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APh 24

VIEWS: 4 PAGES: 11

									APh 24                           Experiment No. 4                          April 4, 2007

                MICHELSON AND JAMIN INTERFEROMETERS

I. Brief Overview

In this experiment you will learn how to set up and align Jamin and Michelson
interferometers. You will use the Jamin interferometer to study the spatial variation
of the phase of light transmitted through simple objects. You will use the Michelson
as a Fourier–Transform Spectrometer to examine the spectrum of the light from a
helium–neon laser and a sodium discharge lamp. Review Hecht’s Sections 9.1, 9.2,
9.4, and 9.8, and your notes from APh 23.

II. Theoretical Background

The general scheme of any two–beam interferometer is to divide the beam from a
source into two separate paths and then recombine the beams to create an
interference pattern. If the source’s frequency spectrum is assumed known, we can
learn something about phase difference between the paths. For example, we can
examine the phase variation across a transparent object by inserting the object in one
interferometer arm and observing the change in the output fringe pattern (Part IV A
below). Conversely, if the details of the two propagation paths are well known, we can
learn something about the frequency spectrum of the source. In Parts IV B and C of
this experiment, we will demonstrate this concept by using the Michelson
interferometer as a high resolution optical spectrum analyzer.

The derivation of the relationship between the interference pattern and the spectrum
of the light source is beyond the scope of this write up. (See Michelson’s original paper
for the details.) However, the basic result is simple: the envelope of intensity
variation or fringe visibility will change with the relative path length (L2 – L1). This
is illustrated below in Figure 1.

                        Ligh t In t en sit y




                                                    (L2 – L1 )


                        F r in ge Visibilit y




                                                    (L2 – L1 )



   Figure 1. Illustration of fringe visibility in an interferometer.

                                                1
   The individual fringes shown in top of Figure 1 have a width of about  2  c 2 ,
   while the fringe visibility changes much more slowly on the scale of c  , where  is
   the spectral width (in frequency) of the light source. The variation in fringe visibility
   is associated with frequency “spread” in the spectral components of the source instead
   of the optical frequency itself. Quantitatively, the mathematical derivation shows
   that the visibility is the magnitude of the Fourier transform of the light
   source’s frequency spectrum. Stated in more formal terms, the visibility is
   V ( )   ( ) , where  ( ) is the so-called “complex coherence function”, a function
   derived from the spectrum’s Fourier transform.

   Given a known frequency spectrum, we can predict the fringe visibility. However,
   there is a uniqueness problem for the reverse process, since the visibility has only
   magnitude information. If we measured both the magnitude and phase of the
   interference pattern, we could uniquely reconstruct the source’s frequency spectrum.
   Unfortunately, this is not easily done. Michelson recognized this problem back in the
   1890’s. Like many resourceful would-be Nobel laureates, he basically “cheated”: he
   guessed a spectrum I () and then compared the magnitude of the spectrum’s
   coherence function with what he measured for V ( ) . We will proceed in a similar
   way: a list of the “usual” frequency spectra suspects are listed in Table I (pp 4-5) with
   their respective visibilities. You will match the observed visibility to the best guessed
   spectrum. The Pre-laboratory questions will give you feeling for this procedure.



   We can illustrate these ideas for some simple cases. Suppose a Michelson
   interferometer is illuminated with light that consists of two monochromatic
   components of equal amplitude:

           Original source: e(t) = E 0  ( a t)  cos ( b t)
                                        cos                                                       (1)

   Suppose the field is split equally at the beam splitter and then recombined. We can
   describe this by the following equations:

                                                  
           Along the paths: e1 (t) = (0.707)E 0 cosa (t   1) cosb (t  1 )  
                                                                                        
                                e2 (t ) = (0.707)E 0   a (t  2 ) cosb (t   2) 
                                                      cos                                        (2)

           Recombined:        e net (t)  (0.5)E 0  a (t  1) cos b (t   1) cosa (t   2 ) cos b (t  2 )
                                                    cos                                                      

                                                                                             ,
e net (t)  (0.5)E 0  a (t  1) cos b (t   1) cosa (t   2 ) cos b (t  2 )
                      cos                                                      
                                                                                                (3)
                      L1             L2
      where  1         c and  2     c are the round trip travel times. (The reason the
      recombined total is only (0.5) rather than (0.707) is due to the fact that the
      recombiner is also a splitter which further reduces each field by an additional (0.707).)


