The Geometrical Theory and of Diffraction Applied to

VOL.
IEEE TRANSACTIONS ON .UTEK3-.&Sah?, PROPAGATION
TABLE I
“IDEAL”FEEDFOR EACHTYPEOF “CONIC”REFLECTOR
Eccentricity
Conic Surface
Sphere
Ellipsoid
Paraboloid
E Electric
= 1
The author wouldlike to thank P. U T . Hannan Qf
ITTheeler Laboratories,Inc.,forhelpfuldiscussions
on
thissubject.
Electricfblagnetic
Dipole
( E D = € AID)
Dipole
REFERENCES
+Magnetic
[ I ] I. Koffman, “Reflectiving surfaces formed by wire grids,” MEE
Report, Polytechnic Institute of Brooklyn, Brooklyn, X. Y . ,
June 1961.
[2] C. C . Cutler, “Parabolic-antenna design for microwaves,” Proc.
IRE, vol. 35, pp. 1284-1294, November 1947.
[3] E. M. T. Jones, “Paraboloid reflector and hyperboloid lens antennas,” IRE Trans. on Antennas and Propagation, vol. AP-2, pp.
119-127, July 1954.
[4] P. W . Hannan,n‘%licrowave antennas derived from the Cassegrain telescope, IRE Trans. o n Antennas and Propagation, vol.
AP-9, pp. 140-153, March 1961.
(ED=M D )
Hyperboloid
Plane
1< E <
E=
m
Electric+Magnetic
Dipole
( E D= € M D )
x
Electric Dipole
(Magnetic Dipole = 0)
1966
ACKKOWLEDGMENT
Magnetic Dipole
(Electric Dipole = 0 )
0<€<1
JANVARY,
polarization characteristics areexpressed in termsof the
polarizationcharacteristics
of elementalelectricand
magnetic dipole radiators in the proper intensity ratio.
Feed
E=O
e-14,NO. 1
The Geometrical Theory of Diffraction Applied to Antenna Pattern
and Impedance Calculations
Abstract-The geometrical theory of diffraction is applied to the
calculation of the radiation pattern and impedance of a monopole
antenna on a perfectly conducting circular ground plane of limited
extent. In this calculation, the radiation problem is resolved into two
components, one being the monopole contribution and one the edge
contribution. The impedance problem is resolved into thecomponents
of a reflection from the monopole in an infinite ground plane and a
reflection from the circular edge as seen through the antenna. The
known solutions of these individual components then permit the
calculation of the overall radiation pattern and impedance by superposition. The techniques described are general and are considered
applicable to a large class of similar radiating structures.
s
I.INTRODCCTION
IKCE T H E introduction [ l ] of the geometrical
theory of diffraction in the early1950’s, it hasbeen
employedsuccessfully in the solution of various
types of diffraction problems. In this paper, the theor\is applied to the problem of determining the radiation
pattern and impedance
of a monopole antenna operating
over a finite perfectly-conducting circular ground plane
(disk-monopole antenna). This configuration is shown
in Fig. 1. The purpose of this present paper is not primarily to solve this particular problem, but rather
to
demonstrate the usefulness of the geometrical theory of
Manuscript received June 21, 1965; revised September 20, 1965.
The author is with Uleeler Laboratories, Inc., Smithtoan, N. Y .
diffraction in the solution of a class of radiation problems, which arise often in practice.
Simply stated, the geometrical theory
of diffraction
proposes that the total electromagneticfield, excited by
sources, can be described in terms of a superposition of
raysemanatingfromthesourcesandraysdiffracted
from edges or other diffracting obstacles. T h e essential
feature of the theory is t h a t i t is applicable t o all types
of complicated problems, some of which are, in practice,
impossible to solve rigorously. The approach is t o resolve a complicatedproblemintosimplercomponent
problems, each of which have relatively simple rigorous
solutions.Inthepracticalapplication
of thetheory,
there are available many known rigorous solutions
of
diffraction problems (e.g., n-edge, cone, cylinder, sphere,
etc.) which then serve as basic building blocks
for the
construction of the solutions for the more complicated
problems.
