  In a recent paper we proposed to build an x-ray laser oscillator (XLO) in the
6-10 keV range providing intense, stable, transform-limited, x-ray pulses based
on population inversion driven by an x-ray pulse train generated by an x-ray
free-electron laser (XFEL) operated at a repetition rate of about 100 Hz. Here
we present an analysis of recent experimental results on x-ray lasing with
population inversion, damage caused by the pump on the lasing medium, and
optical cavities, together with theoretical/numerical simulations. Our findings
suggest that it is possible to build and operate a second-generation x-ray
laser oscillator, XLO-II, operating at up to 125 kHz repetition rate. XLO-II
will be pumped by 6-10 keV x-ray SASE pulses, generated by the new LCLS-II-HE
XFEL now under construction at SLAC National Accelerator Laboratory, utilizing
a CW superconducting linac and capable of running at 1 MHz repetition rate.
XLO-II will generate transform-limited, coherent x-ray pulses with an average
power in the tens of mW range. It will open new experimental capabilities, for
instance in fields like imaging, interferometry, and quantum x-ray optics. The
main characteristics of XLO-II and its main components, including the optical
cavity, will be discussed.
