Download Michelson Interferometry and Measurement of the Sodium Doublet Splitting

Michelson Interferometry and Measurement of the Sodium Doublet Splitting
PHYS 3330: Experiments in Optics
Department of Physics and Astronomy,
University of Georgia, Athens, Georgia 30602
(Dated: Revised October 2011)
In this lab we will use a Michelson interferometer to measure a the small difference in wavelength
between two closely spaced spectral lines of a Sodium lamp.
I.
INTRODUCTION TO MICHELSON
INTERFEROMETERY
In the Michelson interferometer, an incident beam of
light falls on a beam splitter which reflects roughly half of
the incident light amplitude. Reflected and transmitted
beams follow different paths, are reflected, and recombined to producing an interference pattern. The structure of the interference pattern depends upon differences
in the length and alignment of the two arms of the interferometer, and also upon the surface smoothness of the
optical components. One can make quantitative measurements of the interference pattern for the accurate
comparison of wavelengths, to measure the refractive index of unknown substances, and measure the quality of
optical components. The LIGO gravity wave detector is
a Michelson interferometer with 2.5 mile long arms.
You will need to complete some background reading
before your first meeting for this lab. Please carefully
study the following sections of the “Newport Projects in
Optics” document (found in the “Reference Materials”
section of the course website): 0.4 “Interference” Also,
the excerpt from Melissinos’ “Experiments in Modern
Physics,” included as an appendix to this manual. Finally read section 8.1 of your textbook “Physics of Light
and Optics,” by Peatross and Ware. Your pre-lab quiz
cover concepts presented in these materials AND in the
body of this write-up. Don’t worry about memorizing
equations – the quiz should be elementary IF you read
these materials carefully. Please note that “taking a
quick look at” these materials 5 minutes before lab begins
will likely NOT be adequate to do well on the quiz.
The optical circuit of a Michelson interferometer is
shown schematically in Figs. 2 and 3. Light from a source
S passes through a ground glass plate DB (optional) and
strikes the beam splitter P. The beam splitter P is a partially silvered mirror (50% reflecting). Half of the incident light amplitude toward mirror M1 and transmits
half of the incident amplitude toward mirror M2. A micrometer adjuster screw is attached to the movable mirror
M1, permitting it to be moved toward or away from the
beam splitter in small, precise steps. The two mirrors,
beam splitters, and compensating glass are flat to about
a 1/4 of an optical wavelength. The compensating glass,
CG, of identical composition and thickness to the beam
splitter, is included so that each of the two beams (paths
P-M1-P-O and P-M2-P-O in Figure 3) passes through the
FIG. 1: Your Michelson interferometer setup. The interferometer is in the lower left, the sodium lamp power source in
the upper left, and sodium lamp in the upper right portion of
the picture.
FIG. 2: Typical Michelson interferometer. The important
parts of a Michelson interferometer include a sturdy base,
a diffusing glass, a beam splitter, a movable mirror with a
micrometer screw for measuring distance of movement, a fixed
mirror, and compensating glass. The light source can be a
spectral lamp, a collimated laser beam, or even a white light
source.
same integrated thickness of glass. (Note that otherwise
the beam that travels along path P-M1-P-O would pass
through a thickness of glass three times while the beam
that travels along the other path would pass through the
same thickness of glass only once. The compensating
glass is not necessary to produce fringes using laser light,
but it is essential for producing interference fringes with
white light, such as those shown in Figure 9.
Light traveling along trajectories making an angle φ
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FIG. 3: Optical arrangement and light path in Michelson interferometer
FIG. 5: HeNe fringes in a Michelson interferometer from this
lab. Photograph taking by your instructor with an iPhone.
FIG. 4: Condition for interference
with respect to the optic axis accumulate a path length
difference of 2d cos θ between the arms. When this difference is an integer number m = 0, 1, 2... of wavelengths,
destructive interference occurs (dark fringes).
mλ = 2d cos θ, m = 1, 2, 3....
(1)
where m is the “order” of the interference. Note that
the beam in one arm undergoes an additional external
reflection, and thus incurs one additional π phase shift,
relative to the beam in the other arm, which is why the
above condition produces a dark, rather than a bright,
fringe.
If the two mirrors M1 and M2 are not aligned precisely
perpendicular to one another the path difference will depend on the particular region of mirror M1 (and the corresponding region of M2) which we are observing from
the position O. The field of view, then, seen by looking
at mirror M1 from position O will be made up of a series
of alternately bright and dark fringes, nearly straight and
parallel, as shown in Figure 5. If the path difference is
near zero, the fringes will be broad and widely spaced in
the field of view; if the path difference is on the order
of 40 or 50 wavelengths the fringes will be narrow and
closely spaced, so much so that they may be unresolvable by the naked eye. Such fringes are shown in Figure
5.