One such problem is the determination of the characteristics of a disk-monopole antenna. In this paper, the
radiation pattern and impedance of the monopole configuration mill be determined by means of the geometrical theory of diffraction. The pattern and impedance
will beconstructedfromtherigoroussolutions
of a
monopole radiating in a n infinite ground plane and the
diffraction by a straightedge excited by a magnetic line
40
LOPEZ:
THEORY
FOR ANTESNA PATTERir; KXD IMPEDBiiCE CALCULATION
41
PT= transmitted power.
G =antenna power gain in direction of the edges.
1.k =received voltage.
V , = transmitted voltage.
V i= incident voltage in transmissionline.
V ,=reflected voltage in transmission line.
I? =antenna reflection coefficient.
MONOPOL
I
?
,
=antenna reflectioncoefficientfor
a n infinite
ground plane.
I’,=antenna reflectioncoefficient component attributed to edge.
J
rol=antenna transmission coefficient for an infinite
ground plane.
Fig. 1. Monopole on a circular ground plane.
Z = antenna impedance.
Z , =antennaimpedanceforaninfiniteground
source, since the solution to the latter problem is readily
plane.
available. (The magnetic line source and the monopole
Z , =antenna impedance component attributed to
produce the samefields in the near neighborhoodof the
edge.
edge as the source-to-edge distance approaches infinity.
Zo=characteristic impedance of transmission line.
If this distance is large in terms
of wavelengths, the
lz = height of monopole.
fieldsin theedgeneighborhoods,and,therefore,the
I ( x ) = current-distribution function of monopole.
diffracted fields, are approximately the same for the two I(o) =base current or input current.
cases.)
111. FAR-FIELDDIFFRACTION
PATTERN
OF A N EDGE
The pattern calculationwill consist of a superposition
of a ray emanating from the monopole
itself and two
Ls\ indicated in Section I, the problem of determining
rays emanating from the edge of the circular ground
the far-field pattern of a monopole radiating in a finite
plane. In a similar manner, the impedance calculation
circular ground plane can be constructed from the rigorwill be a superposition of areflectioncaused
by the
ous solutions of amonopoleradiatingin
a n infinite
monopole operating in a n infinite ground plane and a ground plane and the diffraction by a straightedge exreflectioncaused by the finitecirculargroundplane.
citedby a magneticlinesource.
Inthissection,the
The magnitude and phase of the edge reflection mill be straightedgeproblemisreviewed.
Thedetails of the
determined by utilizing some common radar principles. solution can be found in the literature [SI ; for our purT h e diffraction problem of a straightedge excited by
pose it is sufficient t o summarize the pertinent results.
a magnetic line sourcewill be reviewed, since it is essenFigure 2 is a sketch showing the magnetic line source
tial to the discussion. The radiation patternof the disk- and the diffracting edge. If, in the general solution
of
monopole antenna \vi11 then be calculated and compared this problem, thefar-field condition (kp>>kp’) is imposed,
with experimental results. The impedance of the same thegroundplaneislargeinterms
of wavelengths
configuration mill be calculated and compared to a re(kp’>>l), and the time dependence is expGut), then the
sult found in the literature.
far-field diffraction pattern E@) is related to a complex
Fresnel integral and may be summarized as follows:
11. LIST O F SPJiBOLS
1
p =distance from theobservationpointtothe
E(0) = straightedge.
2
p’ =distance from the source point to the straightedge.
p=angle measured into the shadow region from
the shadow boundary with the origin
at the
straightedge.
k = free-space propagation constant.
X = free-space wavelength.
E@) =ratio in the far field of the diffracted field to
the value of the incident (geometrical optics)
fields at the shadow boundary.
p 3 dB=angle at which diffracted signal is down 3 d B
This lastexpression is asymptotic to theFresnel integral
from its shadow boundary value.
and is a good approximationwhen
0 =antenna pattern angle.
