Spectroscopy
Using the Michelson Interferometer
March 20, 1998
Ben Kinnaman,Davidson
College
Ansel
Singer-Barnum, Davidson College
Abstract
The purpose of this experiment was
to experimentally determine the two dominant wavelengths present
in a sodium light using the Michelson Interferometer. The
michelson Interferometer uses a beam splitter to split an input
wave into two waves which are reflected and combined again so
that an interference pattern of minimum and maximum intensities
is observable. The path length of one of the waves is changed by
moving one mirror. This changing path length creates a changing
interference pattern. The wavelength is related to the frequency
of the changes in intensity and the velocity of the mirror by the
expression l = 2*v/f. Using this expression, we computationally
simulated the data to find the two primary wavelengths present in
a Na lamp and their relative intensities. Our data is consistant
with the known wavelengths for sodium.
Theory
Below is the
Michelson interferometer. Light entering the interferometer is
split by the beam splitter. The beam splitter is a
semi-reflective mirror allowing some light to pass directly
through the splitter to the stationary mirror. The rest of the
light is reflected to the moveable mirror. The light is then
reflected back to the splitter where, again some light is allowed
to pass from the moveable mirror to the ditector and some light
from the stationary mirror is reflected to the detector. These
two waves of light from the stationary and movable mirrors
combine at the splitter. This combination is what the detector
detects. (It should be noted however, that this is a simplified
view of what is really happening. We are ignoring internal
reflection and the spreading of the wave. we have discussed what
was important to this experiment.)
When these two
waves of light combine, they interfere constructively or
destructively, depending on the path lengths traveled by each
wave. The path lengths traveled by each wave will affects the
relative phase of the two waves, so that when they combine, they
will combine contructively or destructively. The images below
illustrate constructive and destructive interference with the red
and blue waves being the descrete waves and the purple wave
representing the combination of the two.
The moving mirror on the
interferometer is driven at a constant rate by the motor shown.
We found that the mirror moved about 1666.4 nm/s (the process of
finding this velocity will be discussed later). As the mirror
moves, the path length of the light wave reflected off of that
mirror changes. this changing path length produces a changing
interference pattern that is visible. The interference pattern
forms a series of bright and dark concentric circles, similar to
what is illustrated below. As the mirror moves, the fringes move
to or from the center of the circle.
The path length of the wave is equal to an
integral number of half wavelengths, or x = n*[l/2]. By taking the time-derivative of
this equation, we can derive a fringe frequency and mirror
velocity dependence: v = f *[l/2].
The velocity of the mirror is related to the
frequency of the fringe procession through the expression v = f
*[l/2], or, velocity is equal to the
fringe frequency multiplied by the number of half wavelengths.
Using this relation, if the frequency and velocity are known, we
can find the wavelength of the light entering the interferometer.
If the light has more than one wavelength, a frequency of the
intensity of the fringes emerges. This is due to one wave
processing faster than the other, resulting in periods of high
intensity and periods of low intensity. This forms what is known
as beats, or a beat frequency. the images below illustrate a
monofrequency wave and a beat wave. The beat wave, on the right,
is the sum of two frequencies 0.8HZ different.
Knowing the frequency with which
the beats occur, we could know the wavelengths present in the
wave.
Procedure
The purpose of
this experiment was to experimentally determine the two dominant
wavelengths present in a sodium light using the Michelson
Interferometer. We know that the frequency of the constructive
and destructive interference of the waves of light emitted from
the Michelson Interferometer is related to the wavelength of the
light and the rate at which the path distance of one of the waves
changes (the velocity with which the mirror moves). This
relationship is simply v = f*[l/2]. Because the motor turning the screw which in turn,
moves the mirror is moving the mirror very slowly, it would be
very inaccurate, if not impossible to measure the velocity of the
mirror mechanically. By using the above equation with a light
emitting a known wave length, we very easily calculated the
velocity of the mirror.
In order to
determine the frequency of the fringe pattern we recorded the
intensity of the light going into the detector as a function of
time, the data is in volts per unit of time. Next we found the
frequency of the bright fringes, constructive interference, by
counting how many fringes passed per unit time. The graphical
data is available.
We used a HeNe
laser with a known wavelength of 632.8 nm. We took two data sets
with this laser, finding the fringe frequency to be 5.2631
fringes/sec. Thus, our moved with a velocity of 1666.4 mm/sec. We used this
velocity to find the dominant wavelengths present in sodium
light.
Setting up the
appartus for the sodium light required great care in order to
have the highest intensity of light possible going into the
detector, with as little periphial light going in as possible. in
order to do this we had to set up two aperatures. Each was about
a 1/4 in. hole that when set up successively, focused the light
so that as narrow of a beam was traveling in the interferometer.
The light was weak enough to require a 500V detector voltage.
After the
apparatus was set up and we were sure the sharpest signal was
being received as possible we could begin taking and analyzing
data, which we did in the following manner:
*We know that two waves are
combining in this experiment. A fundamental way to model this is
[A*sin(2*p*f*t)+A1*sin(2*p*f1*t)]. This is simply the
addition of two waves with different frequencies. We set f equal
to the experimental frequency and let f1 vary until
the plot of this function duplicated the experimental data set.
the next step was to vary A and A1 until the intensity
of the simulated plot matched the data.
Data and Results
Below is a data set showing the
intensity, or fringes, as a function of time.
Light Intensity as a
Function of time for a Na Lamp (small scale)
This data was taken during a
period of greatest intensity. A large scale sample of the data
clearly shows the beat pattern of the intensity due to
constructive and destructive interference patterns.
Light Intensity as a
Function of time for a Na Lamp (large scale)
The section indicated above is the region that
the data for the fringe frequency was taken.
We found the frequency of the fringes to be 5.749
fringes/sec. And found the beats to have a period of
about 178.1717 sec. Notice on the large scale
intensity plot that at during the period of smallest Intensity,
the beginning and end of the beats, there is still some
intensity. It does not go al the way to zero. This is because the
combining waves do not have equal intensities. One wave and its
associated wavelength has a greater intensity than the other wave
and its associated wavelength. The relative intensities are
important in modeling the data.
Our simulated data is
shown below.
Simulated Light
Intensity as a Function of time for a Na Lamp (large scale)
We were able to match the period of the beats
exactly, our indication that we had the correct relative
frequecies. We adjusted the intensities of each wave to match the
relatve minimums and maximums with the data. We found the maximum
and the minimum of the beats to have a relationship of 1.6:1.
This is the relative intensities of the waves and their
corresponding wavelengths. Using this data we found the primary
wavelengths of the Na lamp to be 579.674 nm, and
580.23 nm.
Zooming in on our simulated data, we can very
clearly se the contructve interference occurring at the minimums
and the constructive interference occurring at the maximums. the
two plots below were taken out of our simulated plot and show the
discrete waves at the first minimum and second maximum (notice
the t scale). The relative intensities of the waves and their
phases at the minimums and maximums are obvious. The red series
is the lower intensity while the blue series is the higher
intensity. (Note: the red series in the above plot does not
indicate the same series in the lower two.)
Conclusions
We were
able to find the two primary wavelengths of the sodium lamp. Our
findings were consistent with the known values.
We also found the relative
intensity between the two wavelengths to be 1.6:1.
What is more relevant than
the absolute wavelengths that we found is the difference between
the wavelengths. There is a scaling error in our simulated data,
which is introduced by making the frequency of the superposition
of the two discrete waves a constant. It is impossible to detect
the two discrete waves with the Michelson Interferometer so we
had to use the frequecy of the combined wave. Because of this
scaling error, we would not expect our calculations to yield the
known wavelengths exactly. We could expect the difference between
the two experimental wavelengths to be the same as the difference
between the known wavelengths.
There was plenty of room for error in
this experiment that could explain the large error. The
mechanical turning mechanism was suspected of not having totally
constant velocity, since all of our data is based on the
experimental frequency of the fringes, this could have been quite
significant. Extraneous light entering the detector could have
also been a source of error.