Lab 13 - Michelson Interferometer - PZT
I. Objective
To study the principle of interferometry, alignment of an interferometer, and its
application to measure small motion of an object.
II. Introduction
1.5
1.5
1
1
Wave Amplitude
Wave Amplitude
Interference deals with the superposition of light waves. When waves overlap in space
and time, the resulting wave is the algebraic sum of the individual waves. When the
peaks and valleys of two waves coincide (Figure 1) the resulting wave is larger than the
individual waves (constructive interference). When the peaks of one wave coincide with
the valleys of another wave, the resultant wave is small (destructive interference).
0.5
0
-0.5
-1
0.5
0
-0.5
-1
-1.5
-1.5
0
200
400
600
800
1000
Angle (degrees)
0
200
400
600
800
1000
Angle (degrees)
Figure 1: (Left) Two waves that are in phase. The sum of the two waves results in a wave that is roughly
twice as big as the individual waves. (Right) Two waves that are out of phase. The sum of the two waves
would result in essentially no wave.
Figure 2: Experimental setup of Michelson interferometer. M1 and M2 are mirrors, BS is a cubic
beamsplitter, and the lens magnifies the fringes for viewing on a screen. Mirror MW is a mirror mounted
on a PZT structure.
For the Michelson interferometer shown in Figure 2, changes in the optical path
length between “arms” of the interferometer change the resulting interference pattern at
the output from constructive to destructive, to constructive. One “arm” is the optical path
between the beamsplitter (BS) and Mirror M1. The other “arm” is the optical path
between the BS and Mirror M2. Mirror M2 should be mounted on a linear translation
stage. As shown in Figure 1a, when the phase difference is zero, the path difference is
Document1
Page 1 of 9
Last Modified 10/16/2013
zero and the output of the interferometer corresponds to total constructive interference.
As shown in Figure 1b, when the phase difference is π radians, this corresponds to a path
length difference of half a wavelength (  / 2 ) of light. For this path difference, the
interferometer exhibits total destructive interference. As the path length continues to
increase from  / 2 to  , the phase difference corresponds to 2π and totally constructive
interference. In this experiment will count the number of bright and dark fringes of the
interferometer in order to measure the path length change due to a moving mirror.
We will vary the path length between the beamsplitter and mirror M2 using two
methods: (a) the linear translation stage for large scale (10’s of milleters) motion and (b)
a piezo-actuator (10’s of micrometers) for small scale motion. Specifically, Mirror M2 is
a thin piece of silicon mounted on top of a piezo-actuator. The piezo-actuator changes its
thickness (and hence the length of the optical path between the beamsplitter and M2)
when a voltage is applied to the deivce. Piezo-actuators are typically electro-active PZTceramics (PZT: lead (Pb) zirconia (Zr) Titanate (Ti)). Piezo-actuation is the
complementary effect to piezoelectricity, where electrical charges are produced upon
application of mechanical stress to the ceramics. As an analogy the term “piezomechanics” was introduced in the early 1980’s of the past century to describe the
conversion of electricity into a mechanical reaction by a piezo-material.1
For piezo-mechanical conversion in the simplest case a single PZT layer is used. Such
a PZT monolayer structure as shown in Figure 3 acting as a capacitive element defined
by 2 thin conductive electrode coatings enclosing the piezo-ceramic as dielectric. When
this “piezo-capacitor” is charged by applying a voltage, two deformations are created (a)
a change in thickness (called the d33 effect) and (b) a change in area (called the planar
mode d31 effect). For this experiment, we will utilize the change in thickness with
applied voltage.
Figure 3: Schematic of a piezoelectric single layer element.
III. Procedure
A. Alignment
The distance between the mirrors and the beamsplitter should be initially set to be
about 9cm. The lens is used to magnify the laser spot and resulting interference fringes. A
50 or 100mm focal length lens should be sufficient. The screen can be a white piece of
paper. Mirror M2 should be mounted on a linear translation stage. The direction of
motion should be TOWARDS the beamsplitter.
1
For more info, see http://www.americanpiezo.com
Document1
Page 2 of 9
Last Modified 10/16/2013
1.
2.
3.
4.
5.
6.
7.
