Interferogram infinite

We can consider a real interferogram to be the summation of an infinite number of cosine terms, or ... 

Figure 4. Appearance of spectrum obtained by Fourier transformation of (a) an infinite interferogram (b) a finite interferogram and (c) a finite interferogram with...

Equation (3) implies an integration between - oo and -I- oo with an infinitely high resolution. But we have a limit for the maximal retardation of the interferogram and this means that we are multiplying the interferogram by a truncation function T 8), which fulfills the conditions ... 

Fig. 6. Finite interferogram and resolution. (Example Three narrow lines of different intensities) upper infinite interferogram I (v) and corresponding spectrum middle finite interfero-...

Equation (5.9) shows that in order to measure the complete spectrum, we would have to scan the moving mirror of the interferometer an infinitely long distance, with (5 varying between -oo and +cx) centimeters. In practice, the optical path length difference is finite. By restricting the maximum retardation to /, we are effectively multiplying the complete interferogram by the boxcar truncation function (see Fig. 5.3a left)... 

Figure 12.38 Block diagram showing the major components of an FTIR spectrometer. Also shows the spectrum of an infinitely narrow line source and how the interferogram is generated as the moveable mirror is translated. Maxima in the interferogram occur when the retardation is equal to an integral multiple of the wavelength of the source. Minima occur when the retardation is an odd multiple of half wavelengths. Source. Reprinted from technical literature of PerkinElmer Corp.

The second factor is related to the fact that in real life the interferogram is truncated at finite optical path difference. In addition, in the fast Fourier transform (FFT) algorithm, according to Cooley and TTikey [30], which is used to perform the Fourier transform faster than the classical method, certain assumptions and simplifications are made. The result is that the FFT of a monochromatic source is not an infinitely narrow line. 

The spectra in Figure 2.3c and d both have Lorentzian profiles and yield sinusoidal interferograms with an exponentially decaying envelope. The narrower the width of the spectral band, the greater is the width of the envelope of the interferogram. For a monochromatic source, the envelope of the interferogram will have an infinitely large width (i.e., it will be a pure cosine wave). Conversely, for broadband spectral sources, the decay is very rapid. 

It was shown in Eqs. 2.12 and 2.13 that to compute the complete spectrum from 0 to oo cm, the inteiferogram would have to be sampled at infinitesimally small increments of retardation. That is, of course, impossible, as an infinite number of data points must be collected and computer storage space would be exhausted. Even if these data could be collected, the Fourier transform would take forever to be computed. Obviously, interferograms must be sampled discretely. Just how often the interferogram should be sampled is a problem that has been solved mathematically. 

It is a property of Fourier transform mathematics that multiplication in one domain is equivalent to convolution in the other. (Convolution has already been introduced with regard to apodization in Section 2.3.) If we sample an analog interferogram at constant intervals of retardation, we have in effect multiplied the interferogram by a repetitive impulse function. The repetitive impulse function is in actuality an infinite series of Dirac delta functions spaced at an interval 1 jx. That is,... 

Apodization is a mathematical procedure used to overcome the fact that a recorded inter-ferogram is truncated (i.e., does not extend to an infinite distance) and to ensure that the interferogram to be Fourier-transformed terminates smoothly without a step. An explanation of apodization is given in Section 4.4.1. In Figure 5.12 , a trapezoidal apodization function is shown overlaid with a measured interferogram to be weighted by this apodization function. 

The interferogram as described mathematically is continuous and infinite requiring the Fourier integral to be evaluated over the limits of oo. Since the mirrors cannot move over distances of oo, the actual mirror movement is equivalent to multiplying the infinite interferogram by a boxcar function that has a value at all points between the optical displacement distance, L, and a value of zero everywhere else. 

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