Fast Fourier transform propagation

Pig. 4. Photo dissociation of ArHCl. Left hand side the number of force field evaluations per unit time. Right hand side the number of Fast-Fourier-transforms per unit time. Dotted line adaptive Verlet with the Chebyshev approximation for the quantum propagation. Dash-dotted line with the Lanczos iteration. Solid line stepsize controlling scheme based on PICKABACK. If the FFTs are the most expensive operations, PiCKABACK-like schemes are competitive, and the Lanczos iteration is significantly cheaper than the Chebyshev approximation. 

When performing optical simulations of laser beam propagation, using either the modal representation presented before, or fast Fourier transform algorithms, the available number of modes, or complex exponentials, is not inhnite, and this imposes a frequency cutoff in the simulations. All defects with frequencies larger than this cutoff frequency are not represented in the simulations, and their effects must be represented by scalar parameters. 

This expansion is valid to second order with respect to St. This is a convenient and practical method for computing the propagation of a wave packet. The computation consists of multiplying X t)) by three exponential operators. In the first step, the wave packet at time t in the coordinate representation is simply multiplied by the first exponential operator, because this operator is also expressed in coordinate space. In the second step, the wave packet is transformed into momentum space by a fast Fourier transform. The result is then multiplied by the middle exponential function containing the kinetic energy operator. In the third step, the wave packet is transformed back into coordinate space and multiplied by the remaining exponential operator, which again contains the potential. 

An important feature that affects the numerical solution strategy is that these equations are written in the spectral space, either in the three dimensional space of wave-vectors (/-propagated UPPE) or in a two-dimensional space of transverse wave-vectors plus a one dimensional angular-frequency space (z-propagated UPPE). At the same time, the nonlinear material response must be calculated in the real-space representation. Consequently, a good implementation of Fast Fourier Transform is essential for a UPPE solver. 

Standard methods are used to propagate each Om in time. For the z and Z coordinates we make use of the fast fourier transform [99], and for the p coordinate we use the discrete Bessel transform [100]. The molecular component of asymptotic region at each time step, and projected onto the ro-vibrational eigenstates of the product molecule, for a wide range of incident energies included in the incident wave packet [82]. The results for all ra-components are summed to produce the total ER reaction cross section, a, and the internal state distributions. 

Using whatever propagation method, one has to evaluate the action of the Hamiltonian operator on the wavefunction P(r). This is normally carried out by expanding P(f) in a suitable basis set and then evaluates the operator action on basis functions. One can use the FFT (fast Fourier transform) techniques (7,14), discrete variable representation (DVR) (15,16) techniques, or simply calculate matrix elements of the operator in a given basis set. 

It is noted that in order to perform a spectral analj is of P (t, At) with a standard fast Fourier transform routine, one needs to assure that the polarization has decayed completely. In practice, this requirement results either in invoking a phenomenological damping oc of the polarization (which artificially broadens the spectrum) or in a large propagation time of the polarization (which is computationally expensive). To avoid this problem, it has been suggested to employ a filter-diagonalization method " for the spectral analysis of P t, At). ... 

To be more precise, this time propagation scheme consists of two ingredients (i) application of the Trotter product formula to the time propagation operator, and (ii) an efficient use of the Fast Fourier transform (FFT). Recall that the increase of time to perform FFT scales to NlogN, where N is the number of grid points [219]. A time evolution operator for a short time interval At, which is composed of two noncommutable operators A and B, can be approximately represented with the TVotter product (decomposition) formifla as... 

The basic technique used to propagate the wave packet in the spatial domain is the fast Fourier transform method [287, 288, 299, 300]. The time-dependent Schrodinger equation is solved numerically, employing the second-order differencing approach [299, 301]. In this approach the wave function Sit t = t St is constructed recursively from the wave functions at t and t" = t — St. The operator including the potential energy is applied in phase space and that of the kinetic energy in momentum space. Therefore, for each... 

Time-domain BSS. Because counter-propagating acoustic waves produce a standing wave, the diffracted intensity oscillates in time and the oscillation can be recorded by using a fast photodetector and a digitizing oscilloscope a surface phonon spectrum can thus be obtained from a Fourier-transform of the data. 

Usually, the depolarization current is measured to avoid the dc conductivity contribution. The dielectric relaxation spectrum is then obtained by Fourier transform or approximate formulas, e.g., the Hamon approximation [14]. By carefully controlling the sample temperature and accurately measuring the depolarization current, precision measurements of the dielectric permittivity down to 10" Hz are possible [18]. In fast time domain spectroscopy or reflectometry, a step-like pulse propagates through a coaxial line and is reflected from the sample section placed at the end of the line. The difference between... 

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