Structurally stable transitions in optical tristability

If a four-level atomic system with the possibility of J=1/2 to J=1/2 transitions is driven by a lineary polarized input field, first-order transitions from linearly to circularly polarized output fields may occur. In the mean-field limit and the pure absorptive case, the underlying Maxwell-Bloch equations reduce to a pair of coupled algebraic equations for the circularly right and left polarized outputs as functions of a linearly polarized input. These equations are discussed within the framework of imperfect-bifurcation theory. The most degenerate situation,-the collapse of the circularly polarized transmission branch onto a hysteresis point of the linearly polarized branch-is shown to be determined by a codimension-four singularity. The structurally stable unfolded-bifurcation diagrams of this singularity are analysed. Besides describing correctly all transitions to optical tristability that have so far been observed in the absorptive limit, a variety of new generic transitions is predicted which are not accessible in the pure absorptive limit. These include hysteresis in the circularly polarized transmission branch and transitions to self-pulsing.