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Chapter 1 The Michelson Interferometer 1.1 Prelab In this lab you have to nd the position of mirror M1 (see Fig. 1.1) such that the optical path length from the beam ...

2 CHAPTER 1 THE MICHELSON INTERFEROMETER
given in both csv format and as a saved MATLAB workspace
1 2 Introduction
A well known interferometer operating on the principle of division of amplitude
is the Michelson interferometer The incident light beam is divided into two
parts by means of a beam splitter The divided beams traverse different paths
and then are recombined Depending on the difference in optical path lengths
of the two beams the recombined beams may be 180 out of phase produc
ing destructive interference or in phase producing constructive interference
Observation of these interference fringes with changing optical path difference
allows one to determine wavelengths wavelength differences and or small dif
ferences between the two optical paths
Use of this interferometer as a spectrometer to measure the power spectrum
of an emitting source became feasible with the advent of high speed large
storage capacity computers which could calculate the Fourier transform of the
output irradiance of the interferometer You will use this instrument to realize
such a Fourier transform spectrometer
1 3 The Michelson Interferometer
Fig 1 1 illustrates the physical arrangement of the Michelson interferometer
The beam splitter oriented at 45 to the incoming light is partially metallized
on one surface reflecting half of the beam to mirror M1 and transmitting the
other half of the beam to mirror M2 After reflection from these plane mirrors
the two beams are recombined at the beam splitter and part of this light passes
out of the interferometer towards the CCD camera Mirror M1 is mounted on
a computer controlled stage which can be moved via software or the jog toggle
buttons on the DC servo controller You will notice there is a second piece
of glass in one arm of the interferometer What is the purpose of this
1 4 Finding the Fringes
Be sure that the vacuum cell is not in the interferometer for this part Place
the sodium lamp at the input of the interferometer and make sure the diffusing
screen is in front of it see Fig 1 1 Note that the lamp will take a few minutes
to warm up The output of the interferometer will be examined using the CCD
camera In this case use the CCD camera with its lens assembly installed
Also you should reset the translation stage zero so that your measurements are
properly referenced Open up the APTUser program on the desktop Using
the Settings button on the GUI make sure the max velocity is 0 5 mm s for both
the move and jog Next push the Home Zero button to zero the stage The
stage will move to the end of its travel and the computer will note this as the
home or zero position The interferometer has been set so that the position of
zero path difference falls near the centre of the range of motion of the computer
controlled stage that M1 sits on you should ask the TA about where it might
be since the exact position depends on the setup and changes from week to week
1 4 FINDING THE FRINGES 3
Servo Stage Controller
HeNe Laser
White light lamp
Diffusing Orange Filter
Splitter Mirror M1 Stage
Lens Assembly
Figure 1 1 The Michelson Interferometer Note the sodium lamp is not pic
due to misalignments Position the stage near the center of the range around
10 to 12 mm Then align the mirrors by adjusting the two knobs of M2 and
making the two images of the focussing pin coincide Fringes should appear on
the monitor You will need to make sure the CCD camera is not saturated by
adjusting the aperture on the lens assembly Fringes of different shapes may be
produced by appropriate adjustment of the mirror M2
1 4 1 Circular Fringes
When the mirrors are effectively parallel to each other which means they are
actually perpendicular since the arms of the interferometer are perpendicular
see Fig 1 2 circular fringes are produced see Fig 1 3 These fringes normally
formed at infinity may be focussed on a screen or onto the CCD camera chip
with the use of a lens as shown in Fig 1 2
4 CHAPTER 1 THE MICHELSON INTERFEROMETER
Figure 1 2 Formation of circular fringes M20 indicates the effective position
of the mirror M2 When both mirrors are effectively parallel to each other but
at slightly different positions the optical path lengths are different and circular
fringes occur These fringes are focused with a lens on to the CCD camera chip
1 4 2 Straight Fringes
If the mirrors are slowly angled so that they are not effectively parallel to each
other the centre of the circular fringes moves away and the fringes will appear
like a series of curved lines the edges of a set of concentric circles When the
path length is set to be very near the ZPL straight line fringes or fringes of