4 =angle measured along the ground plane.
P R=received power.
IEEE TRLWSBCTIOWS ON AKTENNAS
42
PROPAGATION
OBSERVATIONPOINTAT
(p,p)
P>>P'
i'
MAGNETIC
LINE SOURCE
SEMI-INFINITE
PERFECTLYL
CONDUCTING PLANE
Fig. 2.
EDGE
SHADOW
BOUNDARY
Magnetic line source excitation of a straightedge.
p"
-7r
ILLUMINATED
REGION
SHADOW
REGION
.5
1
3
x
VOL. AP-14, hTO.1
JOTU-ARY, 1966
I t follows that the radiation pattern in any plane containing the monopole consists of the monopole pattern
plus the patterns of the edge-diffracted signalswhen
added in the proper relative phase.
In this case, the
characteristics of the edge-diffracted signals in the plane
of interest are as described in Section I11 for the case of
the two-dimensional straightedge problem.
Figure 1 shows the pertinent rays in a plane containing the monopole; Fig. 4 shows the corresponding radiation patterns for each component. Thus, the problem is
reduced to thesuperposition of the fields of three sources
separated in space, and having different radiation patterns. [The patterns
of the two edge sources are described in Section I11 andthe monopole(ordipole)
pattern is found in many references, e.g., [7].]
Figure 5 shows the measured and calculated patterns
of the entire configuration for the case where the ground
screen has a radius of three wavelengths and the monopole is a quarter wavelength long. It may be observed
that there is good agreement betweenthe measured and
calculated patterns, especially in the periodicity of the
interferingsignalsand also in the pattern amplitude
near 0 = 90'. However, in the0 = 0 and B = 180' directions
thecalculatedsignalissignificantlylowerthanthe
measured value. This is explainable by noting that in
these two directions every point
on the edge contributes
a ray to the total field. These rays are all parallel and
focus at infinity. This focusing action accounts for the
increased signal level in these directions.
I n this application of the theory, the focusing was not taken into
account. In many applications, it is sufficient to determine the patterns to the degree shown in Fig. 5. However, if required, techniques are available to take the
focusing action into account [9].
v. IMPEDANCE OF A 3,lONOPOLE ABOVE A
FINITE
CIRCULAR GROUND
PLANE
Another application of the geometrical theory of diffraction in the realm of antenna theory is in the calculation of antenna impedance. 4 s a n example, impedance
-180 -140 -100 -60 -20
20
60 140
100
180
or the equivalent reflection is calculated for the same
PATTERN ANGLE, p (degrees)
configuration discussed previously. For this particular
Fig. 3. Far-field diffracted-signal patterns of a straightedge excited problem, reflection concepts are employed since they are
by a magnetic line source.
more compatible with the geometrical theory. The calculation of reflection is converted from three dimensions
Figure 3 shows the amplitude and phase patterns of to the calculationof the reflection for a two-dimensional
the radiation pattern as indicated by
(1), (2), and (4) antenna. Thisproblem is then treated as thespecial case
for the case where p'=3X. Note the peak in the ampli- of a two-dimensionalradarantennawiththeedges
tude pattern and the discontinuity in the phase pattern scattering energy back to the antenna.
at the shadow boundary.As is expected for this problem, Figure 6 shows the ray geometryin the near neighborthe discontinuity in the phase pattern of the diffracted hood of straight and circular edges excited, respectively,
by line and point
sources. The rays to the right are
signal is precisely what is required to counteract the
amplitude
discontinuity
in
the
direct
signal (geo- directly excited by the sources and those to the left are
diffracted by the edges. The ratio of dB to dB represents
metrical optics field).
the ratio of power density incident at the edge to the
Iv. RADIATION
PATTERN O F A fi,lOHOPOLE ABOVE X
power density of the diffracted wave and is directly reFINITECIRCULAR
GROUND
PLANE
lated to the two-dimensional radar cross section of the
The calculation of the radiation pattern for the con- edges. According to the geometrical theory of diffraction, this ratio is approximately the same for the two
figuration under consideration can
now be completed
using the notionsof the geometrical theoryof diffraction. cases being considered, provided that Kp'>>l.