                                                      2
The intensity will be proportional to the time–average of the square of e net (t) :

                          
                                   b    a   b  
                                                                   
       I(  )  (0.5 )I 0   cos a
                           1                    cos           ,               (4)
                          
                                   2             2        
                                                                    
where τ ≡ |τ1 – τ2|. The intensity has a rapidly varying component oscillating with
path difference as
                         a   b 
                                    2 c and slowly varying component oscillating as
 a   b  . If we consider only the slow variation, then the visibility of the fringes is
            2 c


                       b  
                                
       V (  )  cos a           .                                                (5)
                       2        

Recall, that the Fourier transform of two delta function is indeed a cosine function.
This is a special case of the second example in Table I. More complicated source
spectra will produce more complicated visibility curves.

The simplest technique for introducing a path length difference is to lengthen one
arm of an interferometer with a manual stage. This is practical for short distances.
For example, in Part IV C the required path changes are a few tenths of a millimeter
for a sodium lamp source. For much longer path length differences, it is more
convenient and accurate to use an automated mechanical stage which can move one
of the mirrors at a constant velocity. This is the approach used in Part IV B.

Consider a Michelson interferometer which is illuminated by a monochromatic source
of frequency  One of the mirrors is moved at a constant velocity, v, so that
(t) ≡ (L2–L1)/c = 2vt/c. The intensity of the light will be
                                                               
           I(t)  0.5I 0   cos (t)   0.5I 0   cos2v( c)t  ,
                          1                       1                              (6)

The “fast” fringe frequency (2v/c) is exactly the same as the Doppler shift from a
mirror at velocity c. By measuring this frequency of intensity variation, we can
determine the velocity of the mirror. This is essentially how a radar/ladar gun works.

Suppose the source had two monochromatic components as described by Equation 1.
We expect to see “slow” changes in the fringe visibility occurring at a much lower
frequency:

        a          
                b  2v c .                                                          (7)

By measuring these slow variations we can determine the frequency difference of the
two components.




                                                     3
III. Preparation

The notebook for this experiment is available on the reserve shelf of the Fairchild
Library and in the laboratory. It contains copies of pages with detailed descriptions of
the Michelson Interferometer and Jamin Interferometer. If you enjoy reading about
the history of technology, you might also look at A. A. Michelson's 1892 article (in the
folders). Michelson was America's first Nobel Laureate.


Do the following prelab–exercises in your notebook:

1. A helium–neon laser ( = 632.8 nm) oscillates on two cavity modes separated by
500 MHz. Assuming each oscillation is completely monochromatic and that the two
are of equal intensity, plot the fringe visibility curve V() for 0 <  < 5 nsec. Also, add
a second scale to the abscissa, showing the variation of V with L, the inequality in
arm lengths of a Michelson interferometer. Recall  = 2L/c, the factor of 2 arising
because of the path is traversed twice by the optical beam. What would the curve look
like for the same laser if the modes did not have equal amplitudes, but instead one
had twice as much power as the other? Sketch this curve on your previous plot.

2. Using the formula for Doppler linewidth given in Table I (the linewidth caused by
thermal motion of the atoms) calculate the width of the sodium yellow lines  = 589
nm for the case when T = 500 Kelvins. M is the weight of a single sodium atom and k
is Boltzmann's constant.

3. The wavelength spacing of the two components of the strong yellow sodium
doublet is 0.597 nm ( = 589.5923 and 588.9953 nm). By how many MHz are these
components separated? How does this compare with the value for the Doppler width
you calculated in Problem 2. Based on these two frequencies, sketch the spectrum and
the fringe visibility function you would expect, showing numerical values of
important features.