If the two mirrors are precisely aligned exactly parallel
to one another, a “bulls-eye” fringe pattern will be seen
FIG. 6: Circular fringes (equal inclination) seen in Michelson
interferometer
by the observer situated at point “O” in Fig. 3, consising
of a series of concentric rings. Each ring corresponds to a
different angle φ, as illustrated in Figure 6. In this case,
when M1 is translated a distance δz along the optic axis,
the number of fringes N that will appear (or disappear)
at the center of the bulls-eye pattern is:
N = 2 δz/λ
Thus, if you can measure the displacement of M1 which
causes a known number of fringes to to appear (or disappear) from the center of the pattern, an unknown wavelength can be measured. Conversely, you can use a known
wavelength to calibrate the micrometer screw; i.e., convert microns of travel of the screw to microns of travel of
the mirror (which are not necessarily equal!).
II.
ALIGNMENT OF THE INTERFEROMETER
USING A LASER
1. Place and orient your steering mirror to direct the expended beam from a HeNe laser into the the input
port of the interferometer.
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2. Observe three discs of light emerging from the output
side. Two of the copies will lie almost on top of each
other, but the third will likely be far to the side (or
even absent), if the mirror M2 is misaligned. M2
is equipped with two screws on the back side that
tilt the plane of the mirror. A slight adjustment
of the mirror tilt screws will cause one of the three
images to move. You can achieve the proper alignment of the mirrors by using the screws to superimposing the (movable) image onto the rightmost
of the two stationary images. You will see interference fringes appear, though initially they may
be very finely space. As you adjust M2 you must
momentarily STOP turning the screws to look for
fringes; you will not see fringes if you are turning
the screws even if the mirrors are perfectly aligned,
as the movement of the mirror blurs the pattern.
4. It turns out there are two orientations of the M2
which produce fringes with a HeNe laser in your
interferometer. It is important for later stages of
this lab that you now pick the correct orientation.
To do this you must carefully observe the output
and compare to Figure 7. In the wrong case, the
strong fringes die out abruptly on the left side of the
disc, when looking at a ground glass plate installed
on the output port. In the correct case the strong
fringes extend all the way to the left edge of the
pattern. The difference is subtle.
5. While observing the fringes, carefully adjust both
screws on mirror M2 so that the fringes take a circular “bulls eye” pattern. See Figure 8 for guidance.
III. CALIBRATING THE MICROMETER
SCREW USING A HENE LASER AS A
WAVELENGTH REFERENCE.
M1 can be translated without disturbing the alignment
of the interferometer. Each tick on the thimble of the
micrometer adjuster of M1 corresponds to 1 micron of
movement of the spindle. One complete revolution of the
thimble advances the the spindle through 50 microns,
and moves the edge of the thimble across the distance of
one tick-mark on the barrel. Thus, 10 ticks on the barrel
is 5mm of movement of the spindle. Make SURE you
are clear on how to read the scales on the the micrometer adjuster before you start taking calibration measurements. (See http://en.wikipedia.org/wiki/File:
Micrometer_caliper_parts_0001.png if you need a
picture.)
1. Set the micrometer screw to approximately 5 mm.
2. Turn the micrometer screw a quarter-turn in the
direction of smaller reading. This is done to
avoid backlash since all readings will be taken with
the screw moving in the same direction (towards
FIG. 7: The right (upper) and wrong (lower) appearance of
the fringes. In the wrong case, the strong fringes die out
abruptly on the left side of the disc, here demarcated with
the blue dashed line. In the right case, the strong fringes go
all the way to the left edge. The difference is more obvious
when viewed in person.
smaller readings). Record the reading of the micrometer.
3. Count the number of fringes that pass through the
center of the field of view as the micrometer screw is
turned slowly in the direction of decreasing reading.
After counting 10 fringes, record the micrometer
reading again. When you stop turning the screw
at the end of 10 fringes, be very careful to NOT
accidentally slip a fringe while you are recording
the micrometer reading.
4. Continue this process for 20 groups of 10 fringes.
You will find that this procedure requires a cer-
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FIG. 8: Successive fields of view in interferometer alignment
tain amount of technique (and patience), since the
slightest movement of the screw will gain or lose a
fringe.
IV.
ANALYSIS
Enter these points into an Excel spreadsheet, export
as CSV, import the data to python, and perform a fit of
the data to a linear model. Consider carefully what free
parameters you want to include in your model. Don’t
worry about including error bars in this fit. You know
that each group of 10 fringes actually moves the mirror by
5 wavelengths of the HeNe laser. Plot the actual mirror
displacement vs reading of the micrometer. Fit this data
to a straight line. The slope of the line is the calibration
constant K that you are seeking
K=
microns of actual mirror travel
microns of travel of the screw threads.
MEASUREMENT OF CLOSELY SPACED
SPECTRAL LINES VIA MICHELSON
INTERFEROMETERY.
