  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.