equal thickness occur see Fig 1 3 They are identical to the fringes formed
by an air wedge the smaller the wedge angle the broader the fringes
1 4 3 Changing from One Type of Fringe to Another
When the interferometer is set up it is unlikely that the mirrors will be aligned
or that stage will be already close to the zero path length Therefore the fringes
may be either horizontal or vertical or anything in between By adjusting the
mirrors the fringes may be changed from one form to another As the wedge
angle between the mirrors is decreased the fringes become more curved until
circular fringes theoretically appear
Find the fringes described above and practice forming straight and circular
fringes until fringes of a specific kind can be produced at will Note that perfectly
straight fringes will only occur right at the ZPL In practice of course the fringes
may also be distorted due to imperfections in the flatness of the mirrors In fact
Michelson and other amplitude division interferometers are used in industrial
fabrication plants exactly for this purpose to verify the flatness or precise
curvature of mirrors Using the WinTV2000 program on the desktop record
some images of various fringe patterns for your lab book The GUI button
SNAP will grab a frame for you Right click on the captured image to save it
1 5 THE POSITION OF ZERO PATH DIFFERENCE 5
Figure 1 3 a Straight fringes and b circular fringes which occur when the
misalignment of the mirrors is corrected so that they are effectively parallel see
1 5 The Position Of Zero Path Difference
For some applications in this lab it will be necessary to locate the position of
zero path difference The following section describes how this is done
1 5 1 Procedure
Be sure that the vacuum cell is not in the interferometer for this part
If you haven t already done so you should reset the stage zero so that your
measurements are properly referenced It will take some careful adjustment to
find the exact position for the translation stage to achieve a zero path length
difference Therefore start by setting up the sodium lamp and diffuser and
search for the location with the strongest highest contrast fringes possible
Do your best to insure the fringes are circular nicely concentric well spaced
and clearly defined during your search
Actually you will find that there is a series of stage positions where the
contrast is high then low and then high again see Fig 1 6 Take note of the
exact stage positions where the contrast is minimum or disappears near the
6 CHAPTER 1 THE MICHELSON INTERFEROMETER
stage position where the contrast appears best You will next look systematically
and precisely between each of these minima for the ZPL using the white light
Another sign that you are near the zero path length difference is that the
radius of curvature of the circular fringes is the largest Why is that At this
location you should be within a fraction of one millimeter from the zero path
length difference To look for changes of the radius of curvature as you move
the stage position you could also look at the curvature of the straight fringes
with the mirrors slightly angled
Be very careful not to bump the equipment at this stage since you can easily
knock the system out of alignment and ruin your pre alignment work
Now replace the sodium lamp with the white light source Note that the
Sodium lamp will be hot so only grab it by the post You will notice that fringes
do not appear When using the white light source they will only appear when
the arms of the interferometer have equal path length Why What does
this tell you about the coherence length of the white light source
What is the definition of the coherence length You must adjust the
stage until the path lengths are equal and fringes are observed
Since finding the white light fringes is like looking for a needle in a haystack
you should cheat by using a narrow band filter with the white light source to
find fringes with filtered light and then remove the filter to find the zero path
length difference with the white light Why would this filter help With the
narrow band filter in place slowly scan the stage with the jog buttons to the
positions precisely between the minima you found before When you finally try
the correct maxima the stage will be near the ZPL and fringes will be observed
If you are spending more than 15 minutes without finding the white light fringes
ask a TA to help check you are doing everything correctly At the point where
the fringes appear with the unfiltered white light source the stage is set for
zero path length difference Write down the stage position for future reference
and note the direction you approached the zero path difference The latter is
important for mechanical backlash in the micrometer screw Since you will be
using a MATLAB script to take data and this script moves the stage forward
i e to higher values on the indexed stage then backwards then forward again
