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Interferometers The true worth of an ... set up a Michelson interferometer as sketched in Fig. 7, using a piece of wire mesh as the beam splitter and the large thin ...

either in phase or out of phase resulting in an increase or decrease of the received amplitude
To analyze the interference pattern we assume monochromatic plane waves incident on
the beam splitter
E E 0 cos s t 1
where s is a distance along the path n is the index of refraction of the material in the path is
the wavelength in vacuum and is any phase shift caused by reflections along the path The
intensity at the detector is then the time average of the sum of amplitudes squared from mirror 1
and from mirror 2
I E1 E 2 T
where the brackets indicate a time average of the amplitude sum After a good deal of algebra
this can be expressed as
I I1 I2 2 I1I2 cos 3
The total phase difference between the two waves can be written in terms of the two geometric
path lengths
D1 s1 2s2 s4
D2 s1 2s3 s4
This simplifies to
Displacing either mirror will cause the intensity to vary as a cosine with period 2 in s2 s3 If I1
and I2 happen to be equal the intensity will vary from 0 to 4I
Physics 231 Interferometers 2
Fig 2 Geometry of a Fabry Perot interferometer
2 The Fabry Perot interferometer
If we arrange two partially transmitting plates as shown in Fig 2 we can cause the beam
to bounce back and forth many times creating a multi beam amplitude division instrument The
analysis is simplified by noting that the net effect of the multiple reflections is to create a
standing wave by superposing two counter propagating waves We feed energy in through one
plate and detect the energy that comes out through the other plate For a quantitative analysis
we need to derive an expression for the amplitude of the electromagnetic wave emerging from
the second plate in terms of the amplitude of the input wave
For a single plate the incident reflected and transmitted electric field amplitudes are
related by reflection and transmission coefficients r and t defined as follows
Eref rEinc Etrans tEinc 7
We have assumed normal incidence and deal only with the vector magnitudes so that r and t are
scalars We also neglect the phase shift due to the plate If in addition there is no conversion of
electromagnetic energy to heat conservation of energy requires that
At any point in space we can write the scalar fields in the complex representation as
E j E j 0 e i kz t 9
Physics 231 Interferometers 3
The plus sign indicates travel to the left in Fig 2 We can relate the reflected and transmitted
waves at z 0 by
E2 0 tE 4 0 rE1 0 11
E3 0 tE1 0 rE4 0 12
and similarly at z d
E4 0e ikd rE3 0e ikd 13
E5 0e ikd tE 3 0e ikd 14
Solving for E50 in terms of E10 we obtain after some fuss
1 r 2 e 2ikd 1 0
The quantity we actually measure is the transmitted intensity which is proportional to E50 2
Doing the algebra we arrive at the ratio of transmitted to incident intensities
E1 0 1 r 4 2r2 cos
1 2 0 1 2 1
Fig 3 Ratio of transmitted to incident intensity as a function of plate separation d for the Fabry
Perot interferometer
Physics 231 Interferometers 4
where we have used the definition 10 of k in terms of wavelength This is the result we need
to interpret the experiment
The right hand side of Eq 16 is plotted in Fig 3 for several values of the reflection
coefficient r of the plates Note that the maxima and minima of transmission occur at half
wavelength intervals as one could also see directly from the periodicity of cos 4 d As r
increases the amount of transmission between maxima decreases sharply and the peaks become
narrower At optical frequencies r can be made very nearly one resulting in very sharp peaks
indeed If several wavelengths are present in the incident illumination each one will produce a
peak at a particular set of separations and the device is useful as a high resolution spectrometer
EXPERIMENTAL PROCEDURE
1 Microwave detector and source
The detector is a diode rectifier mounted in a short piece of waveguide rectangular tube
A horn antenna acts to impedance match EM waves in free space to the waveguide modes A
diode visible as a white cylinder rectifies the component of the electric field parallel to the short
side of the waveguide producing a DC voltage proportional to a good approximation to the
intensity of the input electric field A DMM in voltage mode therefore gives a measure of the
input intensity
The microwave source is a klystron tube Internally a bunched electron beam traverses a
resonant cavity and thereby excites standing waves at about 10 GHz Part of the cavity field is
coupled into a waveguide through the antenna you can see by looking into the output horn with
power off Another horn acts to impedance match the EM waves in the guide to free space
resulting in approximately plane waves with wavelength about 3 cm The electric field of the
waves is parallel to the short side of the waveguide and horn
The klystron supply voltages are lethal so you must be sure that the DC ON switch is in
STANDBY position before changing any wiring Check that the klystron is connected as specified
Connections for klystron
wire terminal
Brown 6 3 V AC either
Brown 6 3 V AC the other one
Red Common either
Orange Common either
Physics 231 Interferometers 5