4. As discussed above in Section II, it is possible to use a Michelson interferometer to
measure the velocity of a moving mirror by the Doppler effect. The Doppler–shifted
reflection from the moving mirror is mixed with the reflection from the stationary
mirror to produce the frequency difference. The rate at which fringes change from
dark–to–light–to–dark is just this frequency. What is the numerical value of the
Doppler shift in Hz if the mirror moves 1 cm/sec? Assume a He–Ne laser with
 = 632.8 nm.

5. The index of refraction of air is about [1+ .0003 (pT 0/p0T)] where p is the actual
pressure in Torr, T is the actual temperature in Kelvins, and p0 = 760 Torr, T0 = 293
Kelvins. What rise in temperature will produce a single fringe (a change from dark to
light back to dark, say) in the warm air rising from your hand under one path of the
Jamin interferometer? (You will have to assume a value for the size of the warm air
plume).
                                            4
IV. Procedure.

Turn on both Helium–Neon Lasers when you start this experiment. (It is important
to allow the laser for the Michelson interferometer to warm up.)

A. Jamin Interferometer (2 hours)
1. After examining Figure 2, identify the Jamin interferometer on the table. Adjust
the interferometer so that there is a single, bright fringe filling the output beam. Make
a sketch of what you see. Have a TA check it over.
2. Make a simple sketch of how the interferometer works and write a simple
description in your notebook. Make descriptive remarks about the details.
3. Observe the interference fringes caused by the thermal plumes from your warm
hand as you hold it under one path. There is also a soldering iron available to create
still warmer air. Sketch the fringe patterns and describe in your notebook.
4. Insert a microscope slide or other piece of glass in one path and estimate the
thickness variation in the glass from the number of fringes you see. Sketch what you
see. Can you tell what part of the glass is thicker and what part it is thinner?
5. Insert the liquid cell in one path (fill it with fresh water first, available from the
drinking fountain in the hallway.) Can you see any turbulence in the cell? Warm the
ground sides of the cell with your fingers. Can you see the fringes form near the walls?
Drop a few salt crystals into the cell. Can you see the tracks of saline water? What
causes the tracks to remain visible after the salt crystals reach the bottom of the cell?
Sketch the fringe patterns you observe and describe the details in your notebook.
6. Use the aerosol can to observe the difference in refractive index of a gas mixed in
air. (The can contains freon and is colder than air due to rapid expansion.) Sketch the
fringes introduced by objects, the "airfoil" etc. This is one way to map fluid flow in a
small wind tunnel.
7. Move the Jamin equipment out of the way and proceed to the Michelson
interferometer.

B. Michelson Interferometer (3.5 hours)
1. Compare the table set up and electronic equipment with Figures 2 and 3 in the
handout sheet. Do you recognize everything and know how it works? If not, be sure to
ask one of the TAs. Most of the optical components and already bolted to the table in
their correct positions, and the beam splitter is adjusted to the correct angle. You may
have to adjust M1, M2, M3, and M4, in order to complete the alignment of the
interferometer.




                                           5
                           Tektronix TDS 360 Oscillos cope


        Pow er
        Supply




        Detector                                               H–P 850C
                                                             Deskjet Printer
        (Located on the
         Optical Table)
                              H–P 34401A
                              Multimeter




Figure 3. Schematic diagram showing the electronic instruments that will be used to
make time dependent fringe measurements.




                                              6
2. Ask a TA to explain what you are going to do and what you expect to see in
general terms, so that the TA will have confidence that you understand the
laboratory setup. This will also give the TA an opportunity to give you some handy
hints.
3. Look at the motor control for the moving carriage on the UniSlide. Note that there
are 3 switches and a knob that control the carriage motion: (a) a 3–position speed
range rocker switch labeled "off–low–high,” (b) a toggle switch labeled motor "run–
stop", (c) a variable motor speed control knob, and (d) a slide switch labeled "forward–
reverse,” which is located on the bottom end of the control box.