back to the starting point you should always approach from below your position
of zero path length difference The fringes will appear and disappear within 2
m so you will probably have to adjust the max velocity and step distance of
the stage so that you don t go too fast and miss the appearance of the fringes
Also use the actuator toggle with care since rapid changes in motion of the
stage might also misalign the interferometer
1 6 Calibration Of The Interferometer
Before the interferometer can be used to measure wavelengths it must first
be calibrated By calibrated we mean that the distance which the stage
actually moves needs to be correlated to the approximate reading provided by
the computer
1 6 CALIBRATION OF THE INTERFEROMETER 7
1 6 1 Procedure
The distance by which the stage travels is found by counting the number of
fringes which pass given a certain mirror movement and given a light source
whose wavelength is well known The Helium Neon Laser 632 8nm is ideal
for this purpose Slide the lamp up and rotate the diffusing screen and orange
filter out of beam path of the laser When using the laser place an imaging
lens L1 in the post just after the output of the laser You should remove
the lens assembly on the CCD camera By adjusting the height and tilt of the
imaging lens in the post holder locate the the main spot of the HeNe laser on
the backplane of the CCD camera Temporarily put a piece of paper just in
front of the CCD and look at the pattern cast on this paper Align the mirror
M2 to see fringes Now take out the paper and use the monitor to view the
CCD output and further adjust mirror M2 so that about five vertical straight
fringes fill the image on the CCD camera see Fig 1 3
Using the monitor to view the CCD output adjust mirror M2 so that about
five vertical straight fringes fill the image on the CCD camera see Fig 1 3
You should adjust the camera aperture and attach some neutral density filters
to the CCD camera so that it isn t saturated
The overall plan is to move the stage a certain distance and count the number
of fringes that pass You should therefore take the data plot it and count the
fringes on the plot The mirror moves a distance of 2 as one fringe passes
a reference point on the CCD Camera Thus if N fringes pass through as the
mirror moves a distance d we have
If the servo controller readings change by D from D1 to D2 then the
mirror travel is linearly related to the stage displacement through the calibration
constant K
d K D K D2 D1 1 2
To analyze the fringe movement i e to count the fringes you will use MAT
LAB to control the translation stage i e mirror position AND to detect the
variation of the intensity on a given pixel in the fringe pattern as the fringe pat
tern moves Several helpful MATLAB files are located in the desktop folder
Michelson files At this point you should NOT be running the WinTV2000
program as it will not allow MATLAB to access the video card By reading
the comments of the M files you will be able to tell what each of their functions
In order to take data for this section you will be using the MATLAB script
called michelson timestream m You will need to quit the APT user program
when you use the michelson timestream file since MATLAB will open APT
user again with the correct parameters When this script is executed it will
prompt you to click on the image the location where the fringe pattern is the
most visible The script will then move the stage back by a distance span
specified in mm then it will move the stage forward by twice that distance
2 x span at a slow velocity speed specified in mm s and then it will move
the stage back to the starting point To use this script your code will look like
8 CHAPTER 1 THE MICHELSON INTERFEROMETER
data michelson timestream span speed
where span and speed are to be chosen by you and the data will be returned
into the array data As an example consider this code
data michelson timestream 0 01 0 001
This will move the stage back by 10 m and then forward by 20 m at a speed of
1 m s which will take 20 seconds and then back 20 m to the starting point
The data will be returned into the array data which will have the intensity on
the pixel you selected for each frame of the video Since the video frame rate is
30 frames per second this means the stage will have moved 301frames s
For a given amount of distance moved by the stage determine the number
of fringes that passed through the reference point region of interest ROI and
then calculate the calibration constant K Repeat this measurement as many
times as necessary to obtain consistent results for K and estimate its accuracy
1 7 Measuring the index of refraction of air
Now that the constant K has been determined you can use this interferometer
to characterize extremely small changes in the optical path length differences
between the two arms You can use this sensitivity to even measure the phase