43
r
EPUIPHASESURFACEFOR
t
/
SOURCE EXCITED RAYS
NOTE:PATTERNSAREPOLARPLOTS.
Fig. 4.
20
0
1
EDGE
Patterns and relativeorientationand
location of three
sources contributing to the totalfield.
PATTERN ANGLE,
,
1
40
1
1
60
1
1
IT
80
#
1
1
120 1 6 0 1 4 0
I
7
'
EOUIPHASE SURFACE
\FOREOGEDIFFRACED
WAVE
EDGE-DIFFRACTED RAYS
8 {degrees)
I
SOURCE
EDGE-DIFFRACTED RAYS
I
180
I
I
I
I
EPUIPHASE SURFACE FOR
SOURCE EXCITED WAVE
-
MEASURED
CALCULATED
CIRCULARGROUND PLnNE RADIUS ($1 = 3 i
POINT SOURCE
FOR EDGE-DIFFRKTED
WAVE
Fig. 6. Ray geometryfor straight and circular edges in the near
neighborhood of the edges.
Fig. 5. Measured and calculatedradiation patterns of a quarterwave monopole operating over a finite circular ground plane.
The voltage ratio of received to transmitted signals is
given by the square root of the power ratio and by the
introduction of a phase factor.
T h e monopole is next considered to act as a transmitting antenna which excites a uniform radial wave in
The constant phase factor, n j 2 radians, is the residual
the azimuth plane that diverges in the elevation plane.
phase of a 3~/4-radian advance, introduced
at the edges
The circular edge acts as a target which reflects a unibecause of diffraction, and a delay of a/4 radians resultform radial wave in the azimuth plane that also diverges
ing from thefocusing of the diffracted rays as they conin the elevation plane as itconverges on the monopole.
verge to the origin. (This latter phase factor is identical
Consequently, because of circular symmetry, only dito the difference t h a t exists between radial-waveguide
vergence in one plane has
to be considered in deterreflectioncoefficientsdefined
at the origin and a t a
mining the power ratio of the received to transmitted
large radius.)
signals. From antenna theory, this ratio is given by
T h e reflection of the disk-monopole antenna at its
PR
G X
1 GX
input
port may now be calculated by adding the per- - -- - - .
(7) turbation introduced by the edge of the ground plane
PT
2irp'
2ir
2irpf 2
to the reflection of a monopole operating in a n infinite
G/2np' =factor which converts transmitted power t o ground plane. For the case of the infinite ground plane,
power density a t t h e edge (beam divergence the equivalent network is shown inFig. 7(a) ; it consists
in the elevation plane and antenna gain).
of a transmissionline of characteristicimpedance 20
X / 2 a = two-dimensional radar cross section of the feeding the antenna input port. The
reflection and transedges. (Thisfactor
is determinedbythe
mission coefficients at the antenna port for the infinite
rigorous solutionof the straightedge problem case are given by
[SI, with the condition that kp'>>l.)
1 / 2 a p f =factor which converts edge-scattered energy
to power density at the antenna(divergence
in elevation plane).
G x / 2 =effective height of the source antenna.
Combining terms, (7) becomes
PII
PT
Figure 7(b) shows the equivalent network for the case
V R is an equivalent
voltage source representing the contribution
of the edge
of a finite circular ground plane.
44
IEEE TR.LUSACl'IOKS O N AKTENhTAS APiD PROPAGATION
JAXXJAFLY, 1966
VOL. 0 - 1 4 , NO. 1
Storer's result has been derived as an input impedance
and is given by
z = z,+ 2,
TRANSMISSION LINE WITH
CHARACTERISTIC IMPEDANCE
where 2, is the impedance of a monopole operating in
a n infinite ground plane and
(a)
-vi
The agreementbetween the resultsof this present paper
and Storer's may beshown by converting the Storer result to the following reflection coefficient:
Fig. 7.
