2222

NH

 

__THS

MEASUREMENT QE DIELECTRIC CONSTANTS
BY 2%223222N 53 02‘" A 2222 EC“ .2-‘2‘2252’ ‘22:? £223 22252522

T212515 17092212 9292* 29 :32 M. S
IC’I‘GH 3:52. 2.7322223?

Yen Fu Bow
1956

MICHIGAN IIIIIIIIIIIIIIIIIIIIIII

lHlIlllllli!llllHlllllllllllHlllilllllllHllmlllIHHlllI

3 1293 01704 0076 2

PLACE IN RETURN BOX
to remove this checkout from your record.
TO AVOID FINES return on or before date due.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ma W14

EBA UQQ"““ “Q B "T“"QQIC ”CIICMRVQQ BY MEANS

LL‘xu' VJ. LLqu. 1 u .L 1. LU

OF A IQQDnAV IJCIITZ7CICT27

by
Yen Fu ng

AS AIDSTRACT
Submitted to the College of Science ani Arts
Hichigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

MASTER OF SCIENCE

Department of Physics and Astronomy
1956

APPROVED

1
YEN FU BOW ABSTRACT

A simple Fabry-Perot interferometer in the microwave
region is discussed. By means of this interferometer, the
ratio of dielectric constants of polystyrene and teflon has
been determined as 0.793, which is comparable with the ac-

cepted value 0.785.

meeetazxeme or DIELECIRIC conseexrs BY meens

or A EICRCFA E Irenernzoxaes.

by

Yen Fu Bow

A THESIS
Submitted to the College of Science and Arts
Michigan State University Of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

EASTER or so 3303

Department of Physics and Astronomy

1956

I.
II.

III.
IV.
V.

TABLE OF CONTENTS

INTRODUCTICI . . . .

O
O
O
O
O
O
O
O
O
O
O
C
I

THEORETICAL CCHS DERATICNS . . . . . . . . . .
(1) Transmitted Intensity and Sharpness of
Fringe: . . . . . . . . . . . . . .

(11) Development of Reflectors . . . . . .
(iii) Diffraction Consideration . . . . . .
DESIGN AND CPERATICN OF THE INTERFERCEETER . .

EEASURTMEHT OF.DIELECTRIC CDNCTANTS . . . . .

DISCUSSION . . . . . . . . . . . . . . . . . .

BIBLIOGMPHY O O O O O O O O O O O O I O C O I O I O

\O\1#\fl

15
17
18

‘

 

 

LIST 0? FIG 333

FIGURE

1.

2.
3.
4.
5.

Arrangement for calculation of reflection co-
efficients . . . . . . . . . . . . . . . .
Schematic diagram of the interferometer . . .
Structure ani dimensions of the reflector . .
Dimensions of the receiving horn . . . . . .
Showing sharpness of fringes on the inter-

ferometer in relation to the wavelength . .

PAGE

10
11
12

14

ACKflfiwLEDGMENT

The author wishes here to express his sincere grati-
tude to Dr. J. A. Cowen who guided the whole work and gave

encouragement constantly.

 

I. INTRODUCTION

The measurement of dielectric constants in the micro—
wave region is of interest because information about the
electric polarizability and relaxation times can be ob-
tained, which, in turn, leads to a better understanding of
liquids and solids. If one defines a complex dielectric
constant 6 I E'- 36: then the usual dielectric constant is
. k a 576., where 6,,is the permitivity of vacuum and the loss
angle 5 is defined by tans - éu/E'. Thus in order to speci-
fy the dielectric preperties, one should have values of k
and the loss tangent.

In the measurement of dielectric constants in the
microwave region, both waveguide and freeSpace techniques
are applicable. Along with the usual shorted line and reso-
nant cavity methods, one may also use microwave interferome-
ters based on optical techniques. The principles of the
Fabry-Perot and Hichelson interferometers have been adapted

(1.2.3)

by several authors to these measurements. In order
to make reliable absolute determinations of the complex die-
lectric constant, carefully designed instruments with well
collimated microwave beams and accurately dimensioned re-

flectors are necessary. Culshaw, in particular, has cares

fully analyzed the Fabry-Perot interferometer from a theo-

retical standpoint and has designed an instrument for use
at 8 mm. wavelengths which gives results with less than one
percent deviation.

