This paper is devoted to the study of the steady-state solutions and their stability for inhomogeneously broadened, unidirectional single-mode ring lasers. For both Lorentzian and Gaussian inhomogeneous-broadening profiles we find that, for appropriate detuning of the laser cavity, as many as three nontrivial steady-state solutions may appear and provide a formal confirmation of a phenomenon termed mode splitting by Casperson and Yariv in 1970 [ Appl. Phys. Lett. 17, 259 ( 1970)]. We show through stability arguments that bistability between trivial (zero-intensity) and nontrivial solutions is possible. This bistability appears experimentally accessible. We analyze the stability of the stationary solutions, especially in connection with its dependence on the detuning and pump parameters. The instability boundary in the plane of these two control parameters can present a fairly complicated structure with alternate ranges of stability and instability. In correspondence with certain points of the instability boundary, two complex-conjugate pairs of eigenvalues of the characteristic equation become simultaneously unstable. This situation is likely to produce spontaneous oscillations with two coexisting fundamental frequencies.
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