In the next part of the lab you will use your calibrated
Michelson interferometer to measure a small difference
in wavelength between two closely spaced spectral lines
of a sodium lamp. The 589nm “yellow” line of sodium
actually consists of two distinct lines, separated by a few
tenths of a nanometer. When a sodium lamp is used as a
source for a Michelson interferometer, each line will produce its own set of fringes with a slightly shifted pattern
relative to the other. At certain positions of mirror M1
the two sets of fringes coincide (bright regions overlapping bright regions), and the total intensity pattern observed is a bulls-eye pattern of moderately high contrast
(a “sharpening coincidence”). When the M1 is moved,
the two sets of fringes evolve slightly differently, and at
some setting will anti-coincide (bright regions overlapping dark regions) so that a total intensity pattern displays no fringes (a “wash-out anticoincidence”). We can
ma λa = 2 δd
(3)
mb λb = 2 δd
(4)
for orders ma and mb which, as integers, must be related
by
mb = ma + M,
(5)
where M is the “order” of the coincidence, or its number
of sharpening coincidences which would have been observed if one had started observing from the white light
condition δd = 0, in which case both interference patterns would have had dark central fringes, as there would
be no path length difference for either (any!) wavelength.
Substituting (2) and (3) into (4) we have:
(2)
The uncertainty in the slope as reported by the fitting
routine will be a useful estimate of your uncertainty in
the calibration procedure, and you will use this information to estimate a systematic uncertainty in your sodium
wavelength measurements of the next section.
V.
use the periodicity of the wash-out phenomenon to measure the sodium line spacing. The theory is described
next.
The two spectral lines whose difference is to be measured are at wavelengths λa , λb . Let δd be the path
length difference between the two interferometer arms at
some sharpening coincidence. At this coincidence each
set of fringes satisfies a dark fringe criterion for the central fringe of each bullseye pattern (Equation (1) with
θ = 0.
2 δd/λb = 2 δd/λa + M
(6)
λa λb
λ̄2
M≈
M
2∆λ
2∆λ
(7)
or
δd =
where λ̄ is the mean wavelength of the two closely spaced
lines. Thus, if we measure the mirror position for several
sharpening coincidence orders M, M + 1, M + 2, ... the
slope of a linear fit to the data will give us ∆λ. Note
that this is true EVEN if we are off in our reckoning of
the absolute order M by some an unknown integer X,
as the slope we infer from the linear fit to the data is
(of course) independent of arbitrary translations of the
horizontal axis M 7→ M + X Therefore it is not critical
to begin the measurement at the white light condition,
although it does help to make the sharpening coincidence
more obvious. Also note that the same equations apply
for “wash-outs”, which are typically easier to identify. In
this lab you will look for wash-outs.
VI.
PROCEDURE
1. Turn on the sodium lamp and wait at least 5 minutes
for the light to reach full intensity.
2. Direct the light from the lamp into the interferometer
using the steering mirror.
3. Position a ground glass diffuser at the output port.
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4. Spin the micrometer screw to nearly its maximumreading.
5. If you have already achieved fringes with the HeNe
laser you should immediately see fringes. If you
do not see fringes it is possible that you have (unluckily) landed on a wash-out — try spinning the
micrometer screw a turn or two. If you still have
no fringes, put the HeNe light back into the interferometer to check to see if something has been
bumped.
6. Use the same fringe counting procedure you did for
the HeNe calibration to measure the (mean) wavelength of the (two) sodium D-line(s). For this calculation, utilize the micrometer screw calibration
you measired earlier with the HeNe laser.
Now you will measure the Sodium D-Line doublet splitting
7. Spin the micrometer screw to nearly its maximumreading.
8. Turn the micrometer toward smaller readings until
you see the first sharpening event. Record the micrometer position.
9.
Continue turing the micrometer toward smaller
readings and record its value for every subsequent
washout you see. It is ok, and indeed recommended, to scan back and forth across a washout
position to determine its location, but remember to
always “finish up” your screw turning by slightly
advancing the screw in the direction of smaller
reading BEFORE recording its value. (This is essential to eliminate errors due to backlash in the
screw threads.) Estimate an uncertainty in determining the location of each washout. Be sure to
describe the procedure for making this uncertainty
estimate in your manuscript.
8. Continue recording the positions of washouts until
you run reach the minimum reading of the micrometer screw.
VII.
ANALYSIS
Use these data, along with your measurement of mean
wavelength, and Eq. (5), to determine the sodium D-line
splitting. Propagate all your uncertainties, including the
systematic uncertainty in the micrometer screw calibration, to get a total uncertainty in your D-line splitting
measurement.
VIII.
EXTRA CREDIT – WHITE LIGHT
FRINGES
Now you will adjust the interferometer to the “white
light position,” when the two arms are exactly equal in
length. Tilt your work lamp down so that you can see
the lightbulb when you look into the output of the interferometer. Put a ground glass diffuser on the input to
the interferometer. Position the screw at approximately
at 1/2 its full range. Look into the interferometer, and,
with patience and care, slowly turn the micrometer screw
towards smaller readings. White light fringes will only
exist for approximately 1/8 of a turn of the micrometer
screw – otherwise you see nothing but the frosted glass.
Fig. 9 proves it can be done. Take turns looking for the
fringes if you get tired. If you do not see any fringes, go
back to your original position and turn the screw towards
increasing reading. When you see the fringes, shout “Eureka!” Count how many fringes of each color can be seen
on either side of the central maximum, and report this
in your write-up. Take a picture with your cell phone!
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FIG. 9: White light fringes in a Michelson interferometer from
this lab. Photograph taken by your instructor with an iPhone.