  Optical amplifiers in all ranges of the electromagnetic spectrum exhibit two
essential characteristics: i) the input signal during the propagation in the
medium is multiplied by the avalanche effect of the stimulated emission to
produce exponential growth and ii) the amplification saturates at increasing
input signal. We demonstrate that the strong-field theory in the frame of high
harmonic generation fully supports the appearance of both the avalanche and
saturation effects in the amplification of extreme ultraviolet attosecond pulse
trains. We confirm that the amplification takes place only if the seed pulses
are perfectly synchronized with the driving strong field in the amplifier. We
performed an experimental study and subsequent model calculation on He gas
driven by intense 30-fs-long laser pulses, which was seeded with an attosecond
pulse train at 110 eV generated in a separated Ne gas jet. The comparison of
the performed calculations with the measurements clearly demonstrates that the
pumped He gas medium acted as an amplifier of the extreme ultraviolet pulse
train.
