  Theoretical studies of superradiant lasing on optical clock transitions
predict a superb frequency accuracy and precision closely tied to the bare
atomic linewidth. Such a superradiant laser is also robust against cavity
fluctuations when the spectral width of the lasing mode is much larger than
that of the atomic medium. Recent predictions suggest that this unique feature
persists even for a hot and thus strongly broadened ensemble, provided the
effective atom number is large enough. Here we use a second-order cumulant
expansion approach to study the power, linewidth and lineshifts of such a
superradiant laser as a function of the inhomogeneous width of the ensemble
including variations of the spatial atom-field coupling within the resonator.
We present conditions on the atom numbers, the pump and coupling strengths
required to reach the buildup of collective atomic coherence as well as scaling
and limitations for the achievable laser linewidth.
