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In this paper, we present the analytical theory of attosecond pulse formationvia optical modulation of an active medium of the hydrogen-like C5+plasma-based X-ray laser at 3.4 nm wavelength in the "water window" range,taking into account a variation of the population inversion caused by radiativedecay of the upper lasing states. We derive an analytical solution for theX-ray field amplified by an X-ray laser with time-dependent populationinversion, which is simultaneously irradiated by a strong optical laser field,and use it to find the optimal conditions for the attosecond pulse formationfrom a narrowband seeding X-ray field. We show that the shape of pulses can beimproved at the cost of reduced pulse peak intensity (i) via externalattenuation of the resonant spectral component of the amplified X-ray field or(ii) by using a resonantly absorbing medium (the active medium of the X-raylaser after the change of sign of the population inversion) for the pulseformation. The results of the analytical theory are in a good agreement withthe numerical solutions of the Maxwell-Bloch equations which account for thenonlinearity, as well as the amplified spontaneous emission, of the activemedium. Both analytically and numerically we show the possibility to produce atrain of attosecond pulses with sub-200 as duration and the peak intensityexceeding 10^12 W/cm^2 at the carrier wavelength 3.4 nm in the "water window"range, which makes them attractive for the biological and medical applications.

Attosecond pulse formation in the "water window" range by optically dressed hydrogen-like plasma-based C5+ X-ray laser