Analytic self-pulsing solutions and their instabilities in a homogeneously broadened ring laser

Analytical traveling-wave self-pulsing solutions of a homogeneously broadened ring laser are presented in the limit that the dipole relaxation rate is much greater than the atomic inversion relaxation rate. The phase velocity v(Λ) as a function of the normalized pump Λ is given explicitly. This function has a simple relation with the upper boundary ${\alpha }_{\max (\mathrm{\Lambda }}$) of the Risken-Nummedal-Graham-Haken (RNGH) instability domain in the (α,Λ) plane, where α is the wave vector of the perturbations. The self-pulsing may appear in different ways, depending on how the stationary solution becomes unstable. If the instability of the stationary solution is caused by an unstable mode touching upon the upper boundary of the RNGH-instability domain in the (α,Λ) plane, the self-pulsing is supercritical; if the unstable mode lies on the lower boundary, a subcritical self-pulsing arises, and the system is bistable. This rule provides new signatures of the self-pulsing phenomena and will be helpful to experimental identification of the self-pulsing arising from the RNGH instability. The linear stability analysis reveals two kinds of instabilities for these self-pulsing solutions. One kind of instability is rooted in the RNGH instability of the stationary solution and may occur even when the amplitude of the self-pulsing solution approaches zero. The other kind of instability occurs when the oscillating amplitude of the self-pulsing solution becomes large. Above this large-amplitude instability threshold no traveling-wave self-pulsing can be stable any longer. The results for the corresponding Lorenz model are also presented.