Double-sided interferograms

Figure A. 3 Double sided interferogram of a polychromatic source.

The position of ZPD (Zero Path Difference) is critical to the Fourier Transform calculation, since the algorithm assumes that the central burst in the interferogram is in fact the ZPD. However, due to the refractive index properties of the beamsplitter material, the ZPD is not at the same position for every wavelength measured. There are several ways to overcome these phase differences. The most common method is to use a correction factor, which is known as phase correction. This correction factor is calculated for every wavelength, based on a double sided interferogram, since this tends to minimize the effects of phase difference. In practice, most infrared spectrometers collect single sided interferograms, since this halves the mirror movement, and consequently the number of datapoints to be Fourier transformed. 

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) ... 

In other words, the correction of a phase error 95 requires a short double-sided interferogram around s = 0 regardless whether the whole interferogram is recorded single-sided or double-sided. The mostly used correction method was first proposed by M. Forman 68.70>, After determining tp v), the next step of this method is to calculate what can be called the Fourier transform of... 

Power spectrum This can be used instead of Mertz or Foreman, but only for double-sided interferograms, when the spectrum has wide ranges of low... 

Flame (air-C2H2) emission spectra of Li, K, Rb, and Cs recorded by Horlick and Yuen are shown in Figure 9. These spectra are calculated from 512-point, double-sided interferograms. The solutions used to obtain these spectra contained about 250 ppm of each element. 

At higher frequencies, FT spectroscopy is generally carried out with a Michelson interferometer rather than by detection of a coherent transient decay. The Michelson interferometer divides the input radiation into two parts with a beamsplitter and then recombines them. As the optical path difference of the two parts is varied, the interference of the recombined beams produces an interferogram. If the optical path difference, x, changes at a constant rate, v, then the interferogram becomes a function of time, f x)=f vt), and the FT yields the desired spectrum, F(v). In general, double-sided interferograms are... 

Most modem FT-IR spectrometers that operate at a resolution of 1 cm or poorer collect full double-sided interferograms (i.e., they are measured from —A2 to +A2). In this case, the phase spectmm is calculated from the entire interferogram, hence no approximations are made in the calculation of 0y. Chase [11] demonstrated the improvement in photometric accuracy when symmetrically double-sided interferograms are acquired. The advantage of collecting a double-sided... 

If a double-sided interferogram has been collected and the centerburst is near the middle of the interferogram, the summation in Eq. 4.5 may be replaced by the summation... 

When is transposed from one side to the other of Eq. 4.33, only the real terms are retained from the trigonometric expansion because the tme and amplitude spectra are real functions. Equation 4.34 represents a phase correction algorithm for double-sided interferograms. 

Phase correction may not always be possible. As stated above, the phase-angle spectrum is defined only where a signal exceeds the noise, which may not be the case for discrete emission spectra. In this instance, a double-sided interferogram must be collected and the magnitude spectrum computed. Discrete emission spectra are not commonly encountered in infrared spectrometry but occur in the ultraviolet-visible region for atomic emission spectrometry. 

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