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This is to certify that the

thesis entitled

THE RADIATION PATTERH 0F CIRCULAR APER'I'URES

presented bg

Robert Edgar Houston. Jr.

has been accepted towards fulfillment
of the requirements for

“"05- degree in Pb 1C.

W19; x4065.

Major professor

Date 18 September 19‘1

0-169

 

 

 

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1/98 c/CIRC/DateDue.p65-p.14

THE RADIATION PATTERN OF
CIRCULAR APERTURES

by

Robert Edgar Houston Jr.

A Thesis

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Physics

1951

ACKNOWLEDGNENT

A problem such as this requires the help of
many persons. To this end I wish to thank Dr. R. D.
Spence, Mr. Lee Bell, and my wife Barbara Houston.
Special appreciation is due Dr. Robert H. Noble for
his invaluable suggestions and direction toward its

We. (W.

successful completion.

288404

I.
II.
III.

IV.

TABLE OF CONTENTS

INTRODUCTION . .

THEORY . . . .

EXPERIEENTAL PROCEDURE ADD EQUIPMENT

RESULTS AND INTERPRETATION

PAGE 1
PAGE 5
PAGE 5

PAGE 21

I. IETRCDUCTION

Any discussion of the pattern created by an
incident plane wave on a circular aperture immediately
brings into focus three cases of particular interest.
Kirchoff's integral and boundary conditions enable a
prediction of the general aspects of the pattern pro—
vided the radius of the aperture is large compared to
the wave—length of the incident wave.1 Should the
radius of the aperture be small in comparison with the
wave-length of the incident wave,the general features
of the pattern may be explained by the Rayleigh-Lamb
method.2

It is well known, however, that neither of these
methods pretends to give the exact solution when the
wave-length is of the same order of magnitude as the
radius of the aperture. Recently Bethes has develOped
a theory which may be extended to cover this case.
Bethe's theory has the advantage of being vectorial in
character. Since an electromagnetic field is essentially
vectorial this would seem to suit the conditions of the
problem more closely. The theory depends upon the use
of fictitious magnetic charges and currents in the
diffracting aperture. The conclusion drawn clearly
shows that the field near the edge of the aperture will

tend toward infinity. This is the problem to be

investigated experimentally.

The overall approach to the problem was to
construct equipment which would allow an electro-
magnetic field detection probe to be small in
comparison with the wave-length and yet retain definite
limits to the physical size of the apparatus.

Andrews4:5 has made a very through investigation
of the diffraction pattern of apertures in a manner
similar to the method used in this paper. However,
he made no close study of the edge effects of the

aperture.

II. THEORY

DevelOpment of electromagnetic radiation in
the microwave region led to theoretical considerations
of the effect of a small hole in a cavity upon the
oscillation of that cavity. The solution of this
problem is valid for a circular aperture in a perfectly
conducting infinite plane screen. The theory was
develOped for the case of a hole small in comparison
with the wave-length. Extension of the theory to the
condition where the wave-length and radius of the
aperture are the same order of magnitude has not been
completed. However, the type of solution to be
expected is indicated. .

Previous theoretical methodsl’2 in the treatment
of diffraction phenomenon are inadequate in a problem
involving electromagnetic radiation. This is immediately
apparent since an electromagnetic field is essentially
vectorial in character while the previous theories were
scalar in form. A theory involving a vector formulation
has the distinct advantage of fulfilling the
divergence conditions:

cflW’ £?:= ¢div 23"CD (I)
This is identical with saying that the vectorial theory
gives transverse waves in the wave zone which would not
necessarily be the case in the scalar form.