The purpose of the present investigation is to test
the possibility of determining relative values of the die—
lectric constant with a somewhat less carefully designed in-

strument.

II. THECRTTICAL CGHSIDERATICES

The detail theoretical deveIOpment of the Fabry-
perot interferometer in the microwave region can be found
in the paper presented by Culshaw.(3) A brief review with
certain clarifications is given here.

(i) Transmitted Intensity and Sharpness
of Fringes

The operation of this interferometer is due to mul—
tiple reflections between two surfaces or films. Consider
incident plane waves, and let P and R be the transmission
and reflection coefficients of intensity of the films; then
the resultant intensity transmitted through both films is
' (4)

given by the wellknown Airy formula .

P2

T a
1-23 Cos§+ R2

(1)

where é - (2T7/Js.)2}lt CosG is the phase difference between

consecutive beams, t is the distance apart of the films,}1
the refractive index of the medium between the films, and 9
the angle of incidence on the films. For maximum transmit-

ted intensity coed! - 1 or

2ft cose- nA (2)

where n is the order of interference, and

2
P
T - ~27... . (5)
For zero loss in the films, P - l-R, then Tmax becomes uni-
ty.
By equation (1), the phase difference between points

of half maximum intensity either side of resonance can be

found as

A - 2951 (4)

where c08431 - {23 - (l—R)2}'/23. Defining the Q—value
measured on the interferometer as the wavelength divided by
.Zflagives
2.1T ,
e - air/A - "W1 (5)

Sinceé1 is a decreasing function of R, thus it is clear
that to obtain sharp fringes reflectors of high reflectivity
are required. Actually, this is the very important fact
which led to the construction of the FebryaPerot interferome-
ter in the Optical region, as pointed out by Boulouch in

1906.(“)

(ii) DevelOpment of Reflectors
If the electric field in a plane wave traveling in
the x-direction is given by the real part of E - Eoexp(nwt47iL
where 7"- oi + 3? is the proPagation constant, and if Rn+l,n
is the amplitude reflection coefficient at the boundary be-

tween infinite media n and n—1 (see Fig. 1), it can be
shown by the method of multiple reflections, or by impedance
considerations, that the amplitude reflection coefficient

due to the layer of medium n is given bycs)

. Rn+l,n + [QXpC'237ndn)]Rngn-l
1 * Rn+l,an,n~l°xP<'237;dn)

A

 

n+1 (6)
where dn is the thickness of the layer n. This is the fun-
damental formula for the transfer of reflection coefficients
through dielectric layers.

Applying equation (6) to determine the reflection
from a quarter wave sheet of dielectric bounded on either
side by free spaCe, and changing the notation so thatA1

refer: to one sheet, gives
' 2
A, - 301m?) / (“3017”) (7)

where 7}“- exp(- Ftan5’), tan 5’ - {3/4 , and ROI the ampli-

tude reflection coefficient at freespace-dielectric bounda-
I

ry. For no dielectric loss, tan.5 - 0, equation (7) reduces

to

A . 2R (1+agl) (a)

1 01/

Noting that 301 . l_:;£g; .(5) we have
1 + If? '

Medium (n+1) Medium m) Mediumm- n

 

1 a 't ' R '

flC‘ 8Y1 "M,“

WdVe " Re:
. 6-——- uni-I

Am.

 

 

Fig.1, Arrangement for Calculation

of reflection coefficients.

It is not difficult to prove by using equation (8) and
mathematical induction that

An . LL13; (10)

l oik

Thus, neglecting dielectric loss, the amplitude reflection
coefficient of n quarter wave dielectric sheets spaced quar-
ter wavelength in free space is equal to that of a simple
quarter wave sheet having a dielectric constant of k. It is
thus clear that the amplitude reflection coefficient in-
creases with the number of quarter wave sheets used. Equa-
tion (7) also indicates that the dielectric loss tends to

decrease the reflectivity.