Fig 4 Sketch of klystron power output as a function of C voltage
in the table and turn on the AC power switch Allow about a minute for the tube filament to
warm up and be sure that both high voltage knobs are fully counterclockwise lowest voltage
setting Set the meter switch to read B voltage and turn on the high voltage Increase the B to
150 V on the built in meter You will see a small current less than 20 mA on the current meter
Now point the transmitter and receiver horns at each other with their short sides vertical
Switch the meter to read the C voltage and slowly increase it watching the DMM reading The
klystron can oscillate in several modes so you will see the power output change with C voltage
as suggested in Fig 4 Use the C voltage to maximize the output but stay below 75 V Once
you find a good setting for the power supply do not disturb it for the rest of the experiment
The microwave horns produce a broadly divergent beam as you can easily show for
yourself by rotating the transmitter horn so that the beam sweeps across the receiver As a result
it is hard to obtain data of the precision that you might with an optical experiment where the
input beam is very well collimated and of small lateral extent In partial compensation for this
limitation the microwave beam is fully polarized and much more monochromatic than a
conventional light source
When you are through making measurements turn down the B and C voltages on the
klystron set the HV switch to standby and then turn off the AC power switch Leave the klystron
connected for the next group of students
2 Michelson interferometer
Next set up a Michelson interferometer as sketched in Fig 5 using the crossed meter
sticks to keep transmitter receiver and mirrors aligned The beam splitter is a piece of wire
mesh which can be placed across the intersection as shown To accurately set the angle remove
mirror 2 and adjust the beam splitter to maximize the detector reading This ensures that the
transmitted wave bounces off the mesh at 45 and reflects off mirror 1 to the detector Complete
assembly by replacing mirror 2
Physics 231 Interferometers 6
Mirror 1 on slider
Mirror 2 on slider
Transmitter DMM
Fig 5 Arrangement for a microwave Michelson interferometer The mirrors should be 25 30 cm
from the center of the beam splitter Transmitter and receiver can be somewhat closer
By moving the mirror along the meter stick you should see relatively sharp intensity
minima and maxima Plot the intensity volts against scale reading for a distance of 3 4 cm
roughly centered on the starting point According to Eq 3 and 6 you should expect a cosine
variation that you can fit using the LoggerPro cosine function
f x Acos Bx C D 17
Extract the wavelength and uncertainty from parameter B The others A C and D are not
particularly significant
While the Michelson is set up you can use it to demonstrate an application of
interferometry Insert the paraffin plate in either arm and note that the position of the
interference minima and maxima shift For best results you will probably need to rotate the wax
plate so that reflections from its surfaces do not reach the receiver If you do this don t forget to
compute the actual geometric path length through the plate
The shift occurs because a layer of paraffin index np has replaced an equal length layer
of air index 1 in one arm This changes one of the optical path lengths s changing and
moving the whole pattern according to Eq 5 and 6 Measure the shift for a few maxima or
minima and use that data together with the thickness of the plate to estimate the index of
refraction of the paraffin
Physics 231 Interferometers 7
HV Supply DMM
Slotted plates
on sliders
Fig 6 A microwave Fabry Perot interferometer
3 Fabry Perot interferometer
In order to construct a Fabry Perot interferometer we must have partially transmitting
plates to use as mirrors It turns out that a slotted metal sheet does this job quite well in the
microwave frequency range
Align the transmitter and receiver horns on the straight piece of meter stick as in Fig 6
and place the two slotted plates between the horns with their slots horizontal Move the plate
nearest the receiver along the meter stick to find the maxima The maxima are clearest if one
plate is kept within about 2 cm of the transmitter to minimize leakage around it and the
separation d is about 10 cm You will need to keep your hands out of the beam in order to avoid
disturbing the measurement
Record the positions of several maxima and also the maximum and minimum
transmitted intensity When you have seen the pattern with the slits horizontal repeat the
measurements with the slits vertical The maxima for vertical orientation are very sharp so you
will have to look for them carefully Do your results for the two orientations agree qualitatively
with Fig 3 Which orientation has higher reflectance Explain
Plot the positions of the maxima vs an arbitrary mode number and calculate Do your
results for the horizontal and vertical orientations agree Is consistent with your result from the
Michelson interferometer
Your report should include notes about the sharpness of maxima for the Michelson vs the
two Fabry Perot configurations your measurements of and answers to the questions in the
Physics 231 Interferometers 8

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