Download Microwave Interferometry:

Microwave
Interferometry:
Measuring index of
refraction
By: Carissa Erickson
Ryan Forde
Objectives
The objectives of this lab were to become familiar with microwave circuits and to
determine the refractive index of several materials. The microwave circuit used was an
interferometer.
Theory
Microwaves are electromagnetic radiation. The range usually considered to be
microwaves has frequencies between 1.5 GHz and 25.0 GHz. They were generated in
our setup by a reflex Klystron, then sent through an interferometer. The interferometer
allowed the phase shift caused by a sample to be measured.
Reflex Klystron
In a klystron, microwaves are produced by accelerating a beam of electrons. The reflex
klystron is different from other types of klystron in that it uses a reflector plate instead of
an output cavity. The electron beam is modulated by passing it through an oscillating
resonant cavity. The feedback required to maintain oscillations within the cavity is
obtained by reversing the beam and sending it back through the cavity. The electrons in
the beam are velocity-modulated before the beam passes through the cavity the second
time and will give up the energy required to maintain oscillations. The beam is turned
around by a negatively charged electrode called the reflector plate.
Interferometer
In an interferometer, a beam of radiation is split into two paths. The paths are adjusted so
that they are 180 out of phase and no signal is received. A sample introduced into one
of the arms induces a phase shift. The induced phase shift can be measured by adjusting
the path in one of the interferometer arms. Measuring this adjustment provides a measure
of how much phase was induced by the sample.
Induced Phase Shift
The induced phase shift depends on equation (1). From this equation, we can derive an
expression that will allow the index of refraction of the material to be computed.
Equation (4) depends only on the measured phase shift and the thickness, d, of the
material.
d
(1)
   k  k 0 dx  k  k 0 d
0
(2)
(3)
(4)
 k

  k 0 
 1d
 k 0

 c 
2 f
n  1d  2 f
    1d 
c v 
c
c

r
 c 

 
 1
 2f d

2


r 1 d
Index of Refraction (Permittivity)
The materials used as samples in this type experiment must all be dielectrics. This means
that their electrons are not free as in a conductor, but bound to the atoms in the material.
They are affected by an electric field, however. The electric field produces a dipole
moment by shifting the electron orbits in the opposite direction to the electric field. The
shifted electron orbits cause an electric field to form in the dielectric and this field
interacts with the electromagnetic wave passing through. This interaction causes the
difference in permittivity between a dielectric and free space.
(5)
2


ne


 r   0 1 
2 
 m 0 
Where r is the permittivity of the dielectric, 0 is the permittivity of free space, n is the
number of molecules per unit volume, e is the electronic charge, 0 is the frequency of
bound motion and m is the mass of the nucleus of the molecules of the dielectric in kg.
Introduction to Wave Phenomena by A. Hirose and K. E. Lonngren
Apparatus
Waveguide Tuner
E-H
Tuner
Coupler
XXXXXX
Klystron
Gain Horns
Sample
Hybrid
matched tee
Phase
Shifter
Detector
Attenuator
Isolator
Equipment
Klystron: This was the microwave generator. Its operation is described above.
E-H Tuner: This was used to match the radiation generated by the klystron to the
waveguide. It probably consists of some kind of resonant cavity whose properties can be
adjusted. A matched signal has a much higher amplitude and should suffer lower loss in
the waveguide.
Waveguide Tuner:
Gain Horns: Some of the samples attenuate the microwave beam. These amplified the
signal by 5dB so that it was easier to detect.
Sample: This is the piece of material whose index of refraction is to be measured.
Phase Shifter: This was used to adjust the phase of the microwave beam in one arm of the
interferometer. The exact mechanism for this device is not known, but it probably
operates by applying a magnetic field across the waveguide.
Attenuator: Attenuation was used to match the amplitudes of the two signals so that they
cancel as completely as possible when they are out of phase.
Isolator: This was used to prevent radiation from traveling back to the source.
Procedure
1. The corresponding voltage was measured in steps of 0.01 units on the phase
shifter between 0 and 0.11. This included the first maximum as well as the first
minimum on the voltage curve. The voltage curve was plotted and is shown in
Figure 1.
voltage
Voltage vs Position
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
position
Figure 1: Voltage vs Position
2. The formula for the voltage sine wave was determined from Figure 1:
V  .254  .254 sin(  )
(1)
3. Since the voltage, V, was measured experimentally, the phase shift for each
voltage,  , was determined from (1).
 V  .254 
  arcsin 

