  Efficient and cost-effective Si-compatible lasers are a long standing wish of
the optoelectronic industry. In principle, there are two options. For many
applications, lasers based on III-V compounds provide compelling solutions,
even if the integration is complex and therefore costly. However, where low
costs and also high integration density are crucial, group-IV-based lasers -
made of Ge and GeSn, for example - could be an alternative, provided their
performance can be improved. Such progresses will come with better materials
but also with the development of a profounder understanding of their optical
properties. In this work, we demonstrate, using Ge microbridges with strain up
to 6.6%, a powerful method for determining the population inversion gain and
the material and optical losses of group IV lasers. This is made by deriving
the values for the injection carrier densities and the cavity losses from the
measurement of the change of the refractive index and the mode linewidth,
respectively. We observe a laser threshold consistent with optical gain and
material loss values obtained from a tight binding calculation. Lasing in Ge -
at steady-state - is found to be limited to low temperatures in a narrow regime
of tensile strain at the crossover to the direct band gap bandstructure. We
explain this observation by parasitic intervalence band absorption that
increases rapidly with higher injection densities and temperature. N-doping
seems to reduce the material loss at low excitation but does not extend the
lasing regime. We also discuss the impact of the optically inactive carriers in
the L-valley on the linewidth of group IV lasers.