shift induced by the propagation of light through air You will measure the
phase change by counting fringes as air is slowly introduced into one of the
interferometer arms Using the vacuum pump and glass vacuum cell provided
insert the glass vacuum cell between the beam splitter and M2 and measure the
index of refraction of air at 632 8 nm i e with the HeNe laser
For your measurements you can use the vid capture MATLAB script to
capture the intensity on a small area in the CCD image as a function of time
Your code will look like this d vid capture time where you choose
time appropriately try 10 seconds
To do your measurement you should first prepare a vacuum in the cell
close the leak valve and open the pump valve and turn on the vacuum
pump After the pressure in the cell as read by the meter has dropped to the
minimum on the scale close the pump valve and turn off the pump Start
the data acquisition and slowly open the leak valve to let in air while you
count fringes You may need some practice introducing the air fast enough that
the pressure rises to atmospheric pressure during the time you have set while
not rising too quickly so that you miss the passage of fringes due to the finite
sampling time of the video capture Be careful to insert the glass vacuum cell
without hitting or misaligning the mirrors see Fig 1 4
Would you have counted the same number of fringes if you had
used the Na lamp instead of the HeNe laser for this measurement
Why or why not What is your value for the index of refraction of
air for 632 8 nm light Does this agree with the accepted value
What are your sources of error Could you use this apparatus as a
barometer or thermometer
1 8 DETERMINATION OF WAVELENGTHS 9
Figure 1 4 Vacuum vessel inserted into M2 arm of interferometer
1 8 Determination Of Wavelengths
Be sure that the vacuum cell is not in the interferometer for this part
Also knowing the calibration constant K an unknown wavelength may be
measured by the same procedure i e by noting the stage readings for the pas
sage of a certain number of fringes Carry out this measurement for the sodium
source to get the average wavelength for the two Na D1 and D2 lines You
should use the same span and speed as the previous section The fringes for
these two lines coincide at the position of zero path difference and the measure
ment should be made near this position On either side the fringes become
indistinct as they start to overlap This feature will be studied in the next
1 9 Wavelength Differences
In 1887 Michelson and Morley while studying the visibility of fringes in
Michelson s interferometer observed that the H2 line n 3 n 2 is
not one line but actually a doublet with about 0 33 cm 1 separation The
wavenumber is given in units of inverse centimeters and is the spatial analog
of frequency One can convert to frequency by multiplication with the speed
of light f c The wavenumber of electromagnetic radiation is defined as
1 where is the wavelength in vacuum
In this part of the lab you will determine the separation between the D1
and D2 lines in sodium using the Na lamp The fringes from any two lines
10 CHAPTER 1 THE MICHELSON INTERFEROMETER
leak valve Vacuum pump valve pump on off
Gauge switch
Figure 1 5 Vacuum pump valves and vacuum gauge for vacuum cell
wavelengths 1 and 2 such as the Na D1 and D2 lines always overlap at the
position of zero path difference If the stage is moved away from this position
either forwards or backwards then a position will be reached at a distance d
where the bright fringes of one line fall on the dark fringes of the other The
resulting fringe pattern is then very indistinct the fringes are said to have low
visibility This particular distance is
2d N 1 1 N 2 1 2 1 3
where N is an integer The wavelength difference is therefore
2 1 2d 1 4
If 1 and 2 differ by very little it is sufficient to replace their product by
2 where is the average of the two The expected value of 0 6nm
1 9 1 Procedure
Be sure that the vacuum cell is not in the interferometer for this part
Set up a white light source and locate the white light fringes as you did
previously This will locate the stage at zero path difference Recall that due to
backlash the stage position will differ if you approach this position from above
1 10 FOURIER TRANSFORM SPECTROSCOPY 11
versus from below Since you will eventually be using a MATLAB script to
take data and this script moves the stage forward i e to higher values on
the indexed stage then backwards then forward again back to the starting
point you should always approach from below your position of zero path length
difference
Replace the white lamp with the sodium lamp Now move the stage using the
toggle lever on the controller move the stage over a distance of at least 2 mm not
ing the stage readings at several positions of minimum visibility When recording
