**1. Introduction**

The spin polarization of photoelectrons is one of the fundamental characteristics of photoionization processes. Starting from the pioneering paper by U. Fano [1] and the seminal papers by M. Amusia's pupil N. Cherepkov [2,3], it was realized that in spite of the weakness of the spin-orbit interaction, the spin polarization of photoelectrons may be large, of the order of unity. During the last 50 years, a large number of experimental and theoretical works have been devoted to studying the spin polarization of photoelectrons in the photoemission from atoms, molecules and solids (see [3–5] and references therein). There are two main reasons why these investigations are considered important. One is that a high degree of polarization of photoelectrons is an important prerequisite for creating sources of polarized electrons, which in turn may serve as a tool for investigating various aspects of magnetism in solids [6,7]. Another reason is that measurements of spin polarization provide additional information about the mechanism of the photoemission; in particular, they are necessary for the realization of the so-called complete experiment, i.e., the experimental determination of the complex amplitudes of photoionization [8].

Until quite recently, the majority of the experimental investigations of spin polarization of photoelectrons have been performed at synchrotron radiation sources. Since the intensity of the sources is rather low, the interpretation of these experiments was based on the linear single-photon approximation describing the interaction of the photons with a quantum system. Spin polarization in multiphoton processes has been considered theoretically first for relatively weak laser fields where perturbation theory is applicable [9–11]. Here, it was demonstrated that the degree of polarization may be high also in the multiphoton processes. With the advent of free-electron lasers with extremely high intensities of the photon beam, the possibility arises to study experimentally the spin polarization in multiphoton ionization in a wide range of photon wavelengths. A theoretical prediction of high degree of spin polarization in multiphoton strong-field ionization was presented in Ref. [12]. Recently, first experiments of this kind were reported [13–15]. Photoelectron spin polarization, in an interesting particular case of the three-photon bichromatic (*ω* + 2*ω*) ionization, was theoretically considered in papers [16,17].

**Citation:** Kabachnik, N.M.; Sazhina, I.P. Spin Polarization of Electrons in Two-Color XUV + Optical Photoionization of Atoms. *Atoms* **2022**, *10*, 66. https://doi.org/ 10.3390/atoms10020066

Academic Editors: Anatoli Kheifets, Gleb Gribakin and Vadim Ivanov

Received: 6 May 2022 Accepted: 11 June 2022 Published: 20 June 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

A special case of multiphoton processes is the photoionization of atoms and molecules by extreme ultraviolet (XUV) or soft X-ray pulses in the presence of infrared (IR) or optical radiation [18]. In the following, in order to shorten the explanations, we shall discuss the XUV+IR two-color case, although all discussed properties and conclusions are also valid for ionizing by soft X-rays and for dressing by an optical pulse (OP). If the energy of the XUV photons in these two-color experiments is sufficient to ionize the atom, a series of sidebands appear in the photoelectron spectrum at both sides of the photoline, due to the simultaneous emission or absorption of the IR photons [19]. The energy separation between the sidebands is equal to the photon energy of the IR field. This process is convenient for studying the photoinduced transitions in the continuum. The appearance of the sidebands was used for measuring the duration and the arrival time of the XUV pulses from the free-electron lasers (FELs) [20,21]. Sidebands were also used for the determination of the circular polarization of the FEL beams [22,23] by measuring the circular dichroism of the sidebands.

As far as we know, there are no investigations of the spin polarization of photoelectrons generated in two-color XUV+IR processes, although it is clear that the photoelectrons both in the central line and in the sidebands should be spin polarized, provided that the fine structure of the lines is resolved in the experiment [3]. In this paper, we report a theoretical investigation of the spin polarization in such processes. We sugges<sup>t</sup> a simple theoretical model based on the description of the XUV+IR processes in the strong field approximation (SFA) [24], which was widely used in the description of multiphoton processes [25]. We analyze the spin polarization of photoelectrons induced by circularly polarized XUV photons in the presence of collinear circularly polarized IR beams. In particular, we consider the component of the spin polarization parallel to the light helicity, the so-called "polarization transfer". This component is non-zero not only in angle-resolved measurements, but also in angle-integrated experiments [4]. We show that the spin polarization of sidebands is different from the spin polarization of the central photoline and changes with the sideband order. It strongly depends on the helicity of the IR beam.

As a particular example, we chose a short-pulse two-color photoionization of Xe atoms. The ground state of Xe contains a closed 5p electronic subshell. Upon photoionization, the lowest state of the Xe<sup>+</sup> ion is a spin-orbit doublet <sup>2</sup>*P*3/2 and <sup>2</sup>*P*1/2. The spin-orbit splitting is sufficiently large at 1.3 eV, which simplifies the spin-polarization measurements in which it is necessary to resolve the fine-structure components [4]. The spin polarization of photoelectrons from the single-photon ionization of Xe is well investigated both theoretically and experimentally (see, for example, [4] and references therein). Additionally, the spin polarization of emitted electrons in single-color multiphoton ionization of Xe was investigated [13–15] .

In this paper, we consider the two-color multiphoton ionization of Xe by ultrashort (femtosecond) XUV and optical pulses. In the following Section 2.1, a short description of the theoretical approach is given, which is based on the strong field approximation (SFA) [24,25]. Section 2.2 contains the parameters and details of the calculation. Section 3 presents the results of the calculations as well as a discussion including simple approximate formulas for the spin polarization of the sidebands. Finally, Section 4 gives conclusions and an outlook. In the Appendix A, we present a derivation of the approximate expressions for the related matrix elements, which is used for obtaining a simple approximate expression for the spin polarization of photoelectrons in two-color experiments.
