Nonlinear dynamics of continuously tunable optoelectronic oscillators based on stimulated Brillouin amplification

We present a theoretical analysis for tunable optoelectronic oscillators (OEOs) based on stimulated Brillouin scattering (SBS). A pump laser is used to generate a Brillouin gain which selectively amplifies a phase-modulated and contra-propagating laser signal. The radiofrequency beatnote generated after photodetection is filtered, amplified and fed back to the phase modulator to close the optoelectronic loop. Tunability is readily achieved by the adjustable detuning of the pump and signal lasers. OEOs based on stimulated Brillouin scattering have been successfully demonstrated at the experimental level, and they feature competitive phase noise performances along with continuous tunability for the output radiofrequency signal, up to the millimeter-wave band. However, the nonlinear dynamics of SBS-based OEOs remains largely unexplored at this date. In this article, we propose a model that describes the temporal dynamics of the microwave envelope, thereby allowing us to track the dynamics of the amplitude and phase of the radiofrequency signal. The corresponding nonlinear and time-delayed differential equation is then analyzed to reveal the underlying bifurcation behavior that emerges as the feedback gain is increased. It is shown that after the primary Hopf bifurcation that triggers the microwave oscillations, the system undergoes a secondary Neimark-Sacker bifurcation before fully developed chaos emerges for the highest gain values. We also propose a model for the chipscale version of this SBS-based OEO where the delay line is replaced by a highly nonlinear waveguide. The numerical simulations are found to be in excellent agreement with the analytical study.