Networks for a n antenna operating
and finite ground planes.
011
infinite
r
reflection. Theantenna reflectioncoefficient
is now
given by the sum of the infinite ground plane andedgeinduced coefficients.
r = r,
+ r,.
=
r,
+ r:.
',I is the edge induced coefficient according to Storer.
Kote the r, are identicalfor the twocases. For the case
of a quarter-wave monopole,
(11)
I?, is the portion of the edge reflected signal received by
IASo
I(x)
=
the antenna relative to the incident
signal. I t correZ, = 36.59
sponds to the first reflection off the edge and, if the
G = 1.64.
antenna-to-edgedistanceislargeinterms
of wavelengths, i t is the dominant edgeeffect. If the distance is I n order to makea comparison between the two results,
small, multiple edge reflections have to be taken into
the ratio of r,, derived in this paper to I'J, Storer's reaccount, but these are neglected in this case. Consider- sult, is determined to be
ing the antenna as operating in the receive mode, the
re 2Z,G
received signal from the edge relative to the incident is
-=-- 1.13.
given by
r: &60
The ratio is 1.13 n-here perfect agreement would have
been unity. Considering thedifference in approach, this
is reasonably good agreement. The phase is in exact
From (8), solving for V Rand dividing both sides bl: .[Ii, agreement.
Here again, it should be emphasized
that this particuVR
VT
G h
lar exampleof a monopole abovea circular ground plane
4
; I -e x p j
- kp').
(13)
vi
vi
2irp 2n
was chosen for convenience and because a check was
available in the literature. Actually, the concepts
of twoIf (10) is substituted for V T / V ~ ,
dimensional antennas are not as widelyemployed as
thoseforthetypicalthree-dimensionalantennas.In
cases Ivhere ground planes with straightedges are employed, the geometricaltheory of diffraction canbe
of antenna
Finally, substituting this result for VRIT'; in (12), the applied in conjunction with the usual notions
gain
and
effective
area.
For
example,
for
the
case of a
edge reflection coefficient is
monopole centered on a square ground plane, the dominant edge-inducedreflectioncomponentcanbecon2zz,
G
re =
expj($ - 2kp').
(15) sidered as caused by four point scatterers located
in the
(2,
Zd2 d
&'
plane of the ground plane, on lines perpendicular to the
Equation (15) n:aJ- now be substituted in (11) for the edges and passing through the monopole, and at twice
the edge distance from themonopole. For this case, the
desired result.
This result may be shown to
be in good agreement effective area of the monopole is used in the calculation
\\-ith the results derived by
J. E. Storer from the solution andno focusing of the diffractedraysneeds
t o be
of an integral equation ([6], also summarized in [ 5 ] ) . considered.
:(
+
IEEE TFL4XS.4CTIOX.5 ON A
N
T
E
N
N
%
. S
ANJJ PROPdGATION
VI.CONCLUSION
duced the writer to the
geometrical theory of diffraction.
The
writer
is
also
indebted
t o N’heeler Laboratories,
The objective of this paper has been to demonstrate
Inc.,
and
its
staff
for
its
cooperation
and encouragement
that the geometricaltheory of diffractionprovidesa
in
the
completion
of
this
work;
in
particular,
several dissimple technique for calculating the radiation charactercussions
with
H.
137.
Redlien
were
very
helpful.
The
istics andimpedance of complicatedradiatingstrucexperimental
work
described
herein
and
preparation
of
tures.In essence, i t resolves a difficultproblem into
this
paper
were
done
at
these
Laboratories.
component problems. The characteristics of the component problems are usually obtained from the literature or can be calculated and then combined to deterREFERENCES
mine the characteristics of the initial problem.