the positions you should always move the stage in same direction to avoid back
lash errors Repeat this process several times and average the stage readings
You can do this manually or you can use the script from before Your code will
again look like this
d michelson timestream span speed
where you choose span and speed appropriately A span 1 0 and speed
0 01 are probably good starting values for this part
From the average value of D obtained and the measured calibration con
stant K calculate d from equation 1 2 and hence obtain a value for from
equation 1 4 Use the average value of the two Na D1 and D2 lines that you
determined above and find the difference between the two lines
1 10 Fourier Transform Spectroscopy
So far you have investigated the behavior of the interference fringe pattern as
a function of the optical path difference for different optical sources In general
the fringe pattern intensity versus the optical path difference or equivalently
versus the time delay between the two interfering beams is related to the power
spectrum of the light entering the amplitude division interferometer by a Fourier
transform Therefore one can easily measure the optical power spectrum using
the measurement of the intensity variation as a function of stage position
Let s see how this works 1 The electric field incident on the camera is the
sum of the electric fields E1 and E2 arriving from M1 and M2 respectively after
having passed through the beam splitter Assuming for the moment that the
input source is a purely monochromatic wave we can characterize these electric
fields with a vector amplitude encoding the strength and polarization of the
wave and a spatial and time dependent phase
E1 A1 ei k1 r t 1 1 5
E2 A2 ei k2 r t 2 1 6
The phase includes terms 1 k1 l1 and 2 k2 l2 which depend on the
optical path lengths from the beamsplitter along path l1 encountering M1 and
path l2 encountering M2 The intensity on the camera is simply the square of
the total electric field
I E 2 E E E1 E2 E 1 E 2 1 7
A1 A2 2A1 A2 cos 1 8
I1 I2 2 I1 I2 cos 1 9
1 This derivation follows that in Introduction to Modern Optics by Grant R Fowles
12 CHAPTER 1 THE MICHELSON INTERFEROMETER
where we have assumed in the last step that the waves have the same polarization
and that k1 k2 r 1 2 You might be wondering about a white
light source which is probably completely unpolarized Is this assumption still
valid in that case This assumption is justified since the two fields E1 and E2
are copies of the incident field generated by the beam splitter As long as the
beam splitter and the optics which follow preserve the polarization of the light
these two fields will arrive at the detector with exactly the same polarization
whatever it was at the input of the interferometer What could the beam
splitter and following optics do to change the polarization of the input
If the interferometer is well aligned so that the waves are co linear k1 k2
and the 50 50 beamsplitter generates two waves of the same intensity I1
I2 I 2 then the interference pattern is simply
I x I 1 cos kx 1 10
where x l1 l2 is the path length difference and k k1 2 is the
magnitude of the wavevector
Okay now in general the light illuminating the interferometer will be com
posed of many different frequencies So let s model the power spectrum of the
incident light as a continuous distribution G Then the total optical power
emitted into the interferometer in the frequency range from 0 to 0 d
where d is an infinitesimally small frequency band is simply G 0 d For
convenience we can instead use the distribution over wavelength G k where
ck Then the optical power P hitting the detector from illuminating the
interferometer with this polychromatic source will be the sum of the interfer
ence patterns from each monochromatic wave composing G k Assuming the
interferometer is ideal and has no losses we have then
P x G k 1 cos kx dk 1 11
eikx e ikx
G k dk G k dk 1 12
P 0 G k eikx dk 1 13
where P 0 is simply the power measured at zero path length difference We
can rearrange this expression and define the power function W x
W x 2P x P 0 G k eikx dk 1 14
We see then that the power function W x is the inverse Fourier transform of
the power spectral density G k Inverting this expression we have that the
power spectrum is the Fourier transform of the power function
G k W x e ikx dx 1 15
or equivalently Z
G W x e i2 x c dx 1 16
1 10 FOURIER TRANSFORM SPECTROSCOPY 13
Monochromatic
Two wavelengths
close together
Wide band pass
white light
coherence length
Narrow band
pass filter
orange filter
mirror position
Figure 1 6 Fringe patterns plotted as a function of the mirror position Notice
how the fringe visibility collapses and then revives as a function of the mirror
position when the spectrum is discrete Also note that if the mirror is moved a
distance d the optical path length between the two arms in the interferometer
change by 2d