The interferometer must be carefully set up in order to produce localized fringes:
Mount mirrors M1, M2 cubic beam splitter, laser, and the screen as shown in Figure
1. It is important that mirror M2 be placed in the arm of the interferometer which
results from light reflecting from the beamsplitter.
Note that mirror M2 does NOT have any tilt adjustments on its mount. In order to
properly align M2, adjust the height and position of M2 so that the reflection from M2
hits as close to the SAME SPOT in the middle of the beamsplitter as in incoming
beam. NOTE since mirror M2 does not have any tilt adjustments, you will not be able
to get the spots to overlap.
Next adjust the tilt/ rotation of the beamsplitter such that the spot from the incoming
beam and the reflection from M2 overlap in the beamsplitter.
Adjust the reflection from M1 so that the reflected beam hits the SAME SPOT in the
middle of the beamsplitter as in incoming beam.
Make fine adjustments to the tilt of mirror M1 and the tilt/ rotation of the BS so that
two output beams from the interferometer overlap in the middle of the beamsplitter
AND on the screen. The goal is to align the interferometer so that both of the output
beams are collinear and overlap in space. If they do not, repeat steps 2-5.
Place a lens in the path of the output beam to expand the size of the laser beam.
If you have aligned the optics properly, you will see a pattern similar to the figure
shown below. If you do not observe any fringes, slowly adjust the mirrors until you
observe the localized fringe pattern.
8. If you have not already done so, make sure that the optical table is floating on a
cushion of air. Ask your instructor for assistance.
B. Localized fringes
1. Slowly vary the tilt of the mirrors and observe the change in fringe patterns. If you
have a digital camera or a cell phone camera, take a picture of the fringes for
inclusion in your lab report. It will be helpful in further analysis to include in your
picture a ruler so that you have a scale for determining the spacing between the
fringes. IF YOU DO NOT HAVE A CAMERA, make sure that you measure the
spacing between the fringes (Δx) with a ruler in class right now so that you can do
discussion Question 3.
2. Record the focal length f of the lens that you used.
3. Record the distance D from the lens to the screen.
Document1
Page 3 of 9
Last Modified 10/16/2013
C. Relative displacement measurement
1. With localized fringes, place a photodetector near the middle of the fringe pattern.
The mirror M2 should be mounted on a linear translation stage. Connect the output of
the photodetector to the oscilloscope.
2. Manually move the translation stage by rapidly turning the micrometer of the stage
and verify that the detected signal varies as the fringes sweep past the detector. The
signal on the oscilloscope should look similar to the figure below:
0.08
Voltage (V)
0.07
0.06
0.05
0.04
0.03
0.02
0
0.005
0.01
0.015
0.02
Time (s)
3. Bang your fist on the table. You should observe changes in the measured voltage on
the oscilloscope. What are this changes in measured voltage due to? HINT: Why are
the path lengths changing?
4. The next step is to apply a voltage to the PZT structure to make small changes to the
path length of the interferometer. The electrical connections are shown in the figure
below.
15
0.08
10
0.07
0.06
Voltage(V)
5
0.05
0
0.04
-5
0.03
-10
0.02
0
0.005
0.01
0.015
0.02
Time(s)
5. Connect the PZT structure, function generator, and oscilloscope as shown in the
figure. Start by applying a 100Hz, 10V peak-to-peak, triangular waveform to the
PZT. Trigger the scope from Channel 2 (reference). You might find it useful to use
Document1
Page 4 of 9
Last Modified 10/16/2013
Voltage (V)
the “AUTOSCALE” button on the oscilloscope. Adjust the time base so that 1 to 2
oscillations of the reference waveform appear on the oscilloscope. Your data on the
scope should look something like the following.
15
0.08
10
0.07
0.06
5
0.05
0
0.04
-5
0.03
-10
0.02
0
0.005
0.01
0.015
0.02
Time (s)
Figure 4: Typical data for the applied voltage to the PZT (Purple, left axis) and corresponding
interferometer output (blue, right axis scale).
6. Keep the applied frequency at 100Hz for a triangular waveform. Using the supplied
LABVIEW program (Lab13-2008), record in an EXCEL file the reference and
photodetector voltages for an applied voltage to the PZT of 10V. After the
LABVIEW program records the data, you might need to push the “RUN” button on
the front of the oscilloscope to actively view your data on the oscilloscope.
