The Coulomb Green's function and multiphoton calculations

Maquet, Alfred; Véniard, Valérie; Marian, Tudor A.

doi:10.1088/0953-4075/31/17/004

Cited by 71 publications

(78 citation statements)

References 149 publications

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“…Closed form expressions for G λ are known, see for instance [33]. Here, we have used the expression given as an expansion over a discrete Sturmian basis:…”

Section: Exact Calculations Of 2-photon Ati Matrix Elementsmentioning

confidence: 99%

Theory of attosecond delays in laser-assisted photoionization

et al. 2013

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We study the temporal aspects of laser-assisted extreme ultraviolet (XUV) photoionization using attosecond pulses of harmonic radiation. The aim of this paper is to establish the general form of the phase of the relevant transition amplitudes and to make the connection with the time-delays that have been recently measured in experiments. We find that the overall phase contains two distinct types of contributions: one is expressed in terms of the phase-shifts of the photoelectron continuum wavefunction while the other is linked to continuum-continuum transitions induced by the infrared (IR) laser probe. Our formalism applies to both kinds of measurements reported so far, namely the ones using attosecond pulse trains of XUV harmonics and the others based on the use of isolated attosecond pulses (streaking). The connection between the phases and the time-delays is established with the help of finite difference approximations to the energy derivatives of the phases. The observed time-delay is a sum of two components: a one-photon Wigner-like delay and an universal delay that originates from the probing process itself.

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“…Closed form expressions for G λ are known, see for instance [33]. Here, we have used the expression given as an expansion over a discrete Sturmian basis:…”

Section: Exact Calculations Of 2-photon Ati Matrix Elementsmentioning

confidence: 99%

Theory of attosecond delays in laser-assisted photoionization

et al. 2013

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“…It is this form of the transition amplitude which has often been used in the literature to study non-resonant, two-photon processes [10,18].…”

Section: Two-photon Transition Amplitudes and Ionization Cross Sectionsmentioning

confidence: 99%

“…In second-and higher-order perturbation theory, for instance, these functions help to carry out the summation over the complete spectrum in a rather efficient way. But although different analytic representations are known for the Green's functions [6,7,8,9,10], until today, there are almost no reliable codes freely available. Therefore, to facilitate the further application of the 'hydrogen atom' in different contexts, here we present the Greens library which provides a set of C++ procedures for the computation of the Coulomb wave and Green's funtions.…”

Section: Long Write-up 1 Introductionmentioning

confidence: 99%

Relativistic wave and Green's functions for hydrogen-like ions

Koval

Fritzsche

2003

Computer Physics Communications

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The Greens library is presented which provides a set of C++ procedures for the computation of the (radial) Coulomb wave and Green's functions. Both, the nonrelativistic as well as relativistic representations of these functions are supported by the library. However, while the wave functions are implemented for all, the bound and free-electron states, the Green's functions are provided only for bound-state energies (E < 0). Apart from the Coulomb functions, moreover, the implementation of several special functions, such as the Kummer and Whittaker functions of the first and second kind, as well as a few utility procedures may help the user with the set-up and evaluation of matrix elements. Computer for which the program is designed and has been tested: PC Pentium III, PC Athlon Installation: University of Kassel (Germany). KovalOperating systems: Linux 6.1+, SuSe Linux 7.3, SuSe Linux 8.0, Windows 98. Program language used: C++.Memory required to execute with typical data: 300 kB.No. of bits in a word: All real variables are of type double (i.e. 8 bytes long).Distribution format: Compressed tar file. CPC Program Library Subprograms required: None.Keywords: confluent hypergeometric function, Coulomb-Green's function, hydrogenic wave function, Kummer function, nonrelativistic, relativistic, two-photon ionization cross section, Whittaker function. Nature of the physical problem:In order to describe and understand the behaviour of hydrogen-like ions, one often needs the Coulomb wave and Green's functions for the evaluation of matrix elements. But although these functions have been known analytically for a long time and within different representations [1,2], not so many implementations exist and allow for a simple access to these functions. In practise, moreover, the application of the Coulomb functions is sometimes hampered due to numerical instabilities. Method of solution:The radial components of the Coulomb wave and Green's functions are implemented in position space, following the representation of Swainson and Drake [2]. For the computation of these functions, however, use is made of Kummer's functions of the first and second kind [3] which were implemented for a wide range of arguments. In addition, in order to support the integration over the Coulomb functions, an adaptive Gauss-Legendre quadrature has also been implemented within one and two dimensions. 2Restrictions onto the complexity of the problem: As known for the hydrogen atom, the Coulomb wave and Green's functions exhibit a rapid oscillation in their radial structure if either the principal quantum number or the (free-electron) energy increase. In the implementation of these wave functions, therefore, the bound-state functions have been tested properly only up to the principal quantum number n ≈ 20, while the free-electron waves were tested for the angular momentum quantum numbers κ ≤ 7 and for all energies in the range 0 . . . 10 |E 1s |. In the computation of the two-photon ionization cross sections σ 2 , moreover, only the long-wavelength...

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“…As in the case of photoeffect, retardation effects are expected to be more visible in the electron angular distribution than in the total cross section. The formalism i), leads to the precise calculations, see [7] for a review paper and [8] for recent calculations. The latter calculations cover the low frequencies range, i.e., from 10 nm to 90 nm.…”

Section: Introductionmentioning

confidence: 99%

Above-threshold ionization of hydrogen and hydrogen-like ions by X-ray pulses

et al. 2013

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Abstract:This paper adresses the problem of above-threshold ionization (ATI) of hydrogen interacting with an intense X-ray electromagnetic field. Two approaches have been used. In the first approach, we calculate generalized differential and total cross sections based on second-order perturbation theory for the electron interaction with a monochromatic plane wave, with the A 2 and A · P contributions from the nonrelativistic Hamiltonian (including retardation) treated exactly. In the second approach, we solve the time-dependent Schrödinger equation (TDSE) for a pulsed plane wave using a spectral approach with a basis of oneelectron orbitals, calculated with L 2 -integrable B-spline functions for the radial coordinate and spherical harmonics Y for the angular part. Retardation effects are included up to O(1/ ), they induce extra terms forcing the resolution of the TDSE in a three dimensional space. Relativistic effects [of O 1/ 2 ] are fully neglected. The isoelectronic series of hydrogen is explored in the range Z = 1 − 5 in both TDSE and perturbative approaches. Photoelectron angular distributions are obtained for photon energies of 1 keV and 3 keV for hydrogen, and photon energy of 25 keV for the hydrogenic ion B 4+ . Perturbative and TDSE calculations are compared. 32.80.rm, 32.80.Fb, 32.30.Rj PACS (2008):

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The Coulomb Green's function and multiphoton calculations

Cited by 71 publications

References 149 publications

Theory of attosecond delays in laser-assisted photoionization

Theory of attosecond delays in laser-assisted photoionization

Relativistic wave and Green's functions for hydrogen-like ions

Above-threshold ionization of hydrogen and hydrogen-like ions by X-ray pulses

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