In his approach to the solution Bethe finds

an expression which s tisifies haxwell's equations as
well as certain of the boundary conditions. To fulfill
the remainder of the boundary conditions and to allow
for the simplest solution the concept of fictitious
magnetic surface charge and density in the aperture

was introduced. The magnetic and electric fields in

the aperture are then given by the expressions

W ’

’ 7" T 2 = . ""‘* )7 3)
EQ'CPTuf-g : 5: (mo/LEO + 7(d-r) nx . (

where n is the unit vector in the direction of the
inward normal to the surface, a is the radius of the
aperture, E0 and HO are known field components, and r

.1

is the varying radius of the aperture. h and H0 are

0
usually of the same order of magnitude and consequently
as r-+ a, Eaperture approaches infinity. It is quite
apparent that Hapertupe approaches infinity as:r-+2L
Hence it is to be expected that at the edge of an
aperture the field will tend toward infinity. Such a
result is shown experimentally in this dissertation.
The theory has not been extended to include
apertures of the order of magnitude of the wave-length.
However, the method of solution has been indicated and

.1. .
the (a2 - r2)'2 dependency remains as an integral

part of the solution.

III. EXPERIMENTAL PROCEDURE AND EQUIPKENT.

The experimental equipment consisted of a source
of electromagnetic radiation, a cylindrical parabolic
reflector, a plane conducting screen containing the
aperture, and a detection system.

The circuit diagram for the source of electro-
magnetic radiation is shown in figure 1. The power
supply contained a rectifier tube and two 'L' filter
sections. 'This reduced the ripple in the output voltage
to a satisfactory level. The Operating values were:
input voltage, 110 volts; oscillator plate current,

65 milliamps at 250 volts; and oscillator grid current,
8 milliamps. The oscillator and power supply are
shown in figure 2.

In the oscillator circuit there were two
Westinghouse 568A.S. door-knob tubes in a push-pull
arrangement with tuned grid and tuned plate circuits.
A push-pull oscillator produced more output than could
have been attained by use of the same tubes in a
parallel arrangement. Tuning of plate and grid tank
circuits resulted in an additional increase in power
output at the desired frequency.

The wave-length of the oscillator was set for
50 cm. This wave-length was chosen because it

represents a compromise between the two conditions

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which must be satisified. Certainly the equipment
could not be so large that moving and handling could
not be accomplished easily by a few persons. Secondly,
to insure meaningful results the detector of the probe
circuit had to be small compared to the wave-length.
This insured that the effect upon the radiation pattern
would be negligible. A wave meter was utilized to
check the frequency and from the well known relation
ht = c. (‘0
the wave-length was determined. A system of standing
waves could also be utilized in a wave-length calibration.

The oscillator output was connected by coaxial
cable to a half-wave dipole antenna located at the
center of the parabolic reflector. The coaxial cable
ran through a brass tube one-half inch in diameter
which supported the antenna.

Radiation from the antenna alone was not
sufficiently powerful to be detected with the available
equipment. Hence it was necessary to construct a
reflector to direct most of the energy to the conducting
screen as well as produce a plane wave. Due to the
radiation pattern of a dipole antenna a cylindrical
parabolic reflector presented no more distortion than
that of a paraboloidal reflector. This choice also
enabled a greatly simplified construction. A View of

the reflector is shown in figures 5 and d.

(O

 

Figure 5

10

A. .Hdru. “FL.

 

11

The reflector size was dictated by the fact that energy

c_;
loss is great if the reflector dimensions are less
than four times the wave—length.

The parabolic curve was plotted from the polar

equation

 

7:
//9= /—cao<+' (59

where /0 is the distance from the focus to the plotted
point, p is the distance from the directrix to the
focus, and 9 is the angle formed by the line perpendicular
to the directrix and passing through the focus with the
line along which/f7is measured. The surface of the
parabolic cylinder was one-sixteenth inch aluminum
fitted to the parabolic wooden frame. Details of
construction of the frame can be seen in figure 5.

One half-wave dipole constructed by the author
and a commercially built antenna were utilized in the
experiment. Due to matching problems between the
Oscillator output and the dipole more power was available
from the commercial antenna. Consequently, the
commercial antenna was used exclusively to obtain the
final-data although the shape of the patterns obtained
varied only a few percent when the results of each
antenna were compared.

