Physics 321 Fall 1997
The Michelson Interferometer
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The Michelson-Morley Experiment
In this experiment you learn another important spectroscopic technique: interferometry. Interference
fringes from monochromatic and white light are observed in a Michelson interferometer, and the
separation of the sodium D lines is measured. Measurements are then taken to look for effects due to
motion of the Earth through the luminiferous ether.
I. References
The Michelson interferometer is discussed in Experimental Physics - Modern Methods, by R.A. Dunlap (Oxford, 1988),
pp. 254-256, and in Experiments in Modern Physics, by Adrian C. Melissinos (Academic Press, 1966), pp. 45-51.
The Michelson-Morley experiment is discussed in Taylor and Zafiratos and in Lea and Burke.
The experimental evidence for the motion of our solar system relative to the 3 kelvin background radiation is discussed
by Sidney van den Bergh in "Size and Age of the Universe," Science 213, 825 (1981).
Accurate values for the sodium wavelengths were given in the last laboratory and can also be found in the Table of
Persistent Lines of the Elements in the CRC handbook in the laboratory. The specification sheet for the Beck
Interferometer is (may be) available in the laboratory.
II. Theory
The Michelson interferometer permits observation of the interference between two beams of light traveling paths of
about the same length, as shown in figure 1.
The beam is split by the half-silvered mirror, and
the two beams are reflected and recombined.
They will interfere constructively if their paths
are of equal length, or if they differ by an
integral number of wavelengths. The
interference will be destructive if the paths differ
by a half-integral number of wavelengths. The
interference actually appears as a pattern of
concentric bright and dark rings, called fringes.
The bright regions correspond to regions of
constructive interference, and the dark regions,
destructive interference.
One standard use of the Michelson apparatus is
to measure the wavelength of the incident light.
One of the mirrors is moved closer or farther
away while observing the fringe pattern. As the
length of one arm changes, the fringes all move
together, either in towards the center or out away
from the center. When the pattern moves out by
an amount such that each ring lies where the next
one out used to be, the pattern is said to have
moved by one fringe. Suppose that the pattern
moves by N fringes while you move the mirror a
distance l. Then the wavelength of the light is given by
 = 2l/N (1)
The yellow light from a sodium discharge tube is usually used for this experiment. The yellow light from the sodium
lamp is dominated by the Fraunhoffer D lines, the two roughly equally intense, closely spaced yellow lines which you
observed in the previous lab. Since the wavelength is different, the interference patterns produced by each of these two
lines do not necessarily occur at the same mirror setting. When you move away from the equal-arm position, the fringes
will appear to alternately fade out and reappear, as a result of interference between the two lines. . To understand this,
consider the condition for constructive interference for both wavelengths, 1 1 and 2 2:observed
2(LB–LA)= n11 ,
(2)
2(LB–LA)= n22 .
(3)
Physics 321 Fall 1997
The Michelson Interferometer
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where n1 and n2 are integers, and LA and LB are defined in figure 1. Now, if LB is changed to LB' by moving the mirror
until the two lines again interfere constructively, n2 will have increased by one more than n1. Say that n1 increases by
n 4 fringes. Then the interference conditions have become
2(L'B–LA)= (n1+n)1 ,
(4)
2(L'B–LA)= (n2+n+1)2 .
(5)
These four equations can be solved to express  =5 2 – 1 in terms of n6. Combining equations (2) and (3) and
equations (4) and (5) gives
n11=n22
(6)
(n1+n)1 =(n2+n+1)2
(7)
Subtracting these equations and solving for 8 gives
 =2 / n
(8)
(You should repeat, in your notebook, this calculation yourself.)
To reproduce the Michelson-Morley ether drift experiment, interference fringes are observed as the apparatus is rotated
through 90o, so that the arm initially parallel to the motion of the ether becomes perpendicular to it. One expects a shift
of the fringe pattern similar to that observed when the length of one of the arms is changed. The expected fringe shift n
is given by
2L v2
n=
,
 c2
(9)
where L is the length of the arms of the interferometer (one way), and v is the velocity of the motion through the ether.
(See T&Z, Lea and Burke, or Tipler for a derivation (see Section 1.5, pp.7-11, of Taylor and Zafiratos. Also review
problems 1.15 and 1.16.))
