Bidirectional emission from a ring resonator driven by an external field and containing a saturable absorber

Abstract

We formulate a first-principles description of the behavior of a bidirectional ring cavity containing a saturable absorbing medium, and driven by an external coherent field. The setting of interest to this study is an extension of the ordinary optically bistable system, where a ring resonator supports only one of the two possible directions of propagation of the cavity field. In this more general case and in the uniform-field limit, we show that the dynamics of the fields-atoms system is described by an infinite set of coupled equations that can be readily solved by standard numerical means after appropriate truncation of the number of atomic variables. With the help of two different approaches, we find unexpected, long-time stationary solutions such that the forward and backward fields oscillate with different carrier frequencies: the first is based upon the time-dependent equations and the second on a set of nonlinear-algebraic equations describing the steady states. As an added confirmation, we carry out a linear-stability analysis of the unidirectional steady state for the purpose of identifying the conditions under which a backward field can grow from an initial fluctuation. By this approach we also suggest a possible strategy for the experimental observation of stationary, frequency-shifted output fields.