The frame of the conducting screen, vhich contained
the aperture, was formed by four pieces of lumber

" ll ' C O O O
1 x2 x 12 and some diagonal bra01ng as shown in figure 6a.

 

 

.14

The hinged section allowed for removal from.the place
of construction. Binder twine was run around the
edge of the frame to add stability and to provide
support for the covering. The center of the frame
contained a composition material %"x4'x4' with a hole
in the center slightly larger than 100 cm. in diameter.
Hence the covering could be given rigidity around the
aperture and assurance that the aperture was planar
resulted. A schematic diagram of this is shown in
figure 6b.

The covering of the frame consisted of strips
of ordinary wrapping paper glued together to form a
single sheet about 12'x12'. Then strips of aluminum
foil were laid on this with one-half inch overlapping
joints perpendicular to the joints of the wrapping
paper. The aluminum foil was overlapped to provide
a coupling between the strips. The entire surface
was rolled smooth for two reasons. The first was to
reduce interference in the aperture due to reflection
from surface unevenness and the second to try and make
contact between adjacent strips of aluminum foil. It
wasn't absolutely necessary to make contact due to the
overlap of the strips of foil. The overlap allows a
capacitative coupling between each particular foil
strip and consequently the current induced in one strip

would not be restricted but could flow to any other

section Of the screen. Capacitative coupling has a

lfluc dependency wherein is 21f multiplied by the frequency
Of oscillation and c is the capacitance. Obviously at

a frequency of 600 megacycles/second coupling losses

were very small. The screen is shown in figure 7.

The different size apertures were obtained by
placing a one-sixteenth inch sheet Of aluminum.with the
desired size aperture over the 100 cm. aperture.in the
screen. Obviously this thickness does not approach
the desired 'infinitely thin' category to the extent Of
the aluminum foil. However, compared to the wave-length
it provides a good approximation of an infinitely thin
screen.

One serious difficulty occurred with the larger
of these apertures. They were constructed Of a soft
aluminum sheeting which had been rolled rather tightly.
Despite every effort they resisted becoming exactly a plane
but deviated from this ideal by a few percent.

The detection system consisted Of two parts;

(1) the probe circuit and (2) the physical means of
moving the probe in the plane Of the aperture. The
antenna Of the probe circuit was a lNBSA crystal.

This was connected by a twisted pair Of fine #56 wires
to a galvanometer as in figure 8. Part (2), also
shown in figure 8, consisted of an Optical bench with

a l"x2"x 8' piece of wood mounted vertically on two

 

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18

bench slides. This gave transverse movement. A second
piece Of wood with a l"x2" hole would slide up and down
the vertical pole, but with a small amount Of torque
applied to the block it held its position. A three
foot dowel with the crystal mounted on one end was
inserted in the block Of wood. The dowel provided
the necessary torque to hold the block of wood in any
set position. This allowed an easily adjustable vertical
motion. The dowel also supported the twisted pair Of
wires in the direction of propogation. In this position
a minimum disturbance Of the field was Obtained.

The experiment was performed in the loft of the
Sheep Barn on the Michigan State College Campus. This
site was chosen because it Offered several distinct
advantages. There was no steel framework in the
structure to distort or cause reflections of the
electromagnetic radiation. A distance of 68 feet
between the screen and reflector could be obtained
still leaving sufficient work Space. When the loft
doors were Open there were no structures behind the
aperture to cause reflections into the aperture and
invalidate the results. The apparatus was completely
protected so that during inclement weather the apparatus
did not have to be taken down with the resulting problem
of realignment before data could be again taken. Certain

aspects were negative but these will be mentioned later.

19

The screen was erected and light from the aperture
was allowed to fall on the reflector. By this means
the antenna could be located exactly on the focal line
of the reflector. The screen was checked to make
certain the plane of the aperture was parallel to the
antenna. A plumb bob was used to check the vertical
plane of the conducting screen. Preliminary readings
were obtained to ascertain the symmetries of the
patterns. It was extremely poor. After many hours Of
checking and rechecking it was discovered that a local
radio station with a nearby transmitter was affecting
the pattern. Consequently, subsequent readings were
taken as a difference between the readings with the
600 megacycle/second oscillator Off and then on. The
exceptions to this occurred when the radio station
was not in Operation as evidenced by a zero effect
in the aperture.

