Sine wave interferogram

Figure 2.11. (a) Sine wave interferogram (b) result of performing the cosine Fourier transform on this... 

For a finite sampling interval A<5, more than one superposition of cosine/sine waves can give rise to the recorded interferogram. For the transformed spectrum to be unique, the sampling interval A<5 must be sufficiently small to detect modulations in the interferogram due to the shortest wavelength present in the spectrum, the so-called Nyquist criterion [66] ... 

Multiple interferometer types exist in modern spectrometers the reader is referred to Candler (1951), Jamieson et al. (1963), Steel (1983), and Strobel and Heineman (1989) for a more exhaustive description of the optical configurations for these devices. The classical interferometer design is represented by the Michelson interferometer as shown in Fig. 3. A MM is displaced linearly at minute distances to yield an interference pattern (or interferogram) as a series of sine waves when the interference pattern is observed from a specific field of view (Fig. 7A). 

Fig. 7. (A) Interferometric output as a sine wave function. (B) The interferogram is plotted as a function of the light intensity (y axis) versus the mirror position (x axis), thus the signal is a function of time (as the mirror is moved at a constant rate). The raw interferogram is subjected to the multiple-step Fourier transformation and a spectrum results.

The sine waves corresponding to different groups of nuclei when superposed yield an interferogram which constitutes the F.I.D. (Fig. 3d). The best method of identifying the different frequencies present is to carry out a mathematical operation known as the Fourier Transform on the signal S(t). We thus obtain the resonance frequency spectrum of the sample (Fig. 3c and 3e). 

From Section 2.3 we know that when a cosine wave interferogram is unweighted, the shape of the spectral line is the convolution of the true spectrum and a sine function [i.e., the transform of the boxcar truncation function, 0(8)]. If instead of using the boxcar function, we used a simple triangular weighting function of the form... 

The cosine Fourier transform of a truncated sine wave has the form shown in Figure 2.11. In general, the shape of the ILS is intermediate between this function and the sine function that results from the cosine transform of a truncated cosine wave. The process of removing these sine components from an interferogram, or removing their effects from a spectrum, is known as phase correction. 

In the transformation the physical units are inverted. When the interferogram is expressed in optical path difference units (cm), the spectrum is obtained in wave-numbers (cm-1) and when the interferogram is expressed in time units (s) the spectrum is in frequency units (s 1). Apart from sine and cosine functions, box-car and triangular, etc. functions are also known, for which the Fourier transformation can be calculated. When applying the Fourier transformation over the whole area + oo, the arm of the interferometer also would have to be moved from — co to +co. When making a displacement over a distance of +L only, the interferogram has to be multiplied by a block function, which has the value of 1 between + and —I and the value 0 outside. I then influences the resolution that can be obtained. 

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