The Michelson Interferometer
Steve Ash, Mitch Anlicker, Derek Peterson
Department of Physics and Astronomy, Augustana College, Rock Island, IL 61201
Abstract: The Michelson Interferometer is a famous device used to calculate the
wavelength of light. It was used by Albert Michelson to calculate a meter as 1,553,163.5
wavelengths of monochromatic red light, which won him the Nobel prize in 1907. The
Interferometer functions by splitting a beam of light in to two beams, changing the path
length of one of the beams, and then recombining the beams on a view screen. This
causes the correlated beams to interfere with each other, allowing an interference pattern
to be seen, which can then be measured and used to calculate the wavelength of the light.
We have performed this experiment for red laser light, with an experimental value of
667.5nm, and with Sodium D lights, with two wavelengths of 634.5nm and 633.192nm.
I. Introduction
This experiment deals with measuring the wavelength of a particular light source. We accomplish this
measurement by making the light interfere with itself, to do these we use an apparatus known as an
interferometer. An interferometer is a device that takes a light source, splits the light beam into 2 separate
beams, then makes one of the beams travel a measurably larger distance to a viewing screen where both
light sources recombine. Since both beams of light originate from the same source the originally are
completely in phase with each other, making the only difference between the two beams at the screen the
difference in path length each traveled. This small difference in path length makes the two beams out of
phase with each other and causes them to interfere; this allows us to see interference patterns where the two
beams come back together.
The interferometer is most famous for its use in two experiments conducted by Albert Michelson. The first
was Michelson’s attempt to test for the presence of an ether, or a medium in which electromagnetic waves
propagate. This experiment returned negative results, disproving the theory of an ether existing. He also
conducted an experiment to precisely measure the length of a meter based of the number of wavelengths of
a certain frequency of a cadmium light source. This experiment did prove successful and won Michelson
the Nobel prize in 1907.
The interferometer operates by changing the path length of one of the split beams, relative to the unaltered
beam. The path length is changed by moving an adjustable mirror back and forth by a measurable distance
d (See Figure 1) . Because the light must travel the distance d both to the mirror, and on the way back after
reflection we express the total change in path length as:
2d = D
[1]
Where D is the total path length difference between the beams. The goal of the experiment is to determine
the wavelength of the incident light by measuring a distance d that causes the interference pattern to
perfectly repeat itself. This requires that interfering laser must move 1 whole wavelength in order to repeat
the same interference pattern, this means that:
D = λ [2]
In the case where an Na lamp is used the interference patterns that are seen are actually the result of two
closely spaced spectral lines, meaning that the measured wavelength is actually the average wavelength,
which means:
D = λavg = (λ1 + λ2)/2
[3]
To determine the wavelength for each of these we must consider what happens as distance d is changed.
When the lines are in phase with each other they interference patterns are sharp, when they are out of phase
they are blurred. The number of interference lines that pass by for each wavelength, going from blurry to
sharp, can be represented by the following:
(m+1)λ1 = mλ2 [4]
where m is the number of fringes that have passed while moving the mirror a distance d. Solving this last
equation for the difference in wavelengths, and substituting (D/ λ2) for m we obtain:
λ2 - λ1 =( λ1 λ2)/D [5]
We can then solve equation 3 for λ1 and replace λ1 in equation 5 with the result, this produces the following
quadratic equation, which when set equal to 0 allows us to find a value for λ2:
0 = λ22 + 2 λ2D -2 λAvgD-2 λ2 λAvg [6]
Once a value is obtained for λ2 we are able to plug this back into equation 3 and solve for λ 1.
Equations 3 through 6 apply the Sodium D lines only.
The error in this experiment was Gaussian in nature and we were able to calculate the standard deviation
using:
σ = ((1/N-1)*Ʃ di2)1/2 [7]
II. Experimental Setup
The experiment was carried out using a Michelson interferometer setup which was arranged in the
following manner:
Figure 1 - A figure taken from the lab manual used for this experiment. Pictured are of the components arranged as they
were during our experiment.
Using figure 1 as a reference the setup functioned in the following manner. The light source shines a single
beam of light into a beam splitter, which partially reflects and emits some of the light. the transmitted
portion of the light follows a path mirror M1, where it is then reflected back to the beam splitter and once
again reflected into the viewing screen. The portion of the light that is reflected the first time it strikes the
beam splitter follows a path the adjustable mirror M2. The light is then reflected back through the beam
splitter and onto the viewing screen, now out of phase with the second beam. The amount it is out of phase
by is determined by the distance d, which is how far back M2 has been moved from its aligned position
with M1. When the Na source is used we must use our eye instead of a viewing screen, this is due to the
low intensity of the Na light. To help with this we also add a converging lens not pictured to help focus the
interference image in our eyes.
For the laser light we collected data measuring the distance d, measured in micrometers, that it took to
make the interference pattern repeat 10 times. We then divided this value by 10 to approximate the value
of a single repetition. We repeated this process several times to get a range of values. Then the average of
all the individual calculations for d was taken, allowing us to calculate d and in turn λ. The same process
was repeated for the Sodium D lines allowing us to calculate an average wavelength for the two closely
spaced fringes.
To help calculate the wavelengths of each spectral line in Sodium D we took a measurement determining
how far we must move the mirror in order to make the interference patterns go from maximum blurriness to
maximum sharpness. The process was repeated several times and an average taken.
III. Results
Laser Light
The laser light was found to interfere on the average when d was changed by 333.75 +/- 2.32609421
nm, giving us a theoretical wavelength of 667.5+/- 2.32609421 . This compared to the known value of 650
nm produces a 2.62% error between the two.
Sodium D Light
The sodium d light was found to interfere on average when d was changed by 316.9231+/-0.2657838nm,
this resulted in a value of 633.8462+/-2.657838nm for the average wavelength of the two spectral lines.
Then we found the distance required to move the two bands completely in and back out of phase, which we
found to have an average distance of 286383.33+/-55.0324nm . Were able to use the average wavelength
and distance to then back calculate the individual wavelengths which were determined to be:
λ2 :
634.5nm
λ1 : 633.192nm
IV. Discussion
The lab overall appear to be quite successful, with it having low percent errors, mostly small deviations
from the mean. This was not my predicted outcome due to the amount of potential error in the lab. The
measurement device itself was prone to error up to +/-.005 microns, this is inherent in the device and can't
be overcome. I did, however, expect there to be larger error in the measurements taken by eye, such as
counting the number of fringes passing by an area. This was especially true when we were attempting to
determine when the Sodium D lines were in and out of phase. but even that measurement only show a
standard deviation of roughly 55 lines for every 286383 lines that passed, which is reasonable. I was
interesting to see how light when split could interact with itself, this really is an ingenious way to
determine the wavelength of a given light; especially if the light is visible.
References
Taylor, John R. An Introduction to Error Analysis: the Study of Uncertainties in Physical
Measurements. Sausalito, CA: University Science, 1997. Print.