  Harmonic frequency combs, in which the lasing modes are separated by a period
of tens of free spectral ranges from each other, have been recently discovered
in quantum cascade lasers (QCLs). There is an ongoing debate how the harmonic
combs can be formed and stable under continuous pumping. Here we reproduce the
harmonic state of lasing in QCLs by space-time-domain numerical simulations and
show that the corresponding optical wave has a frequency-modulated (FM) nature,
with the fundamental beat note in the electron population being suppressed. To
understand the physics behind the formation and stability of the harmonic
state, we develop a frequency-domain linearized analytic theory which analyzes
the stability of single mode lasing with respect to the harmonic state
formation and the instabilities of the resulting harmonic state. Our analysis
shows how the instability of a single lasing mode can result in the growth of
harmonic side modes. The coupling between the two side modes leads to opposite
gains for FM and amplitude-modulated (AM) waves, and the FM gain is enhanced as
compared to the AM gain. The dependence of the sideband gain spectrum on group
velocity dispersion is also studied. While the instability gain of a single
lasing mode does not prove that the harmonic state can be self-starting, we
analyze the sideband instability gain in the presence of three strong harmonic
modes and demonstrate that the harmonic state can be stable and self-supported.
