Stability analysis of a bidirectional homogeneously broadened ring laser

We study the stability of steady state solutions of a resonantly tuned bidirectional homogeneously broadened ring laser retaining the complete longitudinal spatial dependence. We prove that in two limits, the stable steady states are only weakly stable since the relaxation rates are very large. This implies a high sensitivity to any kind of perturbation which will easily induce a transition from one direction of propagation to the other one. We also analyze a popular approximation which uses a truncated harmonic expansion and the adiabatic elimination of the polarization. When the adiabatic elimination is justified, we show that the harmonic truncation does not further limit the range of applicability of the stability analysis.