Modeling harmonic generation by a zero-range potential

Cutoffs of high-energy plateaux for atomic processes in an intense elliptically polarized laser field

“…According to (12), a strong laser fi eld effectively populates only the Fourier-harmonics of Φ e (r, t) near the origin having k up to k ≈ k c , from which the electron can emit (spontaneous) harmonic photons having a maximum energy Note, that the estimate (13) was already obtained earlier [4] based on a straightforward evaluation of an approximate expression for the HHG amplitude in the complex quasienergy approach for a ZRP model. (It is interesting that in [4] all coeffi cients f 2k were neglected except for f 0 = 1. However, the re-sulting HHG amplitude has a form similar to that of the matrix element M k0 (e = -1), thus yielding the same cutoff as in (12).)…”

“…Since the parameter ζ k depends smoothly on k, the k-dependence of the coeffi cients f 2 k is dominated by that of the matrix elements M k 0 (e = -1). To estimate these matrix elements, we note fi rst that the argument z(τ) of the Bessel function in (5) has maxima and minima at τ = τ n , n = 1, 2, ..., where τ n are roots of the equation [4]:…”

“…The approximation represented by (11) and (6) provides numerical results for | f 2k | that are in excellent agreement with those calculated using the exact equation (4) for k values around the cut-off and beyond the plateau region. An analytical estimate for k c corresponding to the cut-off in the spectrum of | f 2k | may be deduced by estimating the maximum of |M k0 | in (11) as a function of k (cf [4]). The position of this maximum may be estimated as the average of k b and k e , k max ≡ k c = (k b + k e )/2, where k b ≈ |z 1 | corresponds to the maximum of the Bessel function in (11) and k e ≈ |z 1 | + ω -1 corresponds to the maximum of the exponential in (11).…”

“…the plateau cutoffs) for both HHG and ATI have been measured experimentally and agree with results of numerical analyses and classical and quasiclassical (mostly one-dimensional) models [1,2]; they are given by (|E 0 | + 3.17u p ) and 10u p respectively, where |E 0 | and u p are the ionization potential and the ponderomotive shift. For an elliptically polarized fi eld, information on the plateau cutoffs is very sparse [2]: for HHG, an estimate of the cutoff position (see (13) below) was obtained in [4] using an approximate quantum result for the HHG amplitude within a zero-range potential (ZRP) model and in [5] using the semiclassical method for a Gaussian form of the dipole matrix element; general features of plateau effects in ATI/ATD for an elliptically polarized fi eld have been discussed in [6]; plateau structures in LAES for the cases of elliptical and circular polarization have been predicted very recently in [7].…”

Theory of Nonlinear Propagation of High Harmonics Generated in a Gaseous Medium

“…A simple physical interpretation for this scaling law is that λ −2 0 originates from the wave packet spreading (spending much more time in the continuum as the wavelength of the laser increases) while an additional λ −1 0 factor arises from the conversion from photon number to energy. This is only valid for a single harmonic with both I p /ω 0 and U p /ω 0 constant [28]. Quantum-mechanics calculations suggested that single-atom HHG yield drops even more dramatically with laser wavelengths and scales as λ −5.5 0 for the fixed laser intensity and for a same photon energy interval [29][30][31].…”