4 - Hopf bifurcation dynamics

Summary

A Hopf bifurcation marks the transition from a steady state to a time-periodic solution. We already encountered an example of a Hopf bifurcation in Section 3.4 as we analyzed the laser subject to an injected signal.

The emergence of spontaneous time-dependent regimes in lasers is not a purely academic problem because physicists have been confronted by the appearance of “noise-like” intensity fluctuations in the laser's beam since the beginning of the laser. This type of behavior was evident even during the earlier investigations of the laser in the 1960s where it was found that the intensity of the light generated by the ruby laser displayed irregular spiking, as shown in Figure 4.1. Were these spikes the result of a noisy environment or were they coming from the laser itself? A lot of experiments have been undertaken on the ruby laser under various conditions (see). It eventually appeared that the oscillatory output of the ruby laser resulted from the combined effect of several mechanisms. Research on this topic vanished because of the advent of new lasers whose parameters are much better controlled and therefore capable of delivering cw power or pulses with well-defined and reproducible properties.

For many years, attempts to understand the appearance of such oscillatory instabilities in lasers were limited (for instance, the extensive but isolated effort of Lee W. Casperson to describe the pulsations of the Xe laser), until Hermann Haken showed the equivalence of the laser equations with the Lorenz system.