4. We suggest that you use the controls as follows: Leave the "run–stop" toggle switch
in "run". Turn the motor on and off with the "off–low–high" rocker switch. (You may
run the moving carriage at moderate speeds to expedite moving it from one end to the
other, but please avoid exciting nasty mechanical resonances in the sliding
mechanism.)

5. Place the two alignment targets on the UniSlide. (You may have to run the moving
carriage to the far end of the slide.) Mirrors M3 and M4 are used to give you the
necessary four–degrees of freedom needed to adjust the position and angle of the laser
beam to successfully align it to the targets. You will need to carefully adjust M3 and
M4 so that the laser’s beam is parallel to the axis of the UniSlide. The hole in the first
target and the reference mark on the second target define the axis of the slide to an
accuracy of ± 0.2 mm.

6. Check that the beam reflected from the beam splitter will hit mirror M2 about in
its center. Then place a card in front of M2 so you can perform the following steps to
align M1. Remove the optical isolator from its position in front of the laser.

7. Remove the second alignment target and adjust mirror M1 so that the laser beam
is reflected back into the laser through the hole in the first alignment target. Note
that you can see the reflected beam from mirror M1 in the scattered light on the front
lens of the laser telescope. A slight adjustment of M1 will get the reflected beam close
to exact retroreflection and you will see the multiple reflections. When you have the
beam from M1 adjusted for exact retroreflection, all of the multiple reflections will be
coincident with the main spot and you may possibly be able to see that the laser beam
intensity is rather unstable or will flicker as you touch M1. You now have M1 so well
aligned that the laser thinks that M1 is part of its resonator, competing with the
output mirror in the laser to determine the exact laser frequency. This will be an
undesirable situation later, but for now we can use this effect to help us align the
interferometer. Now remove the first alignment target.

The mirror M1 is now very precisely adjusted and you should not touch its
adjustment screws for the remainder of the experiment.

8. Move the card placed in front of M2 into a position in front of mirror M1 so that
you can use the same technique to adjust mirror M2. After you have aligned M2,
remove the card from the other path and correct the adjustment of M2 so that its
beam is precisely on top of the beam from M1. You may possibly see interference
effects in the interferometer’s output beam. In order to eliminate undesirable optical
feedback into the laser, place the optical isolator in the beam path in front of the
laser.


                                            7
9. With the motor control, position the UniSlide’s moving carriage to the middle of
slide. You should now see interference fringes or flickering, which will change if you
adjust mirror M2. (Try it, but adjust the mirror only a small fraction of a turn.) If you
don't see this effect, call in a TA. If you do, call in a TA anyway and demonstrate it.
With lens L3 in the output beam close to the beam splitter as shown in Figure 2,
carefully adjust mirror M2 so that you have a "single fringe" covering the entire
output beam. Then adjust M2 slightly away from this position and note the parallel
fringes that you will see. Note how little motion of the mirror adjustment it takes to
produce a lot of parallel fringes. (The mirror adjustment screws have 80 threads–per–
inch, so 1° of rotation of a screw tilts the mirror approximately 23 µradians.) Now
return M2 to exact the exact alignment position, that is, a uniform, flickering spot. In
your notebook, make a sketch of what the interferometer’s output beam looks like.

