Instability in degenerate two-photon running wave laser with detuning

The previous stability analysis of the degenerate two-photon running wave laser is extended to the inclusion of detuning between frequencies of cavity and atoms. We derive the analytical equation for the critical pumping and prove analytically that for the special case of r (γ_∥/γ_⊥) being unity, there is no Hopf bifurcation instability for the bad cavity. The good cavity case is analysed numerically. The role played by detuning is to raise the critical pumping. In the case of k<r (or <Kγ_∥) where there is no Hopf bifurcation instability for the perfect tuning case, the large detuning can give rise to self-pulsing instability.