%e provide a comprehensive analysis of the standard model of the bidirectional ring laser in which only one mode can be supported in each direction and in the limit that the polarization can be eliminated adiabatically. The interaction between the two counterpropagating modes can be derived and it is most naturally viewed as s coupling between them vis scattering from a spatial grating formed in the population inversion. If the grating is s sufficiently small modulation of the spatial average of the population inversion, it can be approximated by a sinusoidal function. A systematic derivation of the model with only a sinusoidal grating for the homogeneously broadened case is presented that reveals and corrects errors in several previously published analyses. The stability of the steady-state solution is analyzed. The bidirectional steady-state solution is unstable and the unidirectional steady-state solutions may be stable or unstable depending on the parameters. The well-established result of bistsbility between the two modes when the cavity is tuned to resonance is recovered, s result that persists, in part, even when the losses between the two modes are di8'erent. Kith detuning, the unidirectional steady-state solutions can be destabilized, cresting regions in the parameter space where only time-dependent solutions are found. For parameters characteristic of solid-state or molecular gas lasers, the instability can occur for very small detunings and even very close to the lssing threshold. Asymptotic 1imits of the instability boundaries for these parameters are presented. Additional results are derived formally for inhomogeneous broadening with a Lorentzisn line shape and for arbitrary inhomogeneous linewidths, Explicit analytic results are presented in the limits of very small and very large inhomogeneous linewidths compared with the homogeneous linewidth. Time-dependent solutions in the homogeneously broadened case for relaxation rates appropriate to CO& lasers show that there are broad regions in the parameter space of gain snd detuning for which the behavior is dynamically chaotic in the form of nearly "square-wave" alternation between the counterpropagsting modes of the laser with irregular switching times. In other regions of that parameter space we 6nd more irregular pulsations and pulse slternations. Regions of periodic pulsations also have been found.Among the earliest theoretical studies of the hornogeneously broadened bidirectional ring laser are those of
