Two-dimensional Fourier

The general task is to trace the evolution of the third order polarization of the material created by each of the above 12 Raman field operators. For brevity, we choose to select only the subset of eight that is based on two colours only—a situation that is connnon to almost all of the Raman spectroscopies. Tliree-coloiir Raman studies are rather rare, but are most interesting, as demonstrated at both third and fifth order by the work in Wright s laboratory [21, 22, 23 and 24]- That work anticipates variations that include infrared resonances and the birth of doubly resonant vibrational spectroscopy (DOVE) and its two-dimensional Fourier transfomi representations analogous to 2D NMR [25]. 

The magnetization density is recovered by a two-dimensional Fourier transfonn of tlie data with respect to t... 

Merks R P J and de Beer R 1979 Two-dimensional Fourier transform of electron spin-echo envelope modulation. An alternative for ENDOR J. Phys. Chem. 83 3319-22... 

In some Hquid crystal phases with the positional order just described, there is additional positional order in the two directions parallel to the planes. A snapshot of the molecules at any one time reveals that the molecular centers have a higher density around points which form a two-dimensional lattice, and that these positions are the same from layer to layer. The symmetry of this lattice can be either triangular or rectangular, and again a positional distribution function, can be defined. This function can be expanded in a two-dimensional Fourier series, with the coefficients in front of the two... 

Using the two-dimensional Fourier-Bessel transform, the PYl equation (7) becomes (cf. Refs. 30,31)... 

The same reversible appearance and disappearance of the Pt(lll)-(12xl2)-Na overlayer is shown in Figure 5.51, together with the corresponding two-dimensional Fourier-transform spectra and also in Fig. 5.52, which shows smaller areas of the sodium-free and sodium doped Pt(lll) surface. The reversible electrochemically controlled spillover/backspillover of sodium between the solid electrolyte and the Pt(lll) surface is clearly proven. 

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

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. 

Multidimensional image information can be processed in the same way as signal functions in general. In many cases, the basis of image processing is the two-dimensional Fourier analysis... 

Fig. 8.32. Two-dimensional Fourier transformation applied to a rectangle function shown in original 3D representation (a) and 2D contour plot (b) and as Fourier transforms (c,d), (according to Danzer et al. [2001])...

B.R. Patyal, R.H. Crepeau, D. Gamliel and J.H. Freed, Two-dimensional Fourier transform ESR in the slow-motional and rigid limits SECSY-ESR, Chem. Phys. Lett., 1990, 175, 445. 

The applicability of the ESE envelope modulation technique has been extended by two recent publications115,1161. Merks and de Beer1151 introduced a two-dimensional Fourier transform technique which is able to circumvent blind spots in the one-dimensional Fourier transformed display of ESE envelope modulation spectra, whereas van Ormondt and Nederveen1161 could enhance the resolution of ESE spectroscopy by applying the maximum entropy method for the spectral analysis of the time domain data. 

Substantial long-range order was observed for dendrigraft poly(styrene) films using two-dimensional Fourier transformation of the AFM images. The... 

About 1915 W.H. Bragg suggested to use Fourier series to describe the arrangement of the atoms in a crystal [1]. The proposed technique was somewhat later extended by W. Duane [2] and W.H. Zachariasen was the first who used a two-dimensional Fourier map in 1929 for structure determination [3], Since then Fourier synthesis became a standard method in almost in every structure determination from diffraction data. 

In the experimental geometry used in this study, the distribution of scattered intensity measured by the detector is the two dimensional Fourier transform of the cross section of the electron density correlation function with a plane perpendicular to the extrusion direction (11). 

In general, for an oblique lattice in two dimensions with primitive vectors ai and ao, the total conductance G can be expanded into a two-dimensional Fourier series. 

For a solid surface with two-dimensional periodicity, such as a defect-free crystalline surface, all the measurable quantities have the same two-dimensional periodicity, for example, the surface charge distribution, the force between a crystalline surface and an inert-gas atom (Steele, 1974 Goodman and Wachman, 1976 Sakai, Cardino, and Hamann, 1986), tunneling current distribution, and STM topographic images (Chen, 1991). These quantities can be expanded into two-dimensional Fourier series. Usually, only the few lowest Fourier components are enough for describing the physical phenomenon, which requires a set of Fourier coefficients. If the surface exhibits an additional symmetry, then the number of independent Fourier coefficients can be further reduced. 

