Pulse envelope

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 truncated, which causes oscillations in the excitation profile. Another frequently used pulse envelope is a Gaussian frmction ... 

The first experimental measurements of the time dependence of the hydrated electron yield were due to Wolff et al. (1973) and Hunt et al. (1973). They used the stroboscopic pulse radiolysis (SPR) technique, which allowed them to interpret the yield during the interval (30-350 ps) between fine structures of the microwave pulse envelope (1-10 ns). These observations were quickly supported by the work of Jonah et al. (1973), who used the subharmonic pre-buncher technique to generate very short pulses of 50-ps duration. Allowing... 

The theoretical approach is based on the solution to the mixed type linear/nonlinear generalized Schrodinger equation for spatiotemporal envelope of electrical field with account of transverse spatial derivatives and the transverse profile of refractive index. In the quasi-static approximation, this equation is reduced to the linear/nonlinear Schrodinger equation for spatiotemporal pulse envelope with temporal coordinate given as a parameter. Then the excitation problem can be formulated for a set of stationary light beams with initial amplitude distribution corresponding to temporal envelope of the initial pulse. 

Here complex amplitude of the pulse envelope E x, z, t) is also a slowly varying function of z and t. Spatiotemporal distribution of the electric field is described by E x,z,t) = E x,z,t)ex-p[iujQt— i/Sz), f3 being the longitudinal wavenumber of the waveguide mode at the pulse peak. 

In the problem of pulse diffraction on waveguide junctions, the quasistatic approximation is feasible if the diffraction length of the light beam is much shorter than the characteristic length of the pulse variation owing to the mentioned above MD, FTNR and SS effects which influence the pulse envelope. Then the results obtained for stationary light beam can be used in the analysis of the non-stationary beam self-focusing. 

All the above results can be directly used in the problem of optical pulse propagation through the junctions provided that the quasi-static approximation is feasible. As the transmittance of a waveguide junction depends on power of a stationary component of the pulse, variation of an input pulse envelope behind the junction should be observed. 

In the quasi-statie approximation, Eq.(2.9) has been solved numerieally by the FD-BPM for eaeh stationary eomponent of temporal distribution of the light beam. Amplitudes of the stationary eomponents were speeified by the form of pulse temporal envelope. The initial pulse envelope was assumed Gaussian. 

The term that depends on the third power of the frequency shift is known as third-order dispersion (TOD). When a TL pulse acquires a significant amount of TOD, the pulse envelope is distorted and a series of sub-pulses is produced, as shown in Figure 8.1. Unlike a pulse with SOD, a pulse with TOD leads to two-photon excitation with the same efficiency as a TL pulse but only for a particular two-photon frequency. At other frequencies, the amount of excitation is suppressed. The control over TOD would allow for preferential excitation in different spectral regions, while its correction would lead to efficient two-photon excitation over the whole accessed spectral range. Unfortunately, measuring and correcting TOD is not a simple task. 

Figure 7. Dipole signal along the direction of laser polarization (Dz, upper part) and number of emitted electrons (Nesc, lower part) as a function of time during the interaction of a femtosecond laser pulse withNaJ. The laser frequency and fluency are = 2,7oeV and J = 6 10uW/cm2, and the pulse envelope is indicated in dashed line. Both Vlasov and VUU results arc plotted for comparison. Ionic background is treated in the soft jellium approximation. From [44],...

Let the laser field be expressed as s(t) — o(t) sin(coLt), where o(t) is the pulse envelope function, including polarization, and col is the central frequency. For simplicity, consider a 8 function excitation. In the rotating wave approximation, the field is expressed as e(t) = S(t) exp(—icoLt) with field strength sq. Integration over t in Eq. (25) gives... 

The shape of the spectrum significantly depends on the laser pulse duration. For an infinite plane wave, the spectrum consists of well-separated harmonics which are not at integral numbers of the laser wavelength. Their spectral separation depends on the observation angle, the laser intensity, and the initial energy of the electron. In the case of short laser pulses, the motion of the electron becomes strongly modulated in the laser pulse envelope and the consecutive harmonics overlap to form a continuous spectrum. 

Fig. 8.1 Laser distillation control scenario discussed in detail in Section 8.3. Two ljtsfi with pulse envelopes ,(0 and e2(0 couple, by virtue of the dipole operator, the states and L enantiomers to two vibrotational states 1) and 2) (denoted If ,) and E2) in the.t i the excited electronic manifold. A third laser pulse with envelope r.0(i) couples the ex E)) and E2) states to one another. The system is allowed to absorb a photon and relax hack the ground state. After many such excitation-relaxation cycles, a significant cnantionneh excess is obtained, as explained in Section 8.3. ...

Thus, we have shown that there is one energy point (E = E0) where the adiabatic states contribute two equal absorption amplitudes of opposite signs, thereby entirely canceling the absorption process. The location of this point is independent of the strength of the S2(t) pulse envelope or of its time dependence, or of the strength of the (E0 H E, n 1) interaction that confers an energetic width to the resonance. 

Replacing the electric field amplitude by a time-dependent pulse envelope 5, we can write the time-dependent Schrodinger equation, in the approximations fh led to Eqs. (12.44) and (12.45), as two coupled equations of the form tf ... 

In Figures 2 and 3 are shown the GAMMA C++ code for a simulated spin-echo pulse sequence using ideal RF pulses and a simulated PRESS pulse sequence that accounts for a non-ideal RF pulse envelope, crusher gradients and spatially varying RF refocusing due to resonance group chemical shift offset and... 

So-called shaped pulses with a smoothly varying pulse envelope B t) constitute an important subclass of multiple-pulse sequences with variable rf amplitudes. A given pulse shape Bit) can be approximated by a train of square pulses with different rf amplitudes. In principle, the accuracy of the approximation may be increased to any desired degree by increasing the... 

Fig. 4. (A) Parametrization of a shaped pulse envelope with the help of a cubic spline interpolation between a small number of anchor points (circles). (B) Approximation of the smooth pulse envelope by rectangular pulses with piecewise constant rf amplitude. (Adapted from Ewing et al., 1990, p. 123, with kind permission from Elsevier Science—NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)...

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