In [I.R. Khairulin et al., submitted to Phys. Rev. Lett.] we propose a method for amplifying a train of sub-femtosecond pulses of circularly or elliptically polarized extreme ultraviolet (XUV) radiation, constituted by high-order harmonics of an infrared (IR) laser field, in a neon-like active medium of a plasma-based X-ray laser, additionally irradiated with a replica of a fundamental frequency laser field used to generate harmonics, and show the possibility of maintaining or enhancing the ellipticity of high-harmonic radiation during its amplification. In the present paper we describe this process in detail both for a single harmonic component and a sub-femtosecond pulse train formed by a set of harmonics. We derive the analytical theory and describe both analytically and numerically the evolution of the high-harmonic field during its 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 neon-like Ti^{12+} ions irradiated by an IR laser field with a wavelength of 3.9 microns.
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