Periodic attractors in two-photon processes

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Abstract
We consider three nonlinear optical models that display a two-photon process and exhibit a Hopf bifurcation corresponding to a phase instability. The three models describe second-harmonic generation, two-photon optical bistability, and degenerate four-wave mixing in resonant cavities. In all three cases we consider only the resonant configuration and analyze the attractors which emerge when the driving field is increased beyond the Hopf bifurcation threshold. At a finite distance from threshold, we find a hysteresis domain that involves a pair of periodic attractors. For all three models, each of the two solutions has the same ‘‘field portrait’’ i.e., the same representation in a plane whose axes are the imaginary and the real parts of the electric field. This suggests that the scenario is generic for this class of systems. On the other hand, the models for two-photon optical bistability and degenerate four-wave mixing in resonant cavities exhibit additional nongeneric attractors when the driving field is further increased.

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