Generation of an Interferogram

To understand the processes that occur in a Michelson interferometer better, let us first consider an idealized situation where a source of monochromatic radiation produces an infinitely narrow, perfectly collimated beam. Let the wavelength of the radiation be Xq (in centimeters) so that its wavenumber Vq (cm ) is 

We will denote the source intensity (power) at this wavenumber as /(vq). For this example we will assume that the beamsplitter is ideal (i.e., it is a nonabsorbing film whose reflectance and transmittance are both exactly 50%). We first examine the intensity of the beam at the detector when the movable mirror is held stationary at different positions. 

It can be seen that / (5) is composed of a constant (dc) component equal to 0.5/(vo) and a modulated (ac) component equal to 0.5/(vq) cos 2tivo5. Only the ac component is important in spectrometric measurements, and it is this modulated component that is generally referred to as the interferogram, /(8). The interfero-gram from a monochromatic source measured with an ideal interferometer is given by the equation 

In summary, the amplitude of the interferogram as observed after detection and amplification is proportional not only to the intensity of the source but also to the beamsplitter efficiency, detector response, and amplifier characteristics. Of these factors, only 7(vo) varies from one measurement to the next for a given system configuration, while all the other factors remain constant. Therefore, Eq. 2.4 may be modified by a single wavenumber-dependent correction factor, 7/(vo), so that the ac signal, 5(8) (in volts), from the amplifier is 

The parameter B(vq) gives the intensity of the source at a wavenumber vq, as modified by the instrumental characteristics. 

---

Figure 2.3 shows a representation of the generation of an interferogram for a broadband source with a flat spectrum. 

In Chapters 2 and 3 the various factors for the generation of an interferogram were discussed. In this chapter the techniques for computing the spectrum from this digitized interferogram are described. 

Fig. 2.4 Instrumental Line Shape/LA(v) top), which is the Fourier transform of a boxcar function of unit amplitude extending from +A to —A. Fourier transform of an interferogram generated by a monochromatic line at vi = 2/A bottom)...

Nowadays, most instruments use a FT-infrared (FT-IR) system, a mathematical operation used to translate a complex curve into its component curves. In an FT-IR instrument, the complex curve is an interferogram, or the sum of the constructive and destructive interferences generated by overlapping light waves, and the component curves are the IR spectrum. The standard IR spectrum is calculated from the Fourier-transformed interferogram, giving a spectrum in percent transmittance (%T) versus light frequency (cm ). 

An interferogram is generated because of the unique optics of an FT-IR instrument. The key components are a moveable mirror and a beam splitter. The moveable mirror is responsible for the quality of the interferogram, and it is very important to move the mirror at constant speed. For this reason, the moveable mirror is often the most expensive component of an FT-IR spectrometer. The beam splitter is just a piece of semireflective material, usually Mylar film sandwiched between two pieces of IR-transparent material. The beam splitter splits the IR beam 50/50 to the fixed and moveable mirrors, and then recombines the beams after being reflected at each mirror. The Fourier transform is named after its inventor, the French geometrician and physicist Baron Jean Baptiste Joseph Fourier, bom in 1830. 

Linear prediction. A mathematical operation that generates new or replaces existing time domain data points with predicted ones. Linear prediction can add to the end of an array of digitized FID data points, can extend the number of t, time domain data points in a 2-D interferogram, or can replace the initial points in a digitized FID that may have been corrupted by pulse ringdown. 

The symmetric Fourier transformation is used if a phase correction is not necessary, because the data contains one half of a symmetric or antisymmetric interferogram. This may occur for example, if the interferogram is generated by an inverse FT. 

---