Fourier Transform Relationship

Given the unit cell vectors a, b, and c, a lattice point in real space can be written as 

We now want to evaluate the Fourier transform of z(r). Writing the Fourier transform as Z(s), we have 

To see how each of the three infinite sums in Equation (C.7) behaves, we first consider the finite sum extending for u, v, or w from —N/2 to N12 (for N even). Thus 

This infinite lattice in fact coincides with the reciprocal lattice based on a, fr, and c as defined in the previous section. Mathematically, this statement is equivalent to saying that 

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By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

A Fourier Transform Relationship between Time-Domain and Frequency-Domain Excitation Functions. 

The laser used to generate the pump and probe pulses must have appropriate characteristics in both the time and the frequency domains as well as suitable pulse power and repetition rates. The time and frequency domains are related through the Fourier transform relationship that hmits the shortness of the laser pulse time duration and the spectral resolution in reciprocal centimeters. The limitation has its basis in the Heisenberg uncertainty principle. The shorter pulse that has better time resolution has a broader band of wavelengths associated with it, and therefore a poorer spectral resolution. For a 1-ps, sech -shaped pulse, the minimum spectral width is 10.5 cm. The pulse width cannot be <10 ps for a spectral resolution of 1 cm . An optimal choice of time duration and spectral bandwidth are 3.2 ps and 3.5 cm. The pump pulse typically is in the UV region. The probe pulse may also be in the UV region if the signal/noise enhancements of resonance Raman... 

Since Pt is zero outside the limits of integration, the limits may be taken as oo, yielding a Fourier transform relationship. The Fourier transform may be inverted to yield... 

Between X(k) and the radial distribution function around the central atom, there is just a simple Fourier transform relationship. Figure 1 gives the different steps of an EXAFS analysis. 

All of this information about vibrational frequencies and transition intensities is observable directly in the frequency domain absorption spectrum, A"= o(w) The autocorrelation function picture is an alternative way of deriving a ball-and-spring physical picture (or dynamical mechanism) from the raw experimental data. Although there is a simple Fourier transform relationship between Ivn (to) and ( f (f)l F(O)), profoundly different intuitive pictures are used to make sense of experimental results and to guide the design of new experiments. 

Although a simple Fourier transform relationship can exist between a high spectral resolution frequency domain experiment and a time-domain quantum beat experiment, whenever the excitation and detection steps involve electromagnetic radiation of different spectral, spatial, or temporal characteristics, the intrinsic information content of time and frequency domain experiments needs not be identical. The format in which the information is presented may be more transparently interpretable in either the time or frequency domain. 

Figure 8.2 Fourier transform relationships among the van Hove correlation function G (r, 0, the intermediate scattering function F(q, f), and the dynamic structure factor S(q, co).

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

The Fourier transform relationship between a function x(t) and the corresponding frequency function X(i/) is ... 

In the CMD correlation function [Eq. (3.10)], the notation )p means that the exact (normalized) phase-space centroid density is used to average the initial conditions of the centroid trajectories in the usual classical sense. Once the centroid correlation function C (f) in Eq. (3.10) is calculated, the exact quantum position correlation function C(t) can be estimated through a Fourier transform relationship similar to Eq. (3.8),... 

The presence of disorder generally reduces diffraction intensities and hence affects the precision of the crystal structure the largest effect is on the disordered part of the structure itself, but the nature of the Fourier transform relationship is such that precision overall is affected, albeit not significantly in many cases. The problem is compounded if the disorder is difficult to model with partial atoms, and this is especially the case when the two or more components do not have resolved electron density distributions. 

Radar functions often characterized by Fourier transform relationship antenna aperture illumination, far field ... 

The phenomenological theory of the dielectric relaxation behaviour of linear systems is well-established [1-5]. The fundamental relationship joining the frequency-dependent complex permittivity c(cu) measured at frequency / = (ofln and the transient step-response function t) is the Fourier transform relationship... 

We can translate this equation into spatial frequency coordinates, such that u =kal2nf and v=kpl2nf, giving the Fourier transform relationship shown above for the far field region, with an added phase distortion due to the compression of R down to the focal plane/of the positive lens. 

Any function, such as a function f(R) giving the probability of finding an atom at a given position R in real space, can be Fourier transformed to provide an indication of the amplitude f(k) of waves having wave vector k which, if summed or integrated, replicate f(R). Mathematically, the Fourier transform relationship is ... 

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