Gaussian envelope function

A typical result of numerical calculation for a Gaussian envelope function E(t) is shown in Fig. 3 in comparison with the case for a transform-limited (unmodulated) pulse with the same envelope function. 

In order to apply these equations to a femtosecond pump-probe experiment, an additional assumption has to be made regarding the shape of the time resolved signal. We wish to account for the finite relaxation time of the transient polarisation and so the signal must be described by a double convolution of an exponential decay function with the pump and probe intensity envelope functions. We will assume a Gaussian peak shape so that the convolution may be calculated analytically. As we will see, the experimental results require two such contributions, and hence, the following function will be used to fit the experimental data... 

The cumulative envelope function, E s), can be complex and is attributable to a number of instrumental and experimental effects, such as spatial and temporal coherence and specimen motion. It has been shown that in practice a simple Gaussian function with width B adequately describes the cumulative envelope function (Saad et al, 2001) ... 

One convenient form for the envelope function is a Gaussian function. 

Fig.13.12, Saturated absorption on the 6328 A transition of a helium-neon laser. (a) Neon absorption tube inside laser resonator. (b) Output power as a function of oscillation frequency showing saturated absorption resonance superimposed on normal Gaussian envelope. (After Lee and Skolnick (1967).)...

Sometimes it is not possible to improve the resolution of a complex mixture beyond a certain level and, under these circumstances, the use of some de-convolution technique may be the only solution. The algorithms in the software must contain certain tentative assumptions in order to analyze the peak envelope. Firstly, a particular mathematical function must be assumed that describes the peaks. The function used is usually Gaussian and, in most cases, no account is taken of the possibility of asymmetric peaks. Furthermore it is also assumed that all the peaks can be described by the same function (i.e. the efficiency of all the peaks are the same) which, as has already been discussed, is also not generally true. Nevertheless, providing the composite peak is not too complex, de-convolution can be reasonably successful. 

Similarly to non-selective experiments, the first operation needed to perform experiments involving selective pulses is the transformation of longitudinal order (Zeeman polarization 1 ) into transverse magnetization or ly). This can be achieved by a selective excitation pulse. The first successful shaped pulse described in the literature is the Gaussian 90° pulse [1]. This analytical function has been chosen because its Fourier transform is also a Gaussian. In a first order approximation, the Fourier transform of a time-domain envelope can be considered to describe the frequency response of the shaped pulse. This amounts to say that the response of the spin system to a radio-frequency (rf) pulse is linear. An exact description of the... 

Gaussian curves (normal distribution functions) can sometimes be used to describe the shape of the overall envelope of the many vibrationally induced subbands that make up one electronic absorption band, e.g., for the absorption spectrum of the copper-containing blue protein of Pseudomonas (Fig. 23-8) Gaussian bands are appropriate. They permit resolution of the spectrum into components representing individual electronic transitions. Each transition is described by a peak position, height (molar extinction coefficient), and width (as measured at the halfheight, in cm-1). However, most absorption bands of organic compounds are not symmetric but are skewed... 

For a spin system in a solid where the lineshape is determined by the magnetic dipole-dipole interaction, the lineshape function in the FD can be approximated by a Gaussian function gG convolved with a rectangular envelope r 10... 

The Franck-Condon factors of polarizable chromophores in Eq. [153] can be used to generate the complete vibrational/solvent optical envelopes according to Eqs. [132] and [134]. The solvent-induced line shapes as given by Eq. [153] are close to Gaussian functions in the vicinity of the band maximum and switch to a Lorentzian form on their wings. A finite parameter ai leads to asymmetric bands with differing absorption and emission widths. The functions in Eq. [153] can thus be used either for a band shape analysis of polarizable optical chromophores or as probe functions for a general band shape analysis of asymmetric optical lines. 

The exponential factor of FC dominates equation (15). This feature simplifies much of the discussion. This exponential function is in Gaussian form, and it is the basis for a Gaussian analysis of absorption or emission spectra. Equations (15-18) provide the basis for analyzing the absorption or emission spectral envelope by considering some range of values of light frequencies (vobsd), but the nomadiative rate constant corresponds to the zero-photon limit. 

CEMS = conversion electron Mossbauer spectroscopy DFT = density functional theory EFG = electric field gradient EPR = electron paramagnetic resonance ESEEM = electron spin echo envelope modulation spectroscopy GTO = Gaussian-type orbitals hTH = human tyrosine hydroxylase MIMOS = miniaturized mossbauer spectrometer NFS = nuclear forward scattering NMR = nuclear magnetic resonance RFQ = rapid freeze quench SAM = S -adenosyl-L-methionine SCC = self-consistent charge STOs = slater-type orbitals TMP = tetramesitylporphyrin XAS = X-ray absorption spectroscopy. 

The time dependent shape of this field is given by a cos-function with amplitude Eq enveloped by a Gaussian centered at the time t = 0 with full width at half maximum (FWHM) At representing the laser pulse length. 

Much effort has been put into getting round both of these problems. The key feature of all of the approaches is to shape the envelope of the RF pulses i.e. not just switch it on and off abruptly, but with a smooth variation. Such pulses are called shaped pulses. The simplest of these are basically bellshaped (like a gaussian function, for example). These suppress the wiggles at large offsets and give just a smooth decay they do not, however, improve the phase properties. To attack this part of the problem requires an altogether more sophisticated approach. 

A sine-shape has side lobes which impair the excitation of a distinct slice. Other pulse envelopes are therefore more commonly used. Ideally, one would like a rectangular excitation profile which results from a sine-shaped pulse with an infinite number of side lobes. In practice, a finite pulse duration is required and therefore the pulse has to be tmneated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian function ... 

Figure 7 The time auto correlation function and the corresponding spectrum for a Gaussian wave packet propagating on an excited harmonic potential energy surface, (a) The short time decay of C(/) (cf. Eq. (17)) and the broad spectrum (= the Franck Condon envelope (cf. (18)). (b)The longer time dependence of C(r) and the corresponding, vibrationally resolved, spectrum.

Equation (18) is a very simple example of how the very short time dynamics determines the low-resolution (i.e., the envelope) spectrum. The details of the intensities of individual transitions require a longer time propagation and result in the resolved spectrum shown in Fig. 7. The envelope is however given by Eq. (18) and what the longer time dynamics reveals are the details which make up this envelope. A subsidiary lesson is that a broad envelope is not necessarily a signature of IVR. In the present example, and this carries over to the polyatomic case as well, the broad envelope is determined by inertia i.e., by the acceleration of the position of the oscillator. (To quantitatively show this, take a in Eq. (13) to be a function of time. It is the first deviation of x(t) from its initial value that gives rise to the Gaussian approximation. We reiterate that this can be shown in the multidimensional case as well (53).)... 

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