Fourier intensity distribution

Figure 2. Morphology and the corresponding 2D-Fourier intensity distribution (power spectra) ofaPSA/PVME(20/S0) blend obtained at lOO C by (a), continuous irradiation in 75 min (b), periodic irradiation in 150 min with...

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. 

J3—ft. Strictly speaking, it depends on the intensity distribution in a line (Jones, 1938) and the ideal method of obtaining / is by a Fourier analysis of the line shape (Stokes, 1948) in practice it is doubtful whether such elaboration is worth while, and it is usually sufficient to use correction curves given by Jones (1938) for the relation between b/B and jS/JB for different line-shapes, or to use Warren s (1941) relation j32 = J52—6a which gives very similar results (King and Alexander, 1954). 

The microdomain orientation as a function of the electric field strength was monitored by a series of scanning force microscopy (SFM) images taken in the center between the electrodes. The entire electrode length of 6 mm was screened in steps of a few tens of microns. From the azimuthal intensity distribution of the 2D Fourier transformations of the SFM images, the orientational order parameter P2 was calculated according to ... 

The angle (p quantifies the in-plane direction, with cp = 0° corresponding to the direction along the stripe-like electrodes. For an alignment of the lamellae along the field direction (maximum azimuthal intensity distribution of the 2D Fourier transform intensity at (p = 90°), P) ranges from 0 to -0.5 with P2 = -0.5 corresponding to the fully oriented case. 

As is well known, if D(z) is a periodic function with period d, then its Fourier transform is a periodic set of delta functions of period 1/d and the intensity distribution in reciprocal space consists of discrete maxima also spaced 1/d apart. If the distribution function is not periodic but consists of two randomly arrayed periods, then it is necessary to consider all possible combinations and permutations, suitably weighted by their probability of occurrence. 

FIG. 21-9 (a) Diffraction patterns of laser light in forward direction for two different particle sizes, (b) The angular distribution 7(9) is converted by a Fourier lens to a spatial distribution l(r) at the location of the photodetector, (c) Intensity distribution of a small particle detected by a semicircular photodetector. 

The study of the intensity distribution can be achieved by two methods either by fitting this peak with a calculated function or by a direct analysis using the Fourier transform of that peak. The fitting requires prior knowledge of the profile, which, by definition, can never exactly correspond to the experimental profile, and the Fourier analysis can only be achieved if the peaks do not overlap. These aspects will be discussed further in Part 2 of this book, which deals with microstrucmral analysis. 

Effects of the grain size and of the nucrostrains on the peak profiles Fourier analysis of the diffracted intensity distribution... 

Three different methods have been designed to quantitatively study structural volume defects. The integral breadth method, based on the theoretical considerations we discussed in Chapter 5, was introduced in 1918 by Scherrer [SCH 18] and generalized by Stokes and Wilson [STO 42], among others. Later on, Toumarie [TOU 56a, TOU 56b] followed by Wilson [WIL 62b, WIL 63] suggested a different analysis based on the variance of the intensity distribution. We described how Bertaut [BER 49] showed in 1949 that the Fourier series decomposition of the peak profile makes it possible to obtain the mean value and the distribution of the different effects that cause the increase in peak width. This method was further elaborated by Warren and Averbach [WAR 50, WAR 55, WAR 69]. 

An alternative approach consists of working directly with the measured profile h(x), by expressing it as a Fourier series that includes the various components associated with each of the effects that modify the experimental profile [SCA 02] (instrumental function, size effect, microstrains, etc.). The intensity distribution h(x) or 1(20) is then expressed from equation [5.98] which was obtained in Chapter 5 by including in this equation the Fourier transform G(x) of the instmmental function, and finally ... 

In Chapter 5, we showed that the diffracted intensity distribution can be written as a Fourier series with coefficients which are directly related to the microstrain amount. Thus, the peak profile is expressed as ... 

The general idea consists of expressing each microstmctural effect by way of the Fourier coefficients associated with the intensity distribution it generates. As we have already said in section 6.1.2, the integrated intensity is then written according to the following equation ... 

Thus, subsequent Fourier transformation in l yields a spinning sideband pattern, with sidebands separated by coR and intensity distribution determined by d. Simulation of the sideband pattern then yields d, the dipolar coupling constant, or in a mobile system undergoing rapid uniaxial motion, <4ff, where de is the motionally averaged dipolar coupling parameter, given for uniaxial motion by... 

In this thesis the CLEAN algorithm is used for the data synthesis of Double Fourier Modulation data and is described in detail in Chap. 5. In general terms, it is basically a numerical deconvolving process applied in the 0x, 0y) domain. It is an iterative process, which consist of breaking down the intensity distribution into point source responses, and then replacing each one with the corresponding response to a clean beam, this is, a beam free of side lobes. 

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