Error interferogram

Fig. 40. Phase errors a) linear phase error b) nonlinear phase error. — Interferograms /(s) and spectra I v) obtained from a single-sided interferogram (—) and from a double sided one (----- cos transform,------- sine transform). —The dotted line (.. . ) indicates the undis-...

This section very briefly outlines some of the problems connected with the Fourier transformation. For details, textbooks such as Griffiths and de Haset (1986) can be recommended. Problems arise primarily because digital computers perform discrete rather than continuous FT of the interferogram / (,r), an approximation which requires care to avoid errors. As a result of DFT, the continuous variables, i.e., the scan length. v and the frequency T, become the discrete variables n A,v and k AT ... 

Phase correction in contrast to the theoretical expectation, the measured interferogram is typically not symmetric about the centerburst (.v = 0). This is a consequence of experimental errors, e.g., frequency-dependent optical and electronic phase delays. One remedy is to measure a small part of the interferogram doublesided. Since the phase is a weak function of the wavenumber, one can easily interpolate the low resolution phase function and use the result later for phase correction. If there is considerable background absorption, phase errors may falsify the intensities of bands in the difference spectra. To avoid such phase errors for difference spectroscopy, the background absorbance should therefore be less than one. 

Fig 2. Maximum spectral error (E) attributed to temporal aliasing for an interferogram scanned at a rate ki that collects an atmospheric-attenuated Planckian source that changes in time according to an exponentially decaying temperature (kx). 

Fig. 39 a-d. Intensity errors a) Spectrumwrong by a constant factor b) Spectrum I (v) and in-terferogram J s) when /( >) is determined incorrectly c) Interferogram I(s) with linear drift of the mean value /(oo) and the spectra obtained from the single-sided (-—) and the double-sided... 

Let us now turn to the intensity errors and discuss the most frequent ones in detail Figs. 39a)—d) demonstrate some of them in the interferogram and the effect in the spectrum. A minor intensity error is that sometimes the computed spectrum happens to be wrong by a constant factor due to some change in the amplifier gain between background and sample measurement (Fig. 39 a). Tliis does not affect the structures in the spectrum but is important when refractive index or absorption coefficient are evaluated from the reflectance or transmittance, respectively. 

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) ... 

When the Michelson interferometer with finite aperture is not properly adjusted nonlinear phase errors arise These phase errors are no longer linearly dependent on the wave number v, and they cause an asymmetric distortion of the interferogram (Figs, 40b and 41). It should be noted that all illustrations in connection with errors (Figs. 39, 40 and 41) have been produced by computer simulation (cf. Appendix 1). In order to make the essential features as clear as possible the effects of finite resolution etc. are left out where they have not necessarily to be included. In these cases, the resolution width /d is given in the figure (Figs. 39a—c). In Fig. 41, the error correction is demonstrated with finite... 

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