10. Using the motor control, move the mirror carriage to the position nearest the
beam splitter so the two optical paths are approximately equal length. Note that
there is now several fringes across the output beam. This occurs because the UniSlide
is actually warped a bit. Of course it would do no good to realign the mirror since it
will then be misaligned by the same number of fringes when the carriage is at the
center of the slide. Note that the warp is very small, not enough for you to see by eye
if you try sighting along the surface of the slide, for example.
Now move the carriage away from the beam splitter in 10 cm increments, stop the
motor, and check to verify that the mirror alignment has not changed radically. As
you perform this test, you will also no doubt find some positions along the UniSlide
where the fringes become very low in contrast. These are the "washout positions" in
the fringe visibility curve that we will record later. (You also may not see them; it
depends on what the laser is doing at the moment.)
11. Return the carriage to the equal–path–length position. It is now necessary to ask
about the effect of the UniSlide’s warped nature on the measurements you are to
perform. If you think about it, all we want to do is record (photoelectrically) how a
"fringe" changes in time but not in space. Thus, if our photodetector is smaller than
the width of the smallest fringe we have to measure, it won't really matter that there
are many fringes. Since we have used a lens to magnify the size of the fringes, all we
have to do is place the photodetector far enough away from the lens so that the
photodetector sensitive area is always smaller than the smallest fringe width that
will occur due to misalignment during the carriage travel. The photodetector size is
2.29 x 2.29 mm. You should be able to find a spot where this requirement is met. If
not, call in a TA for help.
12. Turn on the two electronic instruments located on the workbench and plug in the
power supply for the photodetector. The instruments are connected together as shown
in Figure 3. The Tektronix TDS 360 Digital Oscilloscope is used to view the
photodetector signal. This signal is proportional to the interferometer output fringe
intensity. The H–P 34401A Multimeter is used to measure the fringe frequency when
the mirror carriage is moving. These instruments should be configured using the
procedures shown in the Experiment 4 Binder that is located on the workbench: you
should recall “Setup 1” for the scope and the multimeter timebase should be set to 50
µsec/div. Turn on the UniSlide drive motor and adjust it to move the carriage at a
slow speed, approximately four mm/sec. (A motor control dial setting of 20 gives about
the proper carriage speed.) Note the sinusoidal signal on the oscilloscope and note
how the amplitude of the signal varies with carriage position. This amplitude
                                           8
variation is the fringe visibility function being swept out as you move the carriage.
Ask a TA if you're puzzled.
13. You will now attempt to measure the HeNe laser’s wavelength by using the
interferometer. As the mirror carriage is moving, you will notice that the frequency is
not exactly constant, a result of the vibration and "sticking" in the carriage motion.
The H–P Multimeter measures the frequency by integrating for 1.67 sec and
displaying the result. Write down a series of these measurements while the carriage
moves a distance of approximately 10 cm and calculate the average value. Reverse the
direction of the carriage and repeat the measurements. Measure the velocity of the
mirror carriage using the length marks on the side of the Unislide and the second–
hand on the wall clock above the door. Be sure to note that the velocity of the carriage
is slightly different for the two directions. Use this velocity and your observed Doppler
shift frequency to estimate the wavelength of the HeNe laser, which should be 632.8
nm. How close did you come to the correct result?
14. Reconfigure the Tektronix oscilloscope to display the envelope of the photodetector
output signal by recalling “Setup 2”. (Refer to the Experiment 4 Binder.) Turn on the
HP 850C Deskjet Printer.
Turn on the mirror carriage drive on the Unislide and observe the display on the
oscilloscope screen. The oscilloscope is displaying the envelope of the photodetector
output voltage as a function of time in a rolling–display mode at a speed of 10 sec per
division on the screen.
15. You should see a "fringe visibility curve" like the one you drew in prelab exercise
1. You won't see a simple [|cos| curve] unless the laser just happens to have two
modes oscillating that are equal in amplitude. Most often you will see a not–quite–
sinusoidal variation that doesn't reach the zero line. Actually, you can see anything
"between" the top two figures in Table I. In fact, before you take any serious data,
make a neat sketch in your notebook of what you expect to see, assuming the laser is
slowly drifting from one mode to two modes to one mode. Call in a TA for an opinion.

The peak–to–peak variation may also change during your scan if the laser happens to
be drifting fast enough to change its spectrum significantly. This change should occur
in the minima (the cusps in the fringe visibility curve), the maxima (the "1" line in
the fringe visibility curve) should all be at the about same level. If not, it indicates
that your alignment is going out as the carriage moves. Call a TA to check on your
results, and help with adjustments if needed.

16. The Tektronix oscilloscope will transfer the screen contents to the H–P printer
whenever the HARDCOPY button is pressed. The screen is cleared and the scrolling
display continues while the printer is printing. It is thus possible to produce a
continuous record of the interferometer fringe visibility function as the mirror
carriage is moved along the entire length of the Unislide.

