Relativistic calculations of two-color two-photon K-shell ionization

THE EUROPEAN PHYSICAL JOURNAL D Eur. Phys. J. D (2022)76:18 https://doi.org/10.1140/epjd/s10053-021-00334-x Regular Article – Optical Phenomena and Photonics Relativistic calculations of two-color two-photon K-shell ionization J. Fan1,2,3,a , J. Hofbrucker3,4 , A. V. Volotka5 , and S. Fritzsche2,3,4 1 Abbe School of Photonics, Albert-Einstein-Straße 6, 07745 Jena, Germany Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany 3 Helmholtz-Institut Jena, Fröbelstieg 3, 07743 Jena, Germany 4 GSI Helmholtzzentrum für Schwerionenforschung GmbH, Planckstrasse 1, 64291 Darmstadt, Germany 5 School of Physics and Engineering, ITMO University, Kronverkskiy pr. 49, St. Petersburg, Russia 197101 2 Received 7 September 2021 / Accepted 19 December 2021 © The Author(s) 2022 Abstract. We investigate the two-color two-photon K -shell ionization of neutral atoms based on the relativistic second-order perturbation theory and independent particle approximation. Analytical expressions for the relativistic and nonrelativistic total cross sections are derived in terms of radial transition amplitudes and Stokes parameters. Particular attention is paid especially to how the two-photon ionization total cross section depends on the energy sharing and polarization of the two incident photons. We construct the nonrelativistic expressions of cross section ratios for different polarization combinations of the two incident photons. The numerical results of total cross section and cross section ratios show that the energy sharing of the two incident photons plays an essential role in two-photon K -shell ionization. Particularly, if the energies of the two incident photons are identical, the total cross section and cross section ratios will reach the minimum or maximum value. Moreover, due to the strong screening effects, we find strong deviations of the cross section ratios near the two-photon ionization threshold of the Ne atom. 1 Introduction The advent of high-intensity x-ray free-electron lasers (XFELs) has opened frontiers to study nonlinear ionization processes, such as the two-photon ionization (TPI) [1–3]. The TPI is one of the most fundamental nonlinear phenomena in the light-matter interaction process, in which an atom absorbs two photons and emits a photoelectron. Current XFEL facilities can produce pulses with keV photon energies [4–6] and offer the possibility to explore the inner shell TPI process. One of the first experiments has been performed for direct two-photon ionization of helium-like ion Ne8+ [7]. Recently, studies were conducted for the two-photon K -shell ionization of solid targets, such as Ge [8], solid Zr [9], as well as metallic Cu [10,11]. From a theoretical point of view, the perturbative nonrelativistic framework has been employed in the calculations of total two-photon K -shell ionization cross section of many-electron systems in Refs. [12–15]. Further studies of TPI cross section have been conducted within the full relativistic framework for ionization of hydrogen-like atoms [16,17], and later for neutral atoms [18,19]. Although the studies of two-photon K -shell ionization of atoms by monochromatic light are well established, the TPI process with two nondegenerate incident photons (bichromatic light) was paid less attention in a e-mail: (corresponding author) 0123456789().: V,-vol the past, which should be considered in the context of this work. Nevertheless, let us first discuss the techniques for the realization of the above-mentioned experimental scenario. Various schemes for generation of two-color beams at XFEL facilities have been proposed [20–25], for instance utilizing variable-gap undulators [21], or using a single monochromatizing crystal as demonstrated in Ref. [22]. Recently, the methods by using a double-slotted foil on a chirped beam [23] or sextupole magnet [24] have been performed. In contrast to single-photon ionization, the total cross section of TPI shows a strong dependence not only on the energies of the photons but also on their polarizations. However, so far, the investigation has been focused on two equally polarized photons. Experimentally, the polarization control at XFEL facilities can be achieved either by various undulator configurations or by periodic temporal modulation [26– 32]. such as by employing crossed planar undulators, an arbitrary photon polarization can be generated as demonstrated in Refs. [26,28,31]. These examples hint that the techniques of generating two beams with tunable energies and polarizations will be available in the near future. In this context, an important step is to investigate how the total cross section of two-color two-photon K shell ionization depends on energy sharing and polarization of the two incident photons. Therefore, in Sect. 2 we firstly employ the relativistic second-order pertur- 123 18 Page 2 of 8 bation theory based on the Dirac equation and independent particle approximation to derive the total TPI cross section. We then apply the nonrelativistic limit to obtain simple expressions of cross section ratios for different polarization combinations. In Sect. 3, numerical computations have been carried out for two-color two-photon K -shell ionization of neutral Ne and Ge atoms. The total cross section and cross section ratios have been calculated as a function of energy sharing between the two incident photons. The results for the total cross section indicate that the minimum values occur when the energies of the two incident photons are identical. Similar behaviors occur in the results of cross section ratios as well. Besides, by comparing the analytical and numerical work of cross section ratios, we confirm the boundary values of cross section ratios for different polarization combinations. Finally, a summary is given in Sect. 4. For convenience, relativistic units ( = c = m = 1) are used throughout the paper, unless stated otherwise. 2 Theoretical background In order to simplify the TPI process from many-electron system to single-electron system, we employ the independent particle approximation. We assume that an initial bound active electron of the K -shell |na κa ma  interacts with the two incident photons. Here, na is the principle quantum number, κa is the Dirac quantum number, and ma is the projection of total angular momentum of the initial bound active electron. The Dirac quantum number κ is defined by the total and orbital angular momenta j and l as κ = ∓(j + 12 ) for j = l ± 12 . In addition, we use κν and κ to represent the Dirac quantum number of intermediate virtual state and continuum state throughout the paper. Since in this work, we investigate the scenario that the electron interacts with two photons γ(k, ) with different wave vectors k and polarization . We also assume the two incident photons propagate along the quantization axis (k̂1 = k̂2 = k̂). The screening effect of all other inactive electrons is accounted (...truncated)