Amplification and ellipticity enhancement of high-order harmonics in a neonlike x-ray laser dressed by an IR field

In Khairulin et al. [Sci. Rep. 12, 6204 (2022)] we proposed a method for amplifying a train of sub-femtosecond pulses of circularly or elliptically polarized extreme ultraviolet radiation, composed by high-order harmonics of an infrared (IR) laser field, in a neonlike active medium of a plasma-based x-ray laser, additionally irradiated with a replica of a fundamental frequency laser field used to generate harmonics. Here we present the analytical theory and numerical study of an amplification process of an individual harmonic with either preservation of its polarization state or ellipticity enhancement. We also analyze in detail the spectral-temporal and polarization properties of a sub-femtosecond pulse train formed by a set of elliptically or circularly polarized harmonics during the propagation through the medium. We discuss also the possibility of an experimental implementation of the suggested technique in an active medium of an x-ray laser based on neonlike ${\mathrm{Ti}}^{12+}$ ions irradiated by an IR laser field with a wavelength of 3.2 \textmu{}m.