Bifurcations and Exceptional Point Boundaries in Time-Delay Coupled Opto-Electronic Oscillators

We present experimental, numerical, and analytical investigations of two time-delayed coupled optoelectronic oscillators (OEOs). Results demonstrate that bifurcation and exceptional point analyses can be combined to yield novel system predictions and behavior of interacting, nonlinear, time-delayed systems. Our model rate equations are developed by generalizing an established single OEO model to incorporate arbitrary coupling terms between the two OEOs. We focus on the case where the OEOs are independently on resonance and the cross-couplings are equal