Sinusoidal interferogram

Figure 10.11—Optical arrangement of a Fourier transform IR spectrometer, a) A 90c Michelson interferometer including the details of the beam splitter (expanded view) b) optical diagram of a single beam spectrometer (based on a Nicolet model). A weak intensity HeNe laser (632.8 nm) is used as an internal standard to measure precisely the position of the moving mirror using an interference method (a simple sinusoidal interferogram caused by the laser is produced within the device). According to the Nyquist theorem, at least two points per period are needed to calculate the wavelength within the given spectrum.

Figure 10.8 The optical assembly of a Fourier transform apparatus, (a) 90° Michelson interferometer with below, some details of the beam-splitter (b) the optical diagram of a single beam spectrophotometer (picture of Shimadzu model 8300). A low power He/Ne laser is used as an internal standard (632.8 nm) in order to locate with precision the position of the mobile mirror by an interference method (this second sinusoidal interferogram which follows the same optical pathway, is used by the software to determine the optical path difference).

In the particular case where the spectrum of a source of monochromatic radiation is to be determined, performing the Fourier transform of a measured interferogram is a trivial operation, since the amplitude and wavelength (or wavenumber) can both be measured directly from the sinusoidal interferogram. If the source emits either several discrete spectral lines or continuous radiation, however, the interferogram is more complex and a computer is required to perform the transform. 

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. 

Figure 2.5. (a) Fourier transform of a boxcar function of unit amplitude extending from +A to —A this function has the shape of a sin x/x or sine x, function, (b) Fourier transform of an unweighted sinusoidal interferogram generated by a monochromatic line at wavenumber Vi the maximum retardation for this interferogram was A centimeters. 

A simplified description of this technique for time-resolved asynchronous FT-IR measurements is shown for a monochromatic source in Figure 19.10. The sinusoidal interferogram of the monochromatic light source shown in part (a) is further... 

Apodization is an operation that is necessary because of the finite length of the interferogram. Monochromatic radiation produces a sinusoidal interferogram, but the length of this is limited by the maximum path difference in the interferometer scan. The Fourier transform of such a signal is a line of finite width with oscillations on either side. This line has the form of a sine function ((sin a/a)) (Figure 12). The oscillations (or side-lobes) that... 

A very-low-frequency sinusoid was superimposed on these spectral lines owing to channeling. This comes about by reflections from the window surfaces that contain the sample gas. These often result in a spike on the interferogram, which produces a superimposed sinusoid on transforming. Rather than removing the sinusoid from the entire data set, we fitted a smooth curve to the base line of each isolated set of lines treated. 

The signal observed by the detector is measured as a function of the displacement or retardation, s, of the moveable mirror. For a monochromatic source, the interferogram, I(s), is a sinusoid exhibiting maxima for s = n/v and minima for s =... 

Figure 2. Signal at the detector from a monochromatic source. This interferogram shows the sinusoidal variation of the signal about a DC offset as a function of the retardation. 

Figure 5 A Fourier-transformed signal in F2, at frequency oj2, is modulated in the indirect dimension (ti) by the incremental delay. This interferogram demonstrates how a sinusoid is created in the indirect dimension when a series of such spectra are collected. Fourier transformation of this sinusoid (along fi) would thus yield a peak with a frequency o>i in Fi.

Fourier transformation is necessary to convert an interferogram into an infrared spectrum, which is a plot of the light intensity versus wavenumber, as shown in Figure 9.18b. The Fourier transform is based on a fact that any mathematical function can be expressed as a sum of sinusoidal waves. All the information of wave intensity as a function of wavelength is included in the sum of sinusoidal waves. A computer equipped with FTIR constructs the infrared spectrum using a fast Fourier transform (FFT) algorithm which substantially reduces the computation time. 

Though a sinusoidal motion of the mirror is easier to carry out and is most often used, let us assume for simplicity that it moves according to a square-wave function. Then the intensity at the detector, the interferogram, is a function of the path difference s and of the time t [for the case of a continuous spectrum, see Eq. (3.2)] ... 

Fig. 25. Interferograms recorded using (a) amplitude modulation (b) sinusoidal phase modulation, vibration amplitude 30 jum and (c) sinusoidal phase modulation, vibration amplitude 100 /um. The corresponding spectra are shown for comparison. All data were to taken from Ref. 56)...

It is important to remove cosmic ray spikes from the interferograms before the data processing, because Fourier transforming an interferogram containing a cosmic ray spike will result in a high frequency sinusoidal modulation added to the recovered spectra. 

If the output of the photomultiplier is displayed on an oscilloscope, the sinusoidal envelope which contains a high frequency sinusoidal modulation is observed and as the path lengths of the two arms approach equality, the modulation depth of the higher frequency sinusoidal modulation in the envelope increases. If the interferometer is adjusted to produce a maximum modulation depth, zero path difference has been located. If a white light is then used to illuminate the interferometer, the photomultiplier should detect the expected eight to ten fringes and present the typical interferogram of a broadband source. Fig, 3. 

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