 .254 
(2)
The error in the phase shift was:
1
 
 V .254 

.
254


2
 V  .254 

 .254 

(3)
1- 
 V  .254  V V (.254)  (.254) .254 .254




2
.254
.254 2
.254
 .254  .254

(4)
4. The phase shifts were linearized and plotted with respect to phase shifter position.
This is shown in Figure 2.
Phase Shift vs Position
Phase Shift
500
400
y = 144876x3 - 32832x2 + 5553.9x - 55.173
300
200
100
0
0.0000
0.0200
0.0400
0.0600
0.0800
0.1000
0.1200
Position
Figure 2: Phase Shift vs Position
5. The best fit line to the Phase Shift vs Position curve was described by the
equation shown on Figure 2, or:
 = 144876x3 - 32832x2 + 5553.9x - 55.173
(5)
Where:  = phase shift
x = phase shifter dial position
The error in the best fit value was:
  434628x 2  65664 x  5553.9
(6)
6. (5) was used to determine the phase shift due to the change in phase shifter dial
position. This value corresponded to the distance between the phase shift due to
the material and the next minimum in the voltage curve. In order to isolate the
phase shift due to the material the value was subtracted from 3600, which
corresponded to the next minimum in the voltage curve.
Observations
Table 1: Thickness and Phase Shifter Position Data
Sample
Thickness
(m +/- .000005)
glass1
0.00228
glass2
0.00217
glass3
0.00444
glass4
0.00582
plates1
0.00571
plates2
0.01266
plates2/water
0.00011
bee's wax
0.00092
ice
0.01600
Trial 1
Trial 2
Trial 3
Trial 4
Avg
(+/- .0005)
0.0777
0.0805
0.0562
0.0450
0.0730
0.0540
0.0162
0.0965
0.0578
(+/- .0005)
0.0460
0.0790
0.0571
0.0460
0.0690
0.0555
0.0155
0.0954
0.0577
(+/- .0005)
0.0795
0.0545
0.0540
0.0431
0.0727
0.0525
0.0128
0.0979
0.0580
(+/- .0005)
(+/- .0005)
0.0786
0.0799
0.0558
0.0447
0.0716
0.0540
0.0146
0.0966
0.0578
0.0803
0.0555
0.0140
Analysis
1. The Dielectric constant was determined for each material using the following
formula:
 c 

  
 1
 2f d

2
(6)
The error was calculated using the following formula:
2
2
 c   2 2 d 2 f
     2 2 

f 2d 3
f 3d 2
 2   f d
2
The results are shown in Table 2:
  c   d f 


   
 (7)
fd 2
df 2 
    fd
Table 2: Analysis Results
Sample Thickness
glass1
glass2
glass3
glass4
plates1
plates2
bee's wax
ice
(mm +/- .0005)
0.00228
0.00217
0.00444
0.00582
0.00571
0.01266
0.00920
0.01600
Phase Shift Error
(degrees)
111
107
182
220
133
188
54
176
(degrees)
15
15
16
17
15
16
16
16
Dielec Const Error
(relative)
11.5
11.7
9.09
8.11
4.57
2.99
1.66
2.37
Accep Value
(relative)
1.3
1.4
0.30
0.075
0.14
0.12
0.18
0.10
(relative)
5-10
5-10
5-10
5-10
?
?
2-3
75
Discussion
The experimental values of the dielectric constants for the samples were compared to the
expected values in Table 3. Most of the values were close to the accepted values. There
was a large discrepancy for the measurement of ice, which can be explained by the
difficulty of correctly measuring its thickness and the difficulty of getting a sheet of
uniform thickness. The beeswax thickness was also difficult to measure because of its
compressibility and this may have contributed to the error in its dielectric constant. The
glass was assumed to all be of the same type and so all the samples should have had the
same dielectric constant. They did not, probably because of adjustments to the
attenuators and other parts of the equipment between measurements.
There were several random errors associated with this experiment. The microwaves
produced by the klystron depended on the beam voltage, which fluctuated. We
minimized the effect of this error by making sure the voltage was zero before and after
each reading. The minimum phase was not sharply defined, so we estimated the middle.
The only systematic error that may have occurred was the possible frequency dependence
of the attenuators. The most significant source of error in this experiment was the voltage
dependence of the klystron beam.
Conclusion
The experimental values for the dielectric constants of the sample materials did not all
agree within the calculated error range with the accepted values. Further research should
be conducted on the possible sources of error with the equipment so that more reliable
data could be collected.