7. Repeat step 4 for voltages of 9, 8, 7, 6, 5, 4, 3, 2, and 1 V. Make sure that you push
the “RUN” button on the oscilloscope as needed. You may also need to adjust the
sensitivity (or use “AUTOSCALE” button) of Oscilloscope channels 1 and 2 as you
decrease the applied voltage. as you vary the PZT voltage from 10V.
8. Return the function generator to a 100Hz, 10V peak-to-peak triangular wave and
observe the photodetector voltage on the oscilloscope. You should observe roughly
EVENLY spaced voltage peaks on the oscilloscope. Now change the waveform on
the function generator to a SINE wave output. You should observe that the voltage
peaks are no longer equally spaced. In one of the discussion questions, you will
discuss WHY this is so. For now, use the LABVIEW program to save the data to a
file for further analysis.
9. Switch the waveform BACK TO TRIANGULAR WAVE. Set the function generator
for 100Hz, 10V peak to peak triangular wave. Your next goal is to determine if the
AMPLITUDE of vibration of the PZT device is CONSTANT as you vary the
frequency. Keeping the voltage FIXED, vary the frequency from 100, 200, 300, 400,
500, 1000 and 2000Hz. For each frequency, use the LABVIEW program to record the
data.
10. Make sure that you transfer all of your data from the computer to a memory stick for
further analysis.
IV. Discussion Questions
Document1
Page 5 of 9
Last Modified 10/16/2013
1. Include a picture of your localized fringes in your lab report.
2. Calculate the lateral magnification of the localized fringes due to the lens. The
magnification will be given by
D f
(1)
Mg 
f
3. Calculate the angular alignment error (in radians) according to:
 Mg
(2)

2 x
4.
5.
6.
7.
where x is the spacing between adjacent magnified fringes. Express the angular
alignment error in Degrees.
Clearly in part III.C.2, you manually translate mirror M2 to change the path length in
one section of the interferometer. Why does the photodetector voltage change when
you bang on the table in Part III.C.3? Why is there a change in path length in the
interferometer when you bang on the table?
Based on your answer to Question 4, why is it necessary to float the optical tables for
this experiment?
Using EXCEL (or Matlab if you like) plot of the oscilloscope data for the 100Hz
AND 10V triangular waveform data ALL ON THE SAME PLOT. Typically, the data
from the photodetector will be on a much different vertical scale compared to the
reference triangular waveform data. In EXCEL, the best way to plot them both is to
plot one data set on one vertical axis and the other data set on another vertical axis.
For this operation, you need a SECONDARY AXIS to plot both waveforms on the
some plot. Since methodology for including the secondary axis depends on which
version of EXCEL you use, you should use the HELP within EXCEL and do a search
in HELP on “add secondary axis”. Your plot should look like Figure 4.
Your next goal is to create from your data a plot of the MAXIMUM
DISPLACEMENT of the mirror M2 as a function of applied voltage. Appendix 1
gives the details of the data analysis. For this discussion question, plot the 10V
triangular waveform, your measured photodector voltage, and best fit on the same
graph. This plot should look like Figure 5. Note that the TOTAL duration of the x
axis (time axis) is only 50ms. This is ½ of the period of the 100Hz oscillation. Cut
and paste your plot here.
If the fit does not CLOSELY match your experimental data (as shown in Figure
5), PLEASE ASK YOUR INSTRUCTOR FOR HELP. You will not be able to draw
meaningful conclusions from your data if you do not correctly fit the data.
8. Repeat the calculation for all of your data from III.C.7. Based on your data, please fill
in the following table:
PZT Voltage (V)
1
2
Document1
Best Fit A1 (s-1)
Page 6 of 9
N=A1* Δt/(2π)
ΔD
Last Modified 10/16/2013
3
.
.
9
10
In the above table, the second column is the best fit A1 values from Eq. (5) to your
data. The units of A1 will be in units of s-1 if your time variable is in units of seconds.
The units of ΔD will be the same unit which you use for the wavelength of the He-Ne
laser.
Note that you only need to fit a subset of your data. See the Matlab homework
assignment from Week 1 of the course for instructions on fitting only a portion of
your data. The third column in the above table is the number of oscillations N from
Eq. (4). To calculate this column, use the data from the second column (A1) and
t  0.5/100 (specifically half of the period of the 100Hz applied voltage
waveform). To calculate the last column of data, use the third column of data, the
known wavelength of light for your laser (632.8nm for HeNe laser) and Eq. (4).
