The basics of an interferometer are as follows. An incident beam of light is split and directed down two different paths. Reflective mirrors at the ends of these paths reflect the light back toward the beam splitter where the two beams recombine. As the optical path length of one is changed, the two beams approach the beam splitter in phase, out of phase, or somewhere in between. If the beams are in phase, then a bright spot will be visible. If the beams are completely out of phase, then a dark spot will result. If a motor is attached to vary this path length consistently over time, a moving fringe pattern can be observed. Placing a photo detector such as a photodiode or a photo multiplier tube (PMT) at the end allows us to record the signal intensity as a function of path length. This recording is called the interferogram.

The shape of these fringes is an interesting feature of this experiment.
One might think that the fringes should be vertical, definately vertical.
However, this is not the case. The fringes are actually circular.
For a detailed proof of why go here.
It's a very well done presentation, and I won't both copying it thereby claiming
it as my own.

Two of the biggest advantages to the use of interferometers as opposed to
spectrometers deal with the brightness of the image and the ability to process
the entire spectrum at once. The
Jacquinot advantage states that in a lossless optical system, the brightness of
the object equals the brightness of the image.
Assuming that the optical losses due the mirrors and the lenses is
negligible, an interferomter’s output will be nearly equal in intensity to the
input intensity, thus making it easier to detect the signal.
The Fellgett advantage or multiplex principle results from the fact that
the interferometer produces interference patterns for all wavelengths of light
entering the device at the same time, as opposed to a spectrometer with only
sees one wavelength at a time.

A Fourier transform breaks down a signal into an infinite number of sine
waves with distinct frequencies which, when added together, can reproduce the
signal. In general, the Fourier
transform of a function is given by the improper integral from negative infinity
to positive infinity:

A(ξ)
= ∫ F(x)exp(i2πξx) dx = F{F(x)}

For the purposes of this lab, we need to know
two things. First, that the Fourier
transform of a sine wave is a single peak, similar to the Dirac delta function.
Second, when the Fourier transform of a signal produces two peaks, there
are two different sine waves of different frequencies that are added together.
When these two frequencies are very close together, we should expect to
observe a beat pattern in the original signal.

A Helium Neon, HeNe, laser emits light only at 632.8 nm.
As a result, the emission spectrum consists of a single peak.
The Fourier transform of a delta function (a.k.a. a single peak) is just
a sine wave. Therefore, we expect
the interferogram to be sinusoidal.

A sodium (Na) light source emits not just one wavelength of light, but
two. The Na spectrum consists of a
doublet peak at 588.9 nm and 589.5 nm. Because
the interferometer sees both of these wavelengths at the same time, the
resulting interferogram should consist of a beat pattern.

References:

1.) Bell, Robert John. Introductory Fourier Transform Spectroscopy. Academic Press. New York. 1972.

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