As we have discussed previously, any function with two-dimensional periodicity can be expanded into two-dimensional Fourier series. If a function has additional symmetry other than translational, then some of the terms in the Fourier expansion vanish, and some nonvanishing Fourier coefficients equal each other. The number of independent parameters is then reduced. In general, the form of a quantity periodic in x and y would be... 

The image intensity /(x, y) at the image plane of the objective lens results from two-dimensional Fourier synthesis of the diffracted beams (the square of the FT of the waves at the exit face of the crystal), modified by a phase-contrast transfer function factor (CTF, sin /), given by Scherzer (1949), as... 

The laser heating technique can be applied to perform temperature jumps by irradiating short laser pulses at the sample container. Ernst et al. (54) used such a temperature jump protocol to perform stop-and-go experiments. After the start of the laser pulse, the temperature inside the sample volume is raised to the reaction temperature, the conversion of the adsorbed reactants proceeds, and the H MAS NMR measurement is performed. After the laser pulse is stopped, the temperature inside the sample volume decreases to ambient temperature, and the C MAS NMR measurement is made. Subsequently, the next laser pulse is started and, in this way, the reaction is recorded as a function of the reaction time. By use of the free-induction decay and the reaction time as time domains and respectively, a two-dimensional Fourier transformation leads to a two-dimensional spectrum, which contains the NMR spectrum in the Ej-dimension and the reaction rate information in the Ts-dimension (54,55). 

It is interesting to note that several of the concepts for improving NMR technology, as listed by Levy and Craik, in 1988, already have been partially or fully achieved (1) two-dimensional Fourier transform (FT NMR) (2) high-resolution NMR in solids (3) new types of pulse sequences (4) chemically induced dynamic nuclear polarization (5) multiple quantum NMR and (6) NMR imaging (MRI). 

Figure 6. A pulse sequence for a two-dimensional Fourier imaging using a spin echo. A Gy gradient for a fixed period is imposed for a "phase" modulation to the signal, encoding position dependent information along the y-axis. The magnitude of Gy gradient is varied with a fixed increment for each scan of the sequence.

It is clear that S(kx, ky) is the two-dimensional Fourier transform of the nuclei density function P (x,y) (i.e. the volume density function p(x,y,z) averaged normal to the slice). Reconstruction of p(x,y) from S(kx, ky) simply requires that we calculate the inverse Fourier transform... 

This process is carried out using a computer, subsequent to obtaining the two-dimensional signal, S(kx, ky). A pulse sequence for two-dimensional Fourier imaging is shown in Figure 6. An example of the two-dimensional Fourier transform process is shown for two circular objects in Figure 7. 

Figure 7. Corresponding time (kx,ky) and frequency (x,y) domain data in two-dimensional image using Fourier transform for two uniform circular objects. The time domain S(kx,ky) is successively one- and two-dimensionally Fourier transformed to arise the spectra S (kx,y) and S"(x,y).

Two-dimensional Fourier transformation of the signal with respect to the jc-and y-gradient values gives the spatial distribution of individual spectra. However, only a finite number of phase-encoding steps are acquired, and the point... 

These molecules were studied by two-dimensional Fourier methods and, with the exception of pyrene, attention was confined to the most favourable projection only. By assuming that these molecules are planar and making allowance for their orientation in the unit cell, bond lengths were estimated which agree with those predicted by molecular... 

There has been no controversy about the structure of fluorene (31) but its true conformation was in doubt for a number of years. From an early X-ray analysis, Iball (1936a) concluded that the fluorene molecule had a folded conformation and, in a review, Cook and Iball (1936) discussed further evidence for a non-planar conformation, provided by optical activity studies of unsymmetrically substituted fluorene derivatives. Later stereochemical studies (Weisburger et al., 1950) suggested that fluorene had, in fact, a planar conformation. A reinvestigation of the crystal structure by Burns and Iball (1954, 1955) and, independently, by Brown and Bortner (1954) showed that the early X-ray work was in error and confirmed the planar conformation. The refinement of the crystal structure (Burns and Iball, 1954, 1955), by two-dimensional Fourier and least-squares methods, reveals that the maximum deviation of the carbon atoms from the mean molecular plane is 0-030 A, the r.m.s. deviation being 0-017 A. This deviation, 0-017 A, is taken by Burns and Iball to be a measure of the accuracy of their analysis, assuming now that the molecule is strictly planar. 

The crystal structure analysis (by two-dimensional Fourier methods) was facilitated by the fact that the crystal space group requires the molecule to have symmetry 222, the asymmetric crystal unit consisting of one-quarter of the chemical molecule. If there were no distortions from a regular planar model with a trigonal arrangement of bonds... 

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