In order to test for motion through the ether, one should orient the apparatus parallel and perpendicular to the motion of
the earth relative to the ether. One could orient parallel and perpendicular to the motion of the earth on its axis. Or
one could orient parallel and perpendicular to the motion of the earth about the sun. More properly, one might orient
relative to the motion of the sun itself. The sun is moving at about 400 km/sec towards the constellation Virgo. Since
you are pretty well constrained to measuring in a plane parallel to the surface of the earth, which you choose is
somewhat arbitrary, so long as you present justification. If you have time and do want to go through what is needed to
locate the direction of Virgo, see the appendix.
III. Procedure
A. Setting up.
Look over the apparatus, and try to understand all of its motions and adjustments. You may need to put in the pin
which pushes the movable mirror forward. Check to see that both ends of the pin are centered on their mating pieces;
otherwise the lever motion will be uncalibrated. PLEASE REMOVE IT AT THE END OF THE DAY, to avoid
fatiguing the spring. Refer if necessary to the Beck specification sheet for the correct configuration of mirrors, etc., if
you need to.
Turn on the sodium lamp. It takes a while to warm up.
Illuminate the ground-glass screen with the sodium light, and look for the fringes. You will probably need to adjust the
tiltable mirror to center the fringe pattern. You will also need to move the translatable mirror. It helps in aligning the
mirror to have something to look at, like a piece of wire, behind the ground-glass plate.
The arms are of equal length when the fringes are as sharp as possible and as widely spaced as possible. As the paths for
the light become equal the width of the fringes increases until only one or two fringes fill the viewing aperture.
B. Measurement of the sodium D-line wavelength.
To measure the wavelength of the D line, you want to move the mirror of the interferometer a known distance, and to
count the number of fringes by which the fringe pattern shifts. The micrometer measures motion of the lever which
moves the mirror. Note that the end of the lever moves exactly five times farther than the mirror does. Also,
remember that the light path is double the distance moved by the mirror!
To make a measurement: (1) Read the micrometer; you will need to interpolate to tenths of divisions to get an
accurate result. Remember always to move the micrometer in the same direction just before reading it, to avoid
backlash error. (2) Move the micrometer slowly, watching the fringe pattern and counting the fringes. It is suggested
that you move exactly 100 fringes. However, many students find that the eyestrain caused by trying to count 100
fringes causes inaccuracies. Thus you may do better by moving say 25 fringes 4 times as many rather than the 100
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The Michelson Interferometer
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fringes. You will probably require a bit of practice before you can accomplish this smoothly. (3) Read the micrometer
again, and subtract readings. (4) Before repeating the procedure, move the micrometer a bit, so that you don't use the
same part of its motion for the next measurement.
The procedure should be repeated several times, in order to estimate the error. Note that you may also make counting
errors, especially on your first tries. When you have a number of readings that you think are reliable, average them to
get your best estimate for the sodium-D wavelength (call this value x9).
You should calculate an error, too. The following is suggested: (a) Determine the error x10 on the motion x for a
single 100-fringe measurement, either by calculating the r.m.s. spread of the values, or by estimating it. (b) Calculate
the error x11 on the mean value x12. (For N independent measurements, x =x /N13.) (c) The error on the
value 14 is the same, percentage-wise, as the error on x15.
C. The separation of the sodium D lines.
As explained in the theory section, the yellow light from the sodium lamp consists mainly of two closely spaced yellow
lines of about equal intensity, the Fraunhoffer D lines. The interference from these two lines is not always in phase.
Look at the change in the fringe pattern as you move away from the equal-arm position. The fringes should alternately
fade out and reappear. This is because the line is double.
To measure the spacing of these lines, start by determining x16, the distance the mirror moves as the pattern changes
from washed-out fringes through bright fringes back to washed-out fringes. If you go through several cycles, the
measurement will be more accurate.
Then calculate n17, the number of fringes of motion for one cycle. It is easy to show that
n = x/(5/2 )
Here you might as well use the accepted value for 19. Then use the formula from the theory section to calculate 20,
the splitting of this line. For error analysis, repeating the measurement is the best method!
Compare your result with accepted values from the CRC Handbook. This is an elegant technique for measuring line
separations too small to see with a spectroscope. Note that the closer the lines, the bigger21 n, and so the more
accurate the measurement!
D. White-light fringes
Using sodium light, adjust the arms to be as nearly equal as possible (widest fringes). Keep comparing fringe maxima
between settings of the arms where the fringes are clearest, i.e., where the interference from the two lines are in phase.
When you find maxima where the central fringe almost fills the field of view, you are probably within a maxima or so,
plus or minus. Then try to find fringes using white light from a tensor lamp. Pass the light through a ground-glass
diffusing plate.) The white-light fringes are elusive, but beautiful.
