Laser diode nonlinear dynamics from a filtered phase-conjugate optical feedback.

The rate equations for a laser diode subject to a filtered phase-conjugate optical feedback are studied both analytically and numerically. We determine the Hopf bifurcation conditions, which we explore by using asymptotic methods. Numerical simulations of the laser rate equations indicate that different pulsating intensity regimes observed for a wide filter progressively disappear as the filter width increases. We explain this phenomenon by studying the coalescence of Hopf bifurcation points as the filter width increases. Specifically, we observe a restabilization of the steady-state solution for moderate width of the filter. Above a critical width, an isolated bubble of time-periodic intensity solutions bounded by two successive Hopf bifurcation points appears in the bifurcation diagram. In the limit of a narrow filter, we then demonstrate that only two Hopf bifurcations from a stable steady state are possible. These two Hopf bifurcations are the Hopf bifurcations of a laser subject to an injected signal and for zero detuning.