X-ray free-electron lasers 1,2 delivering up to 1 X 10 13 coherent photons in femtosecond pulses are bringing about a revolution in X-ray science 3 ' 5 . However, some plasma-based soft X-ray lasers 6 are attractive because they spontaneously emit an even higher number of photons (1 X 10 15 ), but these are emitted in incoherent and long (hundreds of picoseconds) pulses 7 as a consequence of the amplification of stochastic incoherent self-emission. Previous experimental attempts to seed such amplifiers with coherent femtosecond soft X-rays resulted in as yet unexplained weak amplification of the seed and strong amplification of incoherent spontaneous emission 8 . Using a time-dependent Maxwell-Bloch model describing the amplification of both coherent and incoherent soft X-rays in plasma, we explain the observed inefficiency and propose a new amplification scheme based on the seeding of stretched high harmonics using a transposition of chirped pulse amplification to soft X-rays. This scheme is able to deliver 5 X 10 14 fully coherent soft X-ray photons in 200 fs pulses and with a peak power of 20 GW.Over the past 10 years, the emergence of hard (A ~ 0.1 nm) and soft (A ~ 4-30 nm) X-ray free-electron lasers (FEL) with intensities rapidly increasing to 1 x 10 W cm has led to scientific breakthroughs in a diverse number of fields 3 " 5 . This race has led to renewed interest in a specific class of soft X-ray lasers (XRLs) that use a plasma amplifier created by the interaction of a nanosecond, high-energy laser with a solid target. These XRLs routinely produce an extreme number of photons per pulse 7 , up to 1 x 10 15 (that is, 10 mj), which compares favourably with soft X-ray FELs, which emit a maximum of 1 x 10 13 photons per pulse 1 ' 2 . However, because these plasma-based XRLs are running in amplification of spontaneous emission (ASE) mode, they demonstrate weak coherence and long pulse durations (100 ps). Seeding these plasmas, which naturally emit up to 10 mj, holds the greatest promise of producing fully coherent, femtosecond, multi-milljoule soft X-ray pulses. Such an experiment was carried out in 1995 8 , but this showed an as yet unexplained weak amplification of the seed (output, 100 nj) and strong amplification of the ASE (reaching several millijoules). Plasmas created by femtosecond 9 to picosecond 10 infrared lasers have been seeded, but continue to be limited by the generation of long (picosecond) and low-energy (1 |xj) pulses. Numerical studies have shown that these schemes could amplify the seed to generate pulses of up to 70 fs (ref 11), but with energy restricted to <40 |xj (ref. 12). It is only an indepth study of Ditmire's seminal experiment 8 that holds the key to unlocking the path towards millijoule, femtosecond soft X-ray lasers.The modelling parameters of our Maxwell-Bloch model 13 were adjusted on the basis of experiments 8 with a 200 fs, 0.5 nj, 21.2 nm seed, and plasma with an electron density of 4 x 10 20 cm 3 and temperature of 550 eV. In the absence of seed, gain peaks at 25 cm 1 were...