1 10 1 Procedure
Be sure that the vacuum cell is not in the interferometer for this part
In this stage of the experiment you will use the interferometer to realize
14 CHAPTER 1 THE MICHELSON INTERFEROMETER
a Fourier transform spectrometer The idea is to translate the stage around
the zero path length position ZPL record the fringe pattern and use it to
determine the power function W x and then compute numerically the corre
sponding spectral density G k of various optical sources To do this you will
as before use the MATLAB script michelson timestream m which records
an array of power values for each video frame i e each stage position You will
use your calibration constant for the motion K to deduce the stage position at
each frame and therefore generate the power function W x be sure the stage
velocities are set the same or else recalibrate K for this part From this you will
numerically compute the Fourier transform to determine the power spectrum of
the light White Light
power arb units
200 300 400 500 600 700 800 900 1000
wavelength nm
Figure 1 7 Example spectrum from the incandescent bulb measured by the
Michelson Fourier transform spectrometer Your reference white light spectrum
should look something like this one Keep in mind that this measurement of
the spectrum is ultimately limited by the wavelength dependent transmission
of the optical elements in the Michelson interferometer and by the sensitivity
of the CCD detector See Fig 1 8 for a typical response curve of a silicon CCD
image sensor
As before set up a white light source find the fringes at zero path length
difference make them vertical and note the stage setting Before taking each
data run you should carefully adjust the iris on the camera lens to maximize
the fringe brightness without saturating the camera This will optimize the
signal to noise and produce much better data Start by recording the fringe
pattern and computing the spectrum of the white light source You should get
a spectrum similar to that shown in Fig 1 7 This is your reference spectrum
which you will be modifying with a filter Keep in mind that this measurement
of the white light source spectrum is ultimately limited by the transmission of
the optical elements in the Michelson interferometer and by the sensitivity of
the CCD detector See Fig 1 8 for a typical response curve of a silicon CCD
image sensor
To take the data for this section you will again be using the MATLAB
script michelson timestream m only this time you should use a very slow
speed and a very narrow span since the filtered white light has a short coher
ence length and you will need good spatial resolution which will determine the
1 10 FOURIER TRANSFORM SPECTROSCOPY 15
Figure 1 8 The spectral response of a typical silicon CCD image sensor ranges
from 400 to 1000 nm
highest frequency in your spectrum so you can see the full spectrum without
any truncation Recommended values are speed 0 001 and span 0 05
Note that this span might not be wide enough for the narrow band filter but
that it already will take around 100 seconds given the speed So first check with
a higher speed that this is the correct range for your filter and ONLY THEN
take your data at the appropriate slow speed Be aware if you choose a span
smaller than the backlash in the micrometer screw the stage will not return to
where you started You should therefore insure the span is wide enough that
the stage returns to the starting point properly
Carefully without disturbing the interferometer alignment insert the 589 nm
interference filter and record the fringe pattern generated Take your spectrum
and normalize it so that the peak of the spectrum has an amplitude of 1 Do
the same with the reference spectrum and divide the filtered spectrum by the
reference to deduce the transmission function of the filter Does the 589 nm
interference filter have a Gaussian transmission function Why or
You should now take a spectrum of the HeNe laser to check that
your calibration of the stage velocity is correct Is the peak of the
spectrum where you would expect it If not why not You should
use this information to refine and or correct the spectra you took of
the filtered white light What is the measured width of the spectral
peak of the HeNe output in nm The linewidth of the HeNe laser is
well below 100 MHz Is the measured width larger than this If so
1 10 2 Analysis with MATLAB
In order to compute the power spectra from your fringe pattern data you will
need to do some data processing First you will need to properly truncate your
data because the vector of data data may include points when the stage was
moving backwards and then sitting stationary before and after the slow sweep
through the ZPL position Therefore find the points AA and BB which are at
the beginning and end of the data and truncate the data Your code will look

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