Fourier spectroscopy

When the incoming wave consists of two components with frequencies co and 002, the interference pattern varies with time according to 

The spectral resolution can roughly be estimated as follows if Ay is the path difference traveled by the moving mirror in Fig. 4.25, the number of interference 

The equivalent consideration in the frequency domain follows. In order to determine the two frequencies coi and C02, one has to measure at least over one modulation period 

The spectral resolving power Xj AX of the Michelson interferometer equals the maximum path difference AsJX measured in units of the wavelength A. 

When the incoming radiation is composed of several components with frequencies cok, the total amphrnde in the plane B of the detector is the sum of aU interference amplitudes (4.36), 

A detector with a large time constant compared with the maximum period / o)i—cok) does not follow the rapid oscillations of the amplitude at frequencies cok or at the difference frequencies cot — cok), but gives a signal proportional to the sum of the intensities 4 in (4.37). We therefore obtain for the time-dependent total intensity 

The spectral resolution can roughly be estimated as follows if Ay is the path difference traveled by the moving mirror in Fig. 4.25, the number of interference maxima that are counted by the detector is N = 2Ay/Ai for an incident wave with the wavelength and N2 = 2AyjX2 for X2 X. The two wavelengths can be clearly distinguished when N2 This yields with 

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Exponential decay often occurs in measurements of diffusion and spin-relaxation and both properties are sensitive probes of the electronic and molecular structure and of the dynamics. Such experiments and analysis of the decay as a spectrum of 7i or D, etc., are an analog of the one-dimensional Fourier spectroscopy in that the signal is measured as a function of one variable. The recent development of an efficient algorithm for two-dimensional Laplace inversion enables the two-dimensional spectroscopy using decaying functions to be made. These experiments are analogous to two-dimensional Fourier spectroscopy. 

Fourier spectroscopy, 23 137 Fourier transformation, essential equations for, 14 226-227 Fourier transform (FT) infrared (ftir) analysis, 19 563-564, 813. See also Microscopy-ftir technique dichroism, in silicone network characterization, 22 569 instruments, 23 138 advantages of, 14 228 resolution for, 14 227 microscopy, 14 232... 

Tsurui H, Nishimura H, Hattori S, Hirose S, Okumura K, Shirai T (2000) Seven colour fluores cence imaging of tissue samples based on Fourier spectroscopy and singular value decomposi tion. J Histochem Cytochem 48 653 662... 

However, for illustration, only one side of the interferogram and its spectrum will be shown, usually the function of the positive spatial and spectral variable. In other operating modes of the interferometer, asymmetric interferograms are produced that have a complex Fourier transform. Asymmetric interferograms will not be treated in this work. For a more complete discussion of Fourier transform spectroscopy, the reader should consult Bell (1972), Vanasse and Strong (1958), Vanasse and Sakai (1967), Steel (1967), Mertz (1965), the Aspen International Conference on Fourier Spectroscopy (Vanasse et al., 1971), and the two volumes of Spectrometric Techniques (Vanasse, 1977, 1981). A review of early work, which includes several major contributions of his own, is given by Connes (1969). Another interesting paper on the earlier historical development of Fourier transform spectroscopy is that by Loewenstein (1966). 

An effective Hamiltonian for this system has been obtained recently by Herman and co-workers based on the high-resolution Fourier spectroscopy of... 

R. E. Murphy and H. Sakai, Proceedings of the Aspen International Conference on Fourier Spectroscopy, G. A. Vanasse, A. T. Stain, D. J. Baker, Eds., Air Force Cambridge Research Laboratory Special Report No. 114, 1970, p. 301. 

Dieter Ziessow, On-Line Computers in Chemistry. Principles and Use in Fourier Spectroscopy, De Gruyter, Berlin, 1973. 

In some cases cyclic events occur, dependent, for example, on time of day, season of tire year or temperature fluctuations. These can be modelled using sine functions, and are tire basis of time series analysis (Section 3.4). In addition, cyclicity is also observed in Fourier spectroscopy, and Fourier transform techniques (Section 3.5) may on occasions be combined with methods for time series analysis. 

CIS. Connes, P., Advances in Fourier spectroscopy. Phys. Bull. 22, 26-28 (1971). 

Mil. Mertz, L., Fourier spectroscopy past, present and future. Appl. Opt. 10, 386-389... 

The pump-probe configuration provides the classic approach to 2D spectroscopy since it involves two independently tunable frequencies [1]. In NMR the analogous approach is double resonance Fourier spectroscopy. In fact the method... 

But the experimental realization of this concept clearly suffers from the fact that one has to deal with two detectors which generally do not have equal responsivity etc. However, using also the interferogram reflected back to the source offers the possibihty of comparing the spectra of two samples or two sources, i.e. the possibility of a double beam op)eration in Fourier spectroscopy (cf. the double beam interferometer by R. Hanel et and Section 6.2). 

For optical investigations, a certain resolution is required. In the case of Fourier spectroscopy, this is expressed as the smallest clearly resolved difference in wave numbers ... 

Table 1 Examples of properly chosen parameters in Fourier spectroscopy...

Recently, first experimental results were published which have been obtained with a Michelson interferometer especially designed by J. Cast and L. Genzel > for reflection studies on small solid samples by means of asymmetric or amplitude Fourier spectroscopy. The main advantage of the optical layout is that sample and reference mirror are located at focal points which do not take part in the motion to produce the path diflerence in the interferometer. Therefore, these foci can be placed inside a cryostat that allows the sample to be cooled. Another recent development in this field is concerned with the difficulty that, when studying the reflectivity of solids, the determination of the phase depends strongly on the exact positioning of the sample mirror instead of the background mirror. This is... 

But the amplitude or as5unmetric Fourier spectroscopy has not been applied only to solids but also to liquid samples and gases. Here too, it has been proved an important help to obtain the two optical constants n and x simultaneously 62a,b) In gases, for example, the anomalous dispersion of the refractive index in the neighbourhood of rotational absorption lines was determined experimentally by this method. 

At last it remains to point out that phase errors are not only unwanted and under certain conditions unavoidable phenomena in Fourier spectroscopy the elimination of which is a necessary but rather involved and elaborate procedure. In contrast, there are special applications of Fourier spectroscopy where phase errors have been introduced deliberately. In this method which is known as "chirping 86,90-92) g. plane parallel plate called chirping plate is placed in one arm of the Michelson interferometer or two different plates are placed one in each arm. These plates introduce a wave niunber dependent phase shift... 

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