communication Foundation for a foreign
researcher in Japan in 2000. He received the second- and first-class scientific and technical achievement award from the Ministry
of Electronic Industry of China in 1996 and 1997, respectively, and the third-class national scientific and technical achievement
award in 1998.
Makoto Ando received his B.S., M.S., and D.Eng. degrees in electrical engineering from Tokyo Institute of Technology
in 1974, 1976, and 1979, respectively. From 1979 to 1983, he worked at Yokosuka Electrical Communication Laboratory, NTT,
and was engaged in development of antennas for satellite communication. He was a research associate at the Tokyo Institute of
Technology from 1983 to 1985 and is currently a professor there. His main interests have included high-frequency diffraction
theory such as physical optics and geometrical theory of diffraction, the design of reflector antennas and waveguide planar
arrays for direct broadcast from satellite (DBS) and very small aperture terminal (VSAT), and the design of high-gain
millimeter-wave antennas. He received the Young Engineers Award of IEICE Japan in 1981, the Achievement Award and the
Paper Award from IEICE Japan in1993, the 5th Telecom Systems Award in 1990, and the 8th Inoue Prize for Science in 1992.
62