Interferogram amplitudes

Figure Bl.2.7. Time domain and frequency domain representations of several interferograms. (a) Single frequency, (b) two frequencies, one of which is 1.2 times greater than the other, (c) same as (b), except the high frequency component has only half the amplitude and (d) Gaussian distribution of frequencies.

Visibility Amplitude (III.7) This is the most readily measured observable, the maximum contrast of the interferogram. It contains essential photometric information for images of the source and all information for circularly (or el-liptically) symmetric sources. A high precision measurment requires calibrator sources with known visibilities and / or monitoring of system parameters for calibration. 

Fig. 19 Interferogram of Fig. 13(a) with Gaussian noise of rms amplitude 0.1 added, (a) Cosine interferogram of unity amplitude with random noise of rms amplitude 0.1 superimposed. (b) Single spectral line with the oscillatory artifacts. 

Fig. 27 Finite interferogram of the four monochromatic sources of Fig. 26 with Gaussian noise of rms amplitude 0.1 superimposed and the resulting degraded spectral lines, (a) Interferogram of 30 data points, (b) Merged and distorted spectral lines.

Now, the mathematical form of the interferogram / (s) is to be derived. The partial beams in the two arms of the Michelson interferometer (see Fig. 2) are described as plane waves. The amplitude or the electric field of the wave reflected at the fixed mirror is... 

Here the negative sign has been included for theoretical reasons (see Appdx. 3). The interferogram I(p s) is closely related to the derivative of the interferogram I s) obtained with amplitude modulation (see Fig. 24 and Appdx 3). For the computation of the spectrum with the computer, the interferogram has to be digitized in the usual way. 

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)...

Eq. (4.19) simply shows the basic relationship and the influence of flnite interferogram, apodization, and digitizing is not considered in detail. We recall that 7o( ) is the background intensity already determined, and the essential results of the Fourier transform are T v) and from which both optical constants can be evaluated. In other words, the complex amplitude transmission coefficient... 

In order to imderstand this important property of Fourier transform spectroscopy, let us consider a broad spectrum over a wide wave number range with one narrow absorption line in it (cf. Fig. 12). Then, according to the rules of Fourier transformation, the broad spectrum produces an interferogram with highly damped oscillation. The maximum amplitude of the oscillation and also the mean value of the interferogram are equal to the total intensity or to the area under the spectral distribution [see Appdx 1 and Eqs. (A 1.1) and (A 1.2)] ... 

Fig. 43. Noise amplitude N (s) in the interferogram (a) and noise amplitude Ai (ii) in the spectrum (b)...

This definition takes into account that the interferogram with its noise is multiplied by S(s) before the Fourier transform is carried out, and that the effective noise amplitude is N s) S (s) instead of A s). Furthermore, it is presumed that the bandwidth d/ of the electronics is selected to be appropiate for the maximum wave number 5max- We recall from our considerations in Section 4.6 that the time constant t limiting the bandwidth of the electronic system should have a value close to (cf. Section 4.6)... 

Figure 5.5. The variation in peak intensity of the amplitude modulated resonance of Fig. 5.4 produces an FID (interferogram) for the ti domain.

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