9. From your data in the table above, create of plot of the measured maximum
displacement ΔD versus the applied PZT voltage (ie. Column 4 versus column 1).
Based on your plot, is the displacement proportional to the applied voltage?
EXPLAIN YOUR ANSWER.
10. Using your data from III.C.8, plot the measured photodetector output versus time for
the applied SINE wave. Include the plot in your lab report. Compare this plot to your
answer for Discussion Question 6. WHY IS THE SPACING BETWEEN THE
INTERFEROMETER FRINGES NOT EVEN when you use the SINE wave, but they
ARE equally spaced with the triangular wave?
11. Your next goal is to determine from your data if the PZT actuator amplitude is
constant while you change the frequency of the applied voltage. Using your data from
III.C.9, For this discussion question, use the 200HZ data and plot the 10V triangular
waveform, your measured photodector voltage, and best fit on the some graph. This
plot should look like Figure 5 EXCEPT the TOTAL duration of the x axis (time axis)
is only 25ms. This is ½ of the period of the 200Hz oscillation. Paste a copy of your
figure here.
12. Repeat the calculation for all of your data from III.C.9. Based on your data, please fill
in the following table:
PZT Frequency (Hz)
100
200
300
Etc.
Document1
Best Fit A1 (s-1)
Page 7 of 9
N=A1* Δt/(2π)
ΔD
Last Modified 10/16/2013
13. Using your data from the above table create a plot of the Maximum displacement
versus PZT frequency. Based on your plot, is the maximum displacement constant
throughout the range of frequencies tested?
APPENDIX 1:
Theoretically, the displacement of the PZT is proportional to the applied voltage. So
as the voltage to the PZT varies from a minimum voltage (eg. -4.16V in Figure 4) to a
maximum voltage (12.25V in Figure 4), the displacement changes by an amount D .
Using the Michelson Interferometer, we measure that distance using the output voltage of
the interferometer.
The maximum phase difference in the two arms of the interferometer (assuming that
they are equally balanced to start) is given by
  2 k D
(3)
where k is the wavenumber of the light ( k  2 /  ) and D is the maximum
displacement of the mirror on the PZT. If we divide the above equation by 2 , we can
calculate the NUMBER OF OSCILLATIONS in the interferometer output to be
 2 D
N

(4)
2

In order to determine N, we need to count the number of oscillations in the interferometer
output. Figure 5 below shows a portion of the data from Figure 4 corresponding to a time
interval for which the applied voltage to the PZT is always increasing from the minimum
applied voltage to the maximum. From Figure 5, we can see that there are roughly 5
oscillations of the interferometer output. From Eq.(4), we can see that 5 oscillations
corresponds to a D maximum displacement of roughly 2.5 . Since the wavelength of
the light source (HeNe Laser) is 632.8nm, the maximum displacement is roughly 1.6
microns.
In order to get an ACCURATE estimation of the number of oscillations, you will
make a BEST FIT of your data from Figure 5 to the following equation:
V  A0  sin( A1 t  A2)  A3
(5)
where V is the detected voltage from the photodetector, A0 is the amplitude of the sine
wave output, A1 is the frequency of the sine wave output, t is the time (x axis), A2 is a
phase and A3 is an offset. For the given t and V data (x and y data that you measured on
the oscilloscope), fit the data to extract parameters A0, A1, A2, and A3. For the data
shown in Figure 5, the best fit parameters are given in the caption. Of the three
parameters, A1 is what you will need for the rest of data analysis.
Document1
Page 8 of 9
Last Modified 10/16/2013
Voltage (V)
15
0.08
10
0.07
0.06
5
0.05
0
0.04
-5
-10
0.0075
0.03
0.02
0.0085
0.0095
0.0105
0.0115
0.0125
Time (s)
Figure 5: Portion of the data from Figure 4. Purple line is part of the triangular reference waveform
corresponding to ½ the period of a 100Hz wave. The points are the experimental data. The thick purple line
is a best fit to the data. Best fit parameters to Eq. (5) are A0 = 1.753×10-2, A1=6.098×103, A3=5.3515, and
A3=5.032×10-2.
Document1
Page 9 of 9
Last Modified 10/16/2013