Temporal Fourier transform

The above integral is nothing but spatial and temporal Fourier transform, so one can write it in the spectral domain as... 

The spectral density function is defined to be the temporal Fourier transform of G(t) so that... 

We can employ the results of such simulations for both the Dirac and Schitidinger equations in order to calculate the HHG as well as the ATI spectra for the same laser parameters. This allows us to estimate the relativistic effects. An important observable is the multiharmonic emission spectrum S((o). It can be represented as the temporal Fourier transform of the expectation value of the Dirac (SchrOdinger) current density operator j(t) according to... 

In some intermetallic compounds clusters of isoenergetical sites occur which are well separated from other sites, see for instance the case of TaV2H in Fig. 26.4, and which form closed loops on those loops the H atom performs a spatially restricted jump motion (jump rotation) for some time until, by thermal fluctuations, eventually it is able to jump into the neighboring loop. For rotational diffusion over a loop of N sites after sufficiently long time the FI atom can be found on any of the N sites with equal finite probability. The time-independent (i.e. long-time) contribution to Is(Q,t) yields an elastic contribution m = 0) after temporal Fourier transformation ... 

The evaluation of the interaction free energy of Eq. (5.25) requires the dispersion relation for the normal modes in an inhomogeneous medium. We consider the specific case, shown in Fig. 5.3, of a thin film of thickness L and of dielectric function m o>) bounded by two half-spaces of dielectric functions (o) and 2 (o). Following Refs. 3,4, we write the temporal Fourier transform of the electric field (r, t) = (, where the discrete... 

The experimentally measured quantity in a scattering experiment is the dynamic structure factor S(q, co), describing the probability for the photon to acquire a momentum transfer q = k — koandan energy transfer o) = )o — o). In 1954, Van Hove demonstrated that the dynamic structure factor, known as the scattering function, relates to the probability G (r, t) for finding any particle at position r and time t when the particle was at r = 0 and t = 0 before by a spatial and temporal Fourier transforms [89] ... 

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. 

Equation (6a) implies that the scale (dilation) parameter, m, is required to vary from - ac to + =. In practice, though, a process variable is measured at a finite resolution (sampling time), and only a finite number of distinct scales are of interest for the solution of engineering problems. Let m = 0 signify the finest temporal scale (i.e., the sampling interval at which a variable is measured) and m = Lbe coarsest desired scale. To capture the information contained at scales m > L, we define a scaling function, (r), whose Fourier transform is related to that of the wavelet, tf/(t), by... 

The kinetics study [38] utilized a Fourier transform-ion cyclotron resonance (FT-ICR) mass spectrometer to measure the pathway branching ratios. The ability to eject selected masses and the extremely high mass resolution of this technique ensured that the observed CD3CH2 was in fact a primary product of the reaction. Temporal profiles from this reaction are shown in Fig. 1. Noticeably absent from the mass spectrum are the cations C2D2H3 and... 

To introduce the application of ultrashort laser sources in microscopy, we want to review some properties of femtosecond pulses first for a comprehensive introduction the reader may refer to one of the established textbooks on femtosecond lasers (Diels and Rudolph 2006). The most important notion is the Fourier transform relation between the temporal shape of a pulse and the spectrum necessary to create it. This leads to the well-known time-bandwidth product for the pulse temporal width (measured as full width at half maximum, FWHM) At and the pulse spectral width Av. 

The variability of hazardous air pollutants (HAPs) is an important factor in determining hnman exposure to such chemicals, and in designing HAP measurement programs. The factors that contribute to HAP variability in an urban area also affect their global impact. Temporal variation was the major contributor to HAP variability for 19 of the 39 frequently detected compounds (Spicer et al., 1996). In the future, more precise measurement tools will be available to determine HAPs. Open-path monitoring of the atmosphere using Fourier transform infrared spectrometry has recently become... 

We have developed ultrahigh-precision coherent control based on this WPI, in which we have succeeded in visualizing and controlling the ultrafast evolution of a WP interference in a molecule with precisions on the picometer spatial and attosec-ond (as) temporal scales [37-39], This is the cutting edge of coherent control. We have utilized this ultrahigh-precision coherent control to develop a molecular computer that executes ultrafast Fourier transform with molecular wave functions in 145 fs [40,41], More recently, we have extended the target of our coherent control to wave functions delocalized in a bulk solid [42,43], In this account, we will describe these developments of our experimental toolbox and the outlook toward the coherent control around the quantum-classical boundary. 

The result of such a measurement is shown in Fig. 2, where the temporal evolution of the excited A E state was recorded for 39,39K2 (Fig. 2a) and 39,4 fo (Fig. 2b). Both figures show distinct oscillations of similar frequencies, which correspond to the molecular vibration of the excited K.2 molecules. Astonishingly, however, the two species show quite different interference patterns, as they fingerprint the excited-state dynamics of apparently similar isotopomers 39,39K2 and 39,41 K2. The Fourier transforms of the two signals, presented in Fig. 3, provide complementary insight into the dif-... 

A Zenith Z-158 PC computer (8 MHz 20 MB, 640 kB RAM 8087 coprocessor) is interfaced to the experiment and controls the interferometer sampling, data acquisition, and subsequent sorting/Fourier transformation of the data to produce time-resolved spectra. Two digitizers were employed to record the temporal evolution of the IR emission and were fully dedicated... 

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