Description of Interferometer Operation
Most interferometers used today for infrared spectrometry are based on the two beam type originally designed by Michelson in 1891. As such, a design employing this approach is referred to as a Michelson Interferometer.
The Michelson Interferometer divides an incoming beam of radiation into two equal (ideal) parts with each part continuing along a separate path. When the two beams are recombined, a condition is created under which interference can take place. Interference occurs when two beams of radiation are added together or combine to form one summation signal.
Since the radiation is in the form of a sinusoid, many combinations of the beams are possible. As shown in the figure below, the resultant signal can vary between zero and some maximum depending on the relative phase of the two beams. By changing the physical length of one path relative to the other path, the phase between the two beams can be varied. The difference between the two paths is know as retardation.
Description of Michelson Interferometer
The design of many interferometers used for infrared spectrometry is based on the two-beam interferometer originally designed by Michelson in 1891. Many other types of two-beam interferometers have been designed that may be useful for certain specific applications. The general idea behind all scanning two-beam interferometers is similar, and the theory of interferometry can be explained by describing the way a simple Michelson interferometer can be used to measure infrared spectra.
The Michelson interferometer is a device that divides a beam of radiation into two distinct paths and then recombines the two beams after introducing a difference in the two paths. Under these conditions, interference between the beams can occur. The interference creates variations in the output beam intensity as the difference in the path length changes. The intensity variations of the output beam can be measured with a detector as a function of the path difference.
The simplest form of Michelson interferometer is shown in the Figure below. It consists of two mutually perpendicular plane mirrors, one of which is mounted so that it can be moved along an axis perpendicular to its plane (surface). This movable mirror is normally moved at a constant velocity or could be moved and held at equally spaced points for fixed, short time periods and then rapidly stepped between points. Located between the fixed mirror and the movable mirror is a beamsplitter. The beamsplitter divides the input beam of radiation into two beams. That is, the input beam is partially reflected to the fixed mirror M1 (at point A) and partially transmitted to the movable mirror M2 (at point B).
After the beams return to the beamsplitter at O, they interfere and are again partially reflected and partially transmitted. Because of the effect of interference, the intensity of each beam, one passing to the detector and the other returning to the source, depends on the difference of path lengths in the two arms of the interferometer. The variation in the intensity of the beams seen by the detector is a function of the path difference and a graph or plot of this intensity is know as an interferogram. After mathematical manipulation, the interferogram ultimately provides the desired spectral information in a Fourier Transform Spectrometer or FTS.
To understand the processes occurring in a Michelson interferometer better, consider an ideal situation where a source of pure monochromatic radiation produces an infinitely narrow and perfectly collimated beam. Let the wavelength of the radiation be l (in centimeters) and its wavenumber be v (reciprocal centimeters)
Assume that the beamsplitter is ideal and has a reflectance and transmittance which are both 50%. Let's first determine the intensity of the beam at the detector when the movable mirror is held stationary at different positions. The path difference between the beams traveling to the fixed and movable mirrors is 2(OB - OA). This optical path difference or retardation is usually given the symbol 
. When the mirrors are held exactly perpendicular as they should be, and the beam is perfectly collimated, is the same for all parallel input beams.
When the fixed and movable mirrors are equidistant from the beamsplitter (zero retardation), then the two beams travel the same distances through the same materials and are exactly in phase after they recombine at the beamsplitter. At this point, the beams interfere constructively and the intensity of the beam passing to the detector is the sum of the intensities of the two beams passing to the fixed and movable mirrors. Therefore, all the light from the source reaches the detector at this retardation.
If the movable mirror is displaced a distance of /4, the total retardation or path difference is /2 (roundtrip). Therefore the path difference between the fixed and movable mirrors is exactly one-half wavelength. On recombination at the beamsplitter, the beams are 180 degrees out of phase and interfere destructively.
A further displacement of the movable mirror by /4 makes the total path difference or retardation . The two beams are once more in phase on recombination at the beamsplitter, and a condition of constructive interference again exists, (Fig. xxx). This pattern of constructive-destructive-constructive..... interference repeats as the mirror moves further. For monochromatic radiation such as a laser, there is no way to determine if a particular point at which a signal maximum occurs corresponds to zero retardation or a retardation equal to an integral number of wavelengths.
Moving the mirror at constant velocity, the signal at the detector varies sinusoidally, a maximum being registered each time the retardation is an integral multiple of wavelengths, . The intensity of the beam at the detector, measured as a function of path difference, is given the symbol I(). The intensity at any point where = n (where n is an integer) is equal to the intensity of the source I (). At other values of , the intensity of the beam at the detector is given by:
) = 0.5 I(v) cos(2
v)
I () as given above is the AC portion of the signal, which is the part of the interferogram we are interested in. Therefore, for a monochromatic source, the interferogram is a cosine wave of constant amplitude and a single frequency.
If the moving mirror is scanned at a constant velocity V then d = 2 V t. Substituting into equation above we now have:
) = 0.5 I(v) cos(2v · 2 V t) The frequency of this cosine wave is given by :
The above description shows how the interferometer when operating at a constant velocity modulates each wavelength into a unique frequency. If the source is made up of many wavelengths, then the interferogram is the sum of all these sine waves.