Give up? Try putting in a band-pass filter – or ask the instructor. The band-pass filter selects a narrow band of light,
and so increases the coherence length of the light. This results in interference fringes even when the two arms are not
precisely equal.
The white light fringes are an early case of where you should make accurate qualitative observations, and where you
should attempt to explain, in your notebook, what is going on. Why do the colors appear as one goes away from the
equal-arm-length position? Why do the colors appear as complementary, rather than primary? Does your explanation
agree with what you observe? Could you describe in detail to someone else what you saw, based on your notes?
E. Other games. You might be amused to try verifying the following facts about fringes:
As the difference in the length of the arms decreases, the circular fringes shrink inward; as the difference increases, they
expand outward.
The radius of the nth ring out from the center of the pattern follows the law
rn = r0 n
(9)
where
r0 = D (/d)
(10)
here D is the distance from your eye to the image of the fringes, and d is the difference in length of the two arms. You
can see, for example, that the fringe spacing increases as d decreases.
F. Detecting the luminiferous ether. If your apparatus is moving with respect to a luminiferous ether, you should
see a fringe shift when you rotate the apparatus by 90o. (See discussion under `Theory'.) You can do this by mounting
it on a rotating platform. Try this, and estimate a fringe shift, and error, for a 90o rotation. Make sure that one arm of
your apparatus points more or less towards the constellation Virgo. To observe the fringes in a reproducible way you
may want to put a crosshair (thread?), or a pinhole to look through, on the viewing telescope.
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The Michelson Interferometer
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Calculate the fringe shift expected if we are moving through an ether at 400 km/sec. You should be able to predict
which way the fringes should shift when you rotate the apparatus.
Compare your observed fringe shift with what is predicted by the ether theory. Is there a disagreement? By how many
standard deviations?
IV. Apparatus
Beck interferometer
sodium lamp
tensor lamp
ground-glass plate
bandpass filter (505-543 nm works well)
Beck instruction book
Beck interferometer catalogue
rotating table
(one grating spectrometer may be set up in the lab to show the sodium spectrum.)
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Astronomical Appendix. (With thanks to Prof. Charles Hagar!) The location of interesting astronomical objects is
given by specifying their right ascension (22) and declination (23). In brief, the following is intended to give
the minimum of astronomical terms and formulas necessary to figure out where to look to see something, based
on its right ascension and declination.
The right ascension is an azimuthal angle about an axis through the north star, measured in hours, eastward from an
arbitrary reference point (determined by where the sun is found at the vernal equinox). Declination is a polar
angle, in degrees, measured from the celestial equator. What you need to know to find the object, at a given
time and at a given place on the globe, is the hour angle. This is an azimuthal angle to the object, measured like
right ascension but westward from overhead, rather than eastward from the vernal equinox.
Here are some of the terms used:
RA (24) = right ascension, measured in hours eastwards from the vernal equinox
declination (25) = a polar angle, measured from the celestial equator
HA = hour angle, measured along the celestial equator from the meridian
Local Sidereal Time (LST) = hour angle of the vernal equinox where you are.
The hour angle can be calculated from
HA = LST -  .
The local sidereal time can be calculated from
LST=HA of the mean sun + RA of the mean sun.
Here the hour angle of the mean sun is found from
HA of the mean sun= PST–10m–12h.
(The ten-minute correction is specific to San Francisco.) The right ascension of the mean sun depends on the
period of the calendar, and can be calculated from
RA of the mean sun= 2h*N+ 4h*n,
where N is the number of whole months since March 21 (the date of the vernal equinox), and n is the number of
extra days.
Example: For September 1, N=5, n=11, and RA mean sun = 10h44m.
The Sun is thought to be moving at about 400 km/sec towards the constellation Virgo. The speed of the Earth's
motion about the sun, approximately 30 km/sec, can be neglected. The interferometer should be aligned so that
one of its arms points as nearly as possible towards Virgo. The right ascension and declination of the Virgo
cluster are =12h30m, =+1326. The hour angle (or HA, measured westward from overhead) of the Virgo
cluster at typical lab times is given in the table below. The local sidereal time (LST) is also given; the hour angle
is given by HA = LST – 27.
Date
Time
Local Sidereal Time
Hour Angle
September 1
15:30 PST
14h4m
1h34m
February 1
15:30 PST
0h4m
11h34m
Physics 321 Fall 1997
The Michelson Interferometer
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