J. B. Keller, “The geometrical theory of diffraction,” 1953 Proc.
For the radiation problem considered in this paper,
Symp. on Microwave Optics, Eaton Electronics Research Lab.,
McGill University, Montreal, Canada.
the basic components were the diffraction by a straightJ. E. Storer, “The radiation pattern of an antenna over a circular
edge and the radiation by a
monopolein an infinite
ground screen,”J . Appl. Phys., vol. 23, p. 588, May 1952.
A. R. Lopez, “Diffracted fields of a triangular rod excited by a
ground plane. A simple array calculation was used t o
non-uniform cylindrical wave,“ Report required for the MSEE
combine the two components. For the impedance probdegree a t the Polq-technic Institute of Brooklyn, N. Y., 1963.
R. W . P. King and T. T. \$Tu, The Scattering awd Diffvection of
lem, the components were diffraction by a straightedge
Wanes, Cambridge, Mass.: Harvard U.P., 1959, p. 84. (Discusand the impedanceof the monopole in a n infinite ground
sion of the geometrical theory of diffraction.)
Y. Ohba, “On the radiation pattern of a corner reflector finite in
plane. Radar principles were used to combine the two
width,” IEEE Trans. on Antenmzas and Propagation, vol. AP-11,
components to obtain the total impedance of the conpp. 127-132, March 1963. (Geometrical theory of diffraction
applied to a corner reflector.)
figuration considered.
J. E. Storer, “The impedance of an antenna over a large circular
screen,” J. Appl. Phys., vol. 22, p. 1058, August 1951.
XCKNOWLEDGMENT
T h e work reported in this paper
was stimulated by
work on a report project required for the master’s degree
atthePolytechnicInstitute
of Brooklyn, S. Y. Dr.
L. B. Felsen was the advisor for this project and intro-
H. Jasik, Antenna Engineering Handbook. New York: McGrawHill, 1961, pp. 3-21.
3.1. Born and E. i1701f,Piimzciples of Optics. New York: Pergamon,
1959, pp. 577-2231.
J. B. Keller, Diffraction by an aperture,” J . AppI. Phys., vol.
28, p. 435. (Focusing effect of parallel rays considered in this
paper.)
The SignalProduced in a Monopole Antenna by theGamma Flux
from a Nuclear Explosion
Absfract-A monopole antenna is considered to bebombarded by
a flux of gamma rays propagating normally to theaxk of the antenna.
As a result of this bombardment, electrons are scattered from the
antenna, thuscharging the antenna positively.This induces a current
in the load impedance of the antenna-appearing as if the antenna
werereceiving a transient RF signal. An expression for thetime
dependent voltage across the load impedance of the antenna is obtained by consideringthat every electron scatteredfrom the antenna
never returns and that the antenna impedance and capacitance are
their free-space values.
Manuscript received July 1, 1965; revised August 19, 1965. Portions of this work are abstracted from a doctoral dissertation
by
C. D. Taylor, carried out under the supervision of R. H. Duncan a t
the New Mexico State University. This work was supported by the
United States Atomic Energy Commission.
C. D. Taylor is with the Sandia Corporation, Albuquerque, N.
Mex.
R. H. Duncan is at theNew Mexico State University, University
Park, N. Mex.
45
IKTRODCCTION
A
CCORDING to experimental observations,
a transient signal is induced in a receiving antenna, or
probe, by the high-density gamma flux emitted
from a nuclear explosion. \lThen an object is irradiated
by this flux, i t becomes positively charged because of
electronsbeingscatteredfrom
itbyvirtue
of the
Compton effect. -An attempt is made in this paper to
explain the transient signal produced in an irradiated
antenna by considering the temporal charging of the
antenna to be exciting the receiving circuit.
An exact treatment of the interaction of gamma rays
with an antenna is not feasible. The electromagnetic
field generated by the interaction is composed
of two
parts: that attributable to the antenna and thatgenerated by the space current of scattered electrons. How-