Since automatic recording devices were not
utilized it was necessary to take frequent successive
readings to approximate a continuous pattern. The
effect which the experiment was designed to investigate
occurs very close to the edge Of the aperture. Hence
the carriage on the optical bench was moved only two
millimeters between readings near the edge. This
continued for five or six readings when an interval

Of one cm. or one-half cm. was chosen. Then the probe

20

had traversed the entire aperture the crystal was then
raised or lowered depending upon which chord or diameter

Of the aperture was to be plotted.

21

IV. RESULTS AND INTERPRETATION

As has been stated previously the experiment
reported in this dissertation was performed in the
Sheep Barn. In addition to the advantages previously
listed there were certain drawbacks. Possibly the
most serious was the hay sling track suspended from
the roof rafters as may be seen in figure 5. Every
effort was made to insure that the pattern was
symmetrical in all respects. However, it will be noted
that the values on the right side Of the aperture were
always slightly lower than on the left. Interference
from the hay sling track may have been a contributing
factor to this asymmetry.

The terms electric and magnetic diameters have
been used at various points in this paper. Due to the
construction of the reflector it was only possible to
orient the antenna horizontally. The electric field
lines, which run from one end of the antenna to the
other, may be represented by one vector oriented as
is the antenna. Thus, the electric diameter is the
horizontal diameter of the aperture. The magnetic
field lines, which encircle the antenna in a vertical
plane, may be represented by a vector perpendicular
to the antenna in a vertical direction. Hence, the

magnetic diameter is the vertical diameter Of the

aperture.

The three dimensional plots shown in figure 9
allows an insight into the form of the pattern. In all
four diagrams the important item to notice is the sharp
rise near the edge of the aperture. This is the effect
which Bethe has predicted.

Certain other aSpects also bear investigation.
All of the small apertures, i.e., diameters of 12.5 cm.,
15.9 cm., and 25 cm., have essentially the same form
as shown in part a of figure 9. The value along the
electric diameter drOps only slightly from the maximum
value occurring at the center until it approaches the
aperture edge. However, along the magnetic diameter
the value drops until approaching the edge where it
again takes a sharp turn upward.

For the next two apertures, i.e., diameters of
57.7 cm. and 47.7 cm., the maximum value in the center
drOps slightly and there is a greater proportional drOp
along the electric diameter. The magnetic diameter
falls off more than the electric diameter as previously.
One other investigation has obtained the identical
result.5

The fourth diagram (d) in figure 9 shows the
pattern for the largest aperture. The Value at the

center of the aperture'is low and there is a distinct

25

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24

maximum in both the electric and magnetic diameters.
This can be seen more clearly in figure 10. Here the
electric and magnetic diameters for the 100 cm. aperture
are shown in their entirety. With the exception of the
end effects these very closely resemble the results
obtained by Andrews.5 Here again the magnetic diameter
falls to a lower value near the edge of the aperture
than does the electric diameter.

Figure 11 indicates the method by which the three
dimensional plots were made. Chords parallel to the
electric diameter were chosen. These were then plotted.
Upon completion of these plots the magnetic diameter
was then plotted. The plot of the magnetic diameter
was useful in that a check on the values at the center
and each of the chord intersections was obtained.

Figure 12 shows the values obtained for the
15.9 cm. aperture. As mentioned previously the smallest
three apertures had the same pattern except that the
maximum in the centers varied. Again the value of the
magnetic diameter is lower near the edge than it is on
the electric diameter.

The curve shown in figure 15 is a plot of the
radiation intensity per unit of aperture area against
the aperture diameter. Unfortunately, due to conditions

beyond the control of the author, it was possible to

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obtain only the values of the electric dia“cter of

the 75 cm. aperture. Consequently, the area under

the electric diameter curve was used since this was

the only curve common to all apertures. To be complete
and accurate the volume under the entire radiation
pattern should be determined. It was thus assumed

that the area under the electric diameter would give
approximate results since the pattern was similar in
most cases. The greatest error will occur in the two
largest apertures.

To obtain the ordinate values in figure 15 all
the electric diameters were plotted on graph paper
with the same scale. The squares under the curves
were then counted. The total number of squares for
each aperture were then divided by the actual area
of the aperture. The results of this division were
then divided by the largest of these values to make
comparison easier. The predicted maximum should occur
in the aperture whose diameter is one—half the wave-
length. This is clearly shown.

The choice of aperture diameter size is anything
but consistent. Five of them are fractions or multiples
of the wave-length, i.e., 12.5 cm., 25 cm., 57.5 cm.,
75 cm., and 100 cm. Two apertures were chosen to

determine if there was similarity between the acoustic

29

and electromagnetic cases. For the we apertures
15.9 cm. (e=*a=-2—A—”—d=l) and 47.7 cm. (6:3), the
electric diameters resemble the figures shown by Spencel
for the corresponding 6.

As has been previously been indicated in

section II one expects a (a2 - r2)

-% dependency.
Obviously, for small r the expression will be negligible
compared to contributions from other terms. However,
as r —)a.the entire result is dependent upon this factor.
Probe position was difficult to determine in a
precise manner. Horizontal movement was accurate to
:tO.5 mm. since the entire probe moved along the Optical
bench which had a scale on it. Hewever, when the probe
was moved in a vertical direction the accuracy was
reduced since the interval variation was 1:1:mm. For
the two millimeter intervals this is quite large, but
corpared to the aperture diameters it is not too great.
The crystal occupied some volume instead of being a
point and consequently the center of the crystal was
the point from which the measurements were made. Since
the crystal was two cm. in length the res ings started
11 mm. from the aperture edge. With an antenna of this
length the possibility exists that the apparatus was
incapable of detecting effects which might have occurred
in intervals of less than two cm.

The frequency of oscillation was determined by

50

a frequency meter which was accurate to i?%%. This

means that the wave-length was 50:: 0.25 cm. The radii
of the smaller apertures were accurate within one percent
while for the larger apertures the error is one-half
percent.

One other construction error apreared and remained
constant throughout the ex: riment. This was the
deviation of the reflector from a true parabolic curve.
At the edges the deviation was 2fi while close to the
center it was about 1%. As enumerated so far the sources
of these errors act independently of one another.
Consequently they may or may not add to give a larger
error.

There exists an error which must be considered
at all times however. With the conducting screen removed
the field varied 10% over the area in which the apertures
were placed. This may have been caused by the preceding
errors or it may have been due to turning the oscillator
on and off between the readings.

Theoretical values for the (a2 - r2)-% dependency
were cemputed and compared with the values obtained in
the experiment. The variation was 10% or less. To
procure a more accurate description of the curves
automatic recording devices could be used. However,
the percentages are within the experimental error and

the effect which Bethe predicted has been shown to exist.

 

31

REFERENCES

lSpence, R. D., "A Note on the Kirchoff Approximation
in Diffraction Theory", J. Acous. Soc. Am., g; (1949),
pp. 98-100.

2
Lamb, H., Hydrodynamics, Dover Publications, New York,
- 1945, p. 5170

 

5
Bethe, H. A., "Theory of Diffraction by Small Holes",
Phys. Rev., 66 (1944), p. 165.

4Andrew, C. L., "Diffraction Pattern of a Circular
Aperture at Short Distances", Phys. Rev., Zl_(1947),
p. 777.

5Andrews, C. L., "Diffraction Pattern in a Circular
Aperture Measured in the Microwave Region", Jour. App.
PhySo’ a (1950), p. 76]..

Radar Electronic Fundamentals, war Dept. Tech. Manual
TM 11-466, 29 June 1944.

 

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