Interference effects in the high-order harmonic amplification process in the active medium of a plasma-based X-ray laser modulated by an optical field

We study the process of propagation of high harmonics of optical radiation in an active medium of a plasma-based X-ray laser, simultaneously irradiated by an intense optical field of fundamental frequency. It is shown that for moderate plasma dispersion of the active medium at the frequency of the modulating optical field, the energy and relative amplitudes of the harmonics at the output of the medium are determined by their phases at the entrance to the medium, as well as by the time-delay of the harmonics with respect to the modulating field. These dependences are due to interference of high-order harmonics with a set of multi-frequency fields generated by each of the harmonics in the process of coherent scattering in a modulated active medium. The possibilities of using these effects to increase the efficiency of harmonic amplification, to control the harmonic spectrum, and determine the relative phases at the entrance to the medium are discussed on the example of the active medium of hydrogen-like Li²⁺ ions (with a 13.5 nm wavelength of an inverted transition).