Details of schemes for two tabletop size x-ray lasers that require a h
igh-intensity short-pulse driving laser are discussed. The first is ba
sed on rapid recombination following optical-field ionization. Analyti
cal and numerical calculations of the output properties are presented.
Propagation in the confocal geometry is discussed and a solution for
x-ray lasing in Li-like N at 247 angstrom is described. Since the calc
ulated gain coefficient depends strongly on the electron temperature,
the methods of calculating electron heating following field ionization
are discussed. Recent experiments aimed at demonstrating lasing in H-
like Li at 135 angstrom are discussed along with modeling results. The
second x-ray laser scheme is based on the population inversion obtain
ed during inner-shell photoionization by hard x rays. This approach ha
s significantly higher-energy requirements, but lasing occurs at very
short wavelengths (lambda less-than-or-equal-to 15 angstrom). Experime
nts that are possible with existing lasers are discussed.