Cosine wave interferogram

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... 

If a single sharp absorption occurs at a wavenumber v, as shown in the wavenumber domain spectmm in Figure 3.15, the cosine wave corresponding to is not cancelled out and remains in the I 5) versus 5 plot, or interferogram, as it is often called. For a more complex set of absorptions the pattern of uncancelled cosine waves becomes more intense and irregular. 

In FT-Raman spectroscopy the radiation emerging from the sample contains not only the Raman scattering but also the extremely intense laser radiation used to produce it. If this were allowed to contribute to the interferogram, before Fourier transformation, the corresponding cosine wave would overwhelm those due to the Raman scattering. To avoid this, a sharp cut-off (interference) filter is inserted after the sample cell to remove 1064 nm (and lower wavelength) radiation. 

A graph of output light intensity versus retardation. 3, is called an interferogram. If the light from the source is monochromatic, the interferogram is a simple cosine wave ... 

If the unique cosine waves can be extracted from the interferogram, the contribution from each wavelength can be obtained. These individual wavelength contributions can be reconstructed to give the spectmm in the frequency domain, that is, the usual spectmm obtained from a dispersive spectrometer. A Fourier transform is used to convert the... 

The interferogram is actually a series of data points (retardation, intensity) collected during the smooth movement of the mirror. Using a mathematical function known as a Fourier transform, the spectrometer computer is able to deconvolute ( Fourier transform ) all the individual cosine waves that contribute to the interferogram, and so produce a plot of intensity against wavelength, or more usually the frequency in cm that is, the infrared single beam spectrum. All the... 

If the laser source described previously is changed so that it emits monochromatic radiation of a different wavelength with a different intensity, the interferogram will be a cosine wave with a different maximum amplitude and a different retardation length for one detector signal cycle. If the radiation from both laser sources described above enters the interferometer, the interferogram will be the sum of the two individual cosine wave interfero-grams. 

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. 

If the doublet has a separation of Av(= vi — V2), the two cosine waves in Figure 2.4b become out of phase after a retardation of 0.5(A and are once more back in phase after a retardation of (Av) To go through one complete period of the beat frequency, a retardation of (Av) is therefore required. An interferogram measured only to half this retardation could not readily be distinguished from the interferogram of a source with the profile shown in Figure 2.3c. The narrower the separation of the doublet, the greater is the retardation before the cosine waves become in phase. It is therefore apparent that the spectral resolution depends on... 

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

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. 

From above we know that the interferogram of a specific wavenumber of light is a cosine wave. A laser line comes dose to this ideal and its interferogram is a cosine wave, as illustrated in the left-hand side of Figure 2.13. Thus, the Fourier transform of a line is a cosine wave, and the Fourier transform of a cosine wave is a line. There really is a relationship between the wiggles in an interferogram and the peaks in an infrared spectrum. 

The laser interferometer is used to measure the intervals between data point collection within each individual interferogram. The coherent monochromatic light yields an interferogram very unlike that produced by white light. The laser interferogram is virtually a cosine wave, with the period determined by the wavelength of the laser. The most commonly used laser is the helium-neon (He-Ne) laser, with a wavelength of 632.8 nm or 0.6328 ym. 

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] ... 

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. 

Now, let us consider phase errors. As already pointed out, an error arises when the true origin of the interferogram is missed by a small path difference b <. As (Fig. 40 a) where Js is the sampling interval. This error is called a linear phase error because 2nvB means an erroneous phase shift in the interferogram function, which is linear with respect to the wave number v. Including the effects of truncation and apodization, we obteiin for the cosine transform of the double-sided interferogram with a phase error e approximately ss.es.vo) ... 

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