a Stability Analysis of a Nonlinear Optical Ring Cavity.

Since the observations of relaxation oscillations in the first ruby lasers there has been a concerted effort to better understand the nature of instabilities in coherent optical systems. With the recent renaissance in nonlinear dynamics, this effort has intensified. Toward these ends, a semiclassical model of a unidirectional ring cavity containing homogeneously broadened, two-level atoms is considered. Although a plane-wave approximation is assumed, this model is considerably more general than those typically found in the literature, as the electric -field envelope is allowed to vary along the length of the atomic medium. Additionally, the atomic and cavity detunings, and the longitudinal and transverse atomic relaxation rates are arbitrary. By varying the system parameters, this model can describe a unidirectional ring laser or the coherently driven nonlinear ring cavities that exhibit optical bistability. After performing a linear stability analysis of an arbitrary stationary state of the dynamical system, a new transcendental characteristic equation is derived. The roots of this equation are the stability eigenvalues of the chosen stationary state. They can be used to predict intrinsic instabilities in the dynamical model. This general characteristic equation is then used to clarify the results of earlier stability analyses of less general systems. Finally, a numerical investigation of the roots of the characteristic equation reveals an instability in a region of parameter space not accessible to earlier analyses. This instability is characterized by a frequency of oscillation less than the generalized atomic Rabi frequency and the free spectral range of the ring cavity, and involves a strong coupling between several cavity modes.