Take at least four scans of the UniSlide carriage (forward, reverse, forward, and
reverse). By measuring the distance between fringe visibility maxima, determine the
average length of the laser cavity. (The specification sheet says that c/2L = 687 MHz

                                           9
for the Melles–Griot LHP–121 Laser.) How closely do you agree? Estimate the laser
cavity drift rate, i.e., the change in the cavity length vs. time, in wavelengths per
minute. Assume that if/when the distance between fringe visibility maxima returns
to the same value, the cavity length has changed by a half–wavelength. Remember to
calibrate the abscissa on your interferogram by measuring the distance that M1 travels
between the stops on the Unislide.

17. Call in a TA for a discussion of your results. If you and the TA are satisfied, glue
the interferogram into your notebook, making notations, labeling axes, etc. directly on
the charts.

18. Turn off the electronic instruments on the workbench.


C. Sodium Lamp Spectrum (1/2 hour)

The purpose of this part of the experiment is to measure the wavelength separation
between the two spectral components of the sodium doublet spectral line. You will use
a technique similar to the one you used earlier to find the frequency spacings of the
He-Ne laser, namely measuring the distance between fringe fade outs.
1. Move the carriage until the two arms of the interferometer are equal to within a
few millimeters. Check to see that M1 and M2 are still aligned, and that you get a few
fringes showing across the output beam.
2. Turn the laser off, but be careful not to move mirrors M3 and M4 since you may
need to realign the interferometer mirrors in case you bump something.
3. Turn on the sodium lamp and place it in position between M4 and the beam
splitter. (It takes a few minutes for the sodium lamp to warm up and it will gradually
get brighter as it vaporizes the sodium metal condensed on the glass walls.) Remove
lens L3 and place the small adjustable–angle viewing mirror where you previously
had the detector. Stand along side the table between M2 and the east wall of the
room. Adjust the viewing mirror so that you can see the sodium lamp’s light that is
coming out of the interferometer.
4. This will allow you to look into the output of the interferometer while adjusting
the micrometer–driven translation stage under M2. You should see black and yellow
fringes floating in space in front of the image of the lamp. Adjusting the alignment of
M2 will change the number of fringes you see. In this case, you do not need to adjust
for "one big fringe". Instead, you will want to see a few fringes so you can judge their
contrast ("visibility") by eye. In your notebook, make a sketch of what the fringes look
like.

5. Move M2 with the micrometer, a small fraction of a turn at a time. You should see
the fringes fade out, become distinct, fade out, and so on. If you don't see this, call a
TA for help. If you do, proceed. Act efficiently, since the lamp can and will overheat.


                                           10
6. If, during the next step, you should note a complete disappearance of the fringes for
all positions of the micrometer, it is likely the sodium lamp has become too hot. Do
not adjust the interferometer mirrors, but turn off the lamp and let it cool for five
minutes or so, then turn it on again and try to do the next step in no more than
fifteen minutes.

7. Measure the distance between the "fade–outs" of the fringes, since they are
somewhat sharper than the peaks of good contrast. We suggest not trying to set the
micrometer exactly at two adjacent "fade–outs". A better strategy is to set
approximately to a "fade–out", record the reading, then turn the micrometer in short
steps (say 0.01 or 0.02 mm at a time) until you think you are at the next "fade–out" or
have passed it. Don't try to go back, just write down the reading where you stop. Go
on to the next "fade–out" and do the same thing. After going through about 10 cycles,
you can get the distance between "fade–outs" pretty accurately by subtracting the end
readings and dividing by the number of cycles you traversed. The intermediate
readings are not used except to assure you that you didn't miss any "fade–outs" and
have counted the number of cycles correctly. (If you don't know how to read a
micrometer, call in a TA.)

8. From your data and the appropriate function from Table I, compute the separation
of the sodium yellow doublet. It should be 0.597 nm. (For reference, the energy levels
diagram for sodium is in the Lab Binder.) How close did you come?

9. Turned off the main power switch located on the front of the workbench. Put the
sodium lamp back on the shelf.




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