(iii) Diffraction Consideration
Since the dimensions of the instrument are much

smaller in terms of the wavelength than those normally used
in the Optical region, a consideration of the diffraction
which inevitably occurs is very important. As pointed out
by Gooker and Clemmow,(6) the most useful approach to this
problem is to make use of the fact that the field at all
points in front of a plane aperture of any field distribu-
tion may be regarded as arising from the interference of
plane waves in various directions. The amplitude and phase
of these waves expressed as a function of their direction

of travel, constitute an angular spectrum of radiation

8
which, apprOpriately expressed, is the Fourier transform of
the aperture distribution. The theory develOped in (1)
shows that the reflectors will exert a selective action on
these plane waves of the angular spectra. From equations
(1) and (2), it can be shown that the angular widthe of the
spectrum transmitted between points of half maximum intensi-

ty is given by the following formula
c039 - l - l / 2n.“c (ll)

where n is the order of interference and Q is the Q—value
defined above. Thus the higher the Q-value and order, n,

of interference, the greater this selection, the process
being strictly analogous to what occurs in a resonant cavi-
ty which can propagate a number of modes. Consequently,

the diffraction effect may be neglected in an interferometer
with very high Qrvalue. Also because of diffraction effects
energy will be lost outside the reflector system, the amount
lost increasing with reflector separation. Thus the Q-
value should decrease as the separation of the reflectors

increases.

III. 351G- AND OPn.n 3! CF Ti: IN“"YTZQ“”ATER

A schematic diagram of the interferometer is shown in
Figure 2. It consists of the microwave source, reflectors
and detector. A 723A/B klystron generating radio frequency
at.a.wavelength of 3.2 cm. is used as the source. The
kl yetron is coupled di ectly to a short section of rectangu-
lar waveguide which is terminated in a circular plate.
This arrangement gives a fairly uniform phase across the
aperture plane ( E-plane). Each reflector is constructed
with four quarter-wave sheets of polystyrene spaced quarter-
wave length in air. Figure 5 indicates the construction
and dimensions of the reflectors. Polystyrene was used for
its low dielectric loss. Since high Q-value could not be
expected for this simple interferometer, in order to compen—
sate for the diffraction effect, the detecting systemnmsthmmwm
much directivity as conveniently possible; i.e., the re-
ceiving horn should not be too small. Several types of rec-
tangular horns were tried. The dimensions of the one which
was used are indicated in Figure 4.

Fundamentally, the operation of this interferometer
is the same as Eabry—Ierot interferometer in the optical re-
gion. Because of the multiple reflection between the reflec-

tors with high reflectivity, interference fringes can be ob-

Kyshon

(vzauir-a-T; — b

 

\\\\\\\\‘

l“
—

r
a
t
I
o
l
I
a
a

I \ \\ \ \ \ \ ‘ \‘
|\\\\\\\\\\\\‘

 

 

 

 

 

 

10

C775 fal
Detector

  

r1

      

I

t .
l

 

 

Paw"
50W

 

 

 

Orfic-F Semi wirfi SCaTe /

/
y

U
i
L
j

 

‘l

{ Audio- .
' Amplifier

3.....- _r.

*-—-—- .d

Fig. 2. Schematic diagram of the interferometer
(A 12 cm., B 46 cm.)

11

‘:

Hopomammn map mo mGOHmcmSHU dam manpoSHpm .m .mflm

33> uEm

353.3

:< K ¢\.

gosh—$2»

 

Stuomdy . "
&¢\.\\_

0

22> use“.

Tlurll .5 ON llnli

 

 

\
x
\
\ \
\
~\ \\ a
x V ~
‘ \ \ \
\ \ \. r .
.\ \
x

x . \
\ . s\ t \,
\\\ . \ \ ‘ . a x
x x \
\\ K \ \
\ \ \ \ \\

 

 

 
  

 

 

 

 

 

J. 4.

(L

 

 

Dimensions of the receiving horn
18.4 cm., B 15.8 cm.)

12

13
served for the transmitted radiation. However, since our
interferometer operates with esser tially parallel beam, the
circular fringe system of the ogticel model is not obt sined;
instead, the fringe is obsem iby elt ri;13 the flietenoe be-
tween the reflectors. fer oonVerie es, the reflector 3
(Figure 2) is fixei eel the reflectsr seferetiae is chem ed
just by moving 92. A system of fringes oht“.
is shown in Figure 5. TLe configure+ioe of eorrce, reflec-

tors ani detector is ind cetei in Figure E 331 he transmito

amplifier of the iete ctinz system. It is notei that the

U)

P4
H.
(I)
d’
‘

distance between the fringes is n3‘f wavelergW h a prey
ed by equation {2}. The Q-velue estimated from this plot
is approximately 29. Because of the diffraction effect the

harassing reflector

[-1-

trsns mitted intensity decreeee with

{If

separation.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

I .—
a 4— 3/2—-)
( l.6Cm.)
23 I‘
'3 l 1
x
a
c
0-. h ‘H
F
0.5 "
L
1 Movement 0) chlu or R 2
Fig. 5.

 

14

Showing sharpness of fringes on the interferometer

in relation to the wavelength

(initial reflector separation

25 cm.)

3'3
\
V: r.

e

—-l

'1

have
t

M

By means of the interferometer discussed above, we
have measured the rati 3 of the dielectric cons an nts of

polystyrene and teflen. With the confi3uration indicated in

IL!

:3
{J
Le
(D
\J
:8
u)
U}
Ir‘
0
O
5‘.)
(A
(b
‘3.
O‘
(4
m

igur 92, a maximum recronse (fi 1—
justing 32, then a sheet of the material was inser ed be-
tween the reflectors eni the shift of the fringe noted.
However,, Culshaw pointed out that, because of the differ-
ing imm edances, effective path leng h changes occur at the
boundaries of the sheet, the amount depeniing on the posi-
tion of the sheet between the reflectors. In order to can-

cel out this path len 3th change at the boundaries, messure~

ments must be made varyin3 the position of the sheet by
mean fringe shift found.' The dielectric constant k is then
given by

mean shift - d(Jk - 1) (12)

where d is the thickness of the sheet. The measured fringe
shifts due to polystyrene and teflon are collected in the
followin3 table:

 

H

 

 

sample position
(distance in cm.

fringe shifts (cm.)

 

polystyrene

teflon

 

 

from R2) (thickness 0.70cm.) (thickness 0.45cm.)
17-0 0-25 0.05
16.7 0.55 0.12
16.4 0.53 3.53
16.0 0.60 9.35
15-7 0.50 0.12
15.4 0.25 0.05
mean 0.37_ 0.165

 

It is noted that the fringe shift is periodic of the

A. 1 v

.u

J

at

cl

sample position, a fact in agreement with 's predic-

I

tion. From the mean frin3e shift, the calculated dielectric
constants for polystyrene end teflon are 2.341 and 1.856 re-
spectively. Although these values are not in agreement with
handbook values, their ratio 0.79} is comparable with the
accepted value 0.785 (dielectric constants of polystyrene
eni teflon are 2.55 and 2.33 respectively). Furthermore,
with our measured ratio and assuring 2.55 as the dielectric
constant of polystyrene, we get 2.32 for the dielectric
constant of teflon. This is in agreement with the accepted
value within one rercent. Thus, it may be clear that our
simple arrengenent is a very convenient method for compar-

ing dielectric constants.

V. DISCUSSION

It should be pointed out that only the real part of
the dielectric constant has been measured in this experi-
ment. The imaginary part which is associated with the di-
electric loss can be estimated by observing the change of
the Q-value due to the sample; however, this can only be
obtained by means of an interferometer with very high Q-
value.

An obvious extension of the experiment discussed
above is to compare the dielectric constants of liquids.
With this idea in mind, it seems possible that more appli-
cations can be developed, namely, the titration of weak
acids and the measurement of relative temperature variation

of polarization.

(1)

(2)
(3)
(4)

(5)

(6)

l8
BIBLIOGRAPHY
Lengyel, B. A., Free. Inst. Radio. Engrs., N. Y., 37,
1242 (1949).
Culshaw, W., Free. Phys. Soc. B, 65 (1953).
Culshaw, w., Proc. Phys. Soc. B, LXVI (1953).

Tolansky, 8., Multirle~Beam Interferometry of Surfaces
and Films, Oxford at Clarendon Press (1948).

Stratton, J. A., Electromagnetic Theory, McGraw-Hill
Book Company, Inc. (1941).

BOOker, Ho Go, and P. C. Clemmow, F. InStno EleCto
EngI‘So, Pt. III, 97’ 110

HYSI"‘7-"' ‘ T”

"anan