  Citation: Khairulin, I.R.; Antonov, V.A.; Ryabikin, M.Y.; Kocharovskaya, O. Enhanced Amplification of Attosecond Pulses in a Hydrogen-like Plasma-Based X-ray Laser Modulated by an Infrared Field at the Second Harmonic of Fundamental Frequency. Photonics 2022, 9, 51. https:// doi.org/10.3390/photonics9020051 Received: 29 December 2021 Accepted: 17 January 2022 Published: 19 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. 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/). photonics hv Article Enhanced Amplification of Attosecond Pulses in a Hydrogen-like Plasma-Based X-ray Laser Modulated by an Infrared Field at the Second Harmonic of Fundamental Frequency Ilias R. Khairulin 1, * , Vladimir A. Antonov 1 , Mikhail Yu. Ryabikin 1,2 and Olga Kocharovskaya 3 1 Institute of Applied Physics of the Russian Academy of Sciences, 46 Ulyanov Street, 603950 Nizhny Novgorod, Russia; antonov@appl.sci-nnov.ru (V.A.A.); mikhail.ryabikin@ipfran.ru (M.Y.R.) 2 Faculty of Radiophysics, Lobachevsky State University of Nizhny Novgorod, 23 Prospekt Gagarina, 603022 Nizhny Novgorod, Russia 3 Department of Physics and Astronomy, Texas A&M University, 578 University Drive, College Station, TX 77843-4242, USA; kochar@physics.tamu.edu * Correspondence: khairulinir@ipfran.ru Abstract: In a recent work (Antonov et al., Physical Review Letters 123, 243903 (2019)), it was shown that it is possible to amplify a train of attosecond pulses, which are produced from the radiation of high harmonics of the infrared field of the fundamental frequency, in the active medium of a plasma-based X-ray laser modulated by a replica of the infrared field of the same frequency. In this paper, we show that much higher amplification can be achieved using the second harmonic of the fundamental frequency for modulating of a hydrogen-like active medium. The physical reason for such enhanced amplification is the possibility to use all (even and odd) sidebands induced in the gain spectrum in the case of the modulating field of the doubled fundamental frequency, while only one set of sidebands (either even or odd) could participate in amplification in the case of the modulating field of the fundamental frequency due to the fact that the spectral components of the high-harmonic field are separated by twice the fundamental frequency. Using the plasma of hydrogen-like C 5+ ions with an inverted transition wavelength of 3.38 nm in the water window as an example, it is shown that the use of a modulating field at a doubled fundamental frequency makes it possible to increase the intensity of amplified attosecond pulses by an order of magnitude in comparison with the previously studied case of a fundamental frequency modulating field. Keywords: attosecond pulses; X-ray optics; plasma-based X-ray laser; strong optical field; Stark effect; high-order harmonic amplification 1. Introduction The beginning of the 21st century was marked by the emergence and rapid develop- ment of attosecond physics—an interdisciplinary field of research aimed at probing and controlling the ultrafast dynamics of charge carriers in atoms, molecules, and solids on their own time scales [1–7]. The main instrument of attosecond physics is attosecond X-ray and vacuum ultraviolet (VUV) pulses produced due to high-order harmonic generation under the action of optical laser field on a gas medium in the tunneling ionization regime [8–10]. Such sources make it possible to generate a spectrum of harmonics with a width of more than 1 keV and up to 12 octaves [11], as well as to generate pulses with a duration of down to 40–50 as [12–14]. However, the energy of such pulses in the X-ray range, from several hundred eV and above, as a rule, does not exceed hundreds of pJ or, at best, few nJ [15,16], which limits the possibilities of their practical applications, in particular, for single-pulse measurements of ultrafast processes in matter, as well as in measurements based on the “attosecond pump–attosecond probe” scheme [17,18]. In recent work [19], we proposed a method for amplifying a train of attosecond pulses formed by a set of high-order harmonics of the infrared (IR) field in a hydrogen-like active Photonics 2022, 9, 51. https://doi.org/10.3390/photonics9020051 https://www.mdpi.com/journal/photonics