Coherent propagation of a half-cycle unipolar attosecond pulse in a resonant two-level medium

Here we numerically investigate the propagation of unipolar half-cycle attosecond pulses with a duration much shorter than the period of the proper oscillations T0 of the medium and its relaxation times T1 and T2 in a twolevel resonant absorbing and amplifying medium.We find that despite breaking down of the famous McCall and Hahn area theorem, some phenomena described by this theorem are still qualitatively preserved. In particular, it is shown that pulses with small initial electric pulse area become sign-alternating upon the propagation, similar to the traditional 0π pulses; pulses with 4π, 5π, and 6π area are unstable and split into lower-order pulses, exactly as in the long-pulse case. The electric pulse area conservation rule and limitations of unipolar pulse amplification in two-level gain media, which follows fromthis rule, are discussed.