*Fourier Spectroscopy with the Michelson Interferometer*

**Introduction**

The method of deducing spectral lines by recording the intensity of interference fringes was pioneered by Michelson in 1891. He did not experiment in depth with the technique due to the difficulty of taking and analyizing data. At the time, the Fourier transform of the data was taken by means of a harmonic analyzer, a machine with over 80 mechanical elements

Today, Michelson's experimental technique has been refined to employ light sensitive devices for taking data and use computational methods for analysis. These advances have rendered Fourier spectroscopy a topic easily covered in the undergraduate laboratory. Understanding this process requires a knowledge of interference theory and Fourier transformations. It can be shown mathematically that taking the Fourier transform of the fringes in a moving interference pattern gives the spectrum of wavelengths present in the light source.

Interference and the Michelson Interferometer

Interference is a phenomenon that occurs when two photons of light interact with each other in such a way that their waves sum to either increase or decrease their total amplitude. Complete constructive interference occurs when two electromagnetic waves are of the same frequency and in phase; destructive interference occurs when two electromagnetic waves of the same frequency have a phase difference of one-half wavelength. When complete destructive interference occurs, no light can be detected. Similarly, complete constructive interference results in intensity quadrupling (intensity is proportional to the square of amplitude). The following picture demonstrates these effects.

Constructive Interference:the two waves, in phase, add to produce a wave of greater amplitude/intensity. | Destructive Interference: the two waves, with phase difference of ½ wavelength, add to cancel all amplitude. |

For more information on wave interference click here.

The interferometer is a device invented by Michelson which allows study of the effects of interference. It takes a single beam of light and splits it into two perpendicular paths of variable length. The light then recombines and the interference effects are observed.

The interference pattern for a Michelson interferometer is circular-- that is, it produces concentric circles of light and dark "fringes". When one mirror on the interferometer is moving, the path difference between the two split beams of light changes, and the interference pattern is seen to move as in the following animated gif.

Using a photomultiplier tube (PMT) or similar light detecting devices, it is possible to monitor the interference pattern as the path difference changes. The resulting graph of light intensity vs. time at the center of the pattern is known as an interferogram.

The Fourier Transform

The relationship between the motion of the interference pattern
and the Fourier transform can be derived from the equation for
the combined amplitude of the two mutually coherent light beams
in the interferometer^{1}. The equation modeling this interaction
is

where tau is the amplitude, Delta is the path difference between the two beams, k is 2Pi/lambda, and C is a constant. The intensity of the combined beams is proportional to the square of the amplitude:

For light with a spectral distribution of S(k) then the intensity of the light emerging from the interferometer is given by

Meanwhile, a separate reference beam has an intensity of

Thus, the intensity as a function of path difference is

The integral appearing inside the above brackets is known as a Fourier cosine integral. It also coincides with the inteferogram produced by monitoring the intensity of the moving fringes. It is assigned to the Greek letter Phi.

From the Fourier integral theorem, the inversion formula may be obtained,

Thus, the previous two equations constitute a Fourier transform pair. Notice how they correspond with the analogous equations for the Fourier transforms:

Following this derivation, the Fourier transform of the interferogram obtained by monitoring the intensity of a moving interference pattern will give S(k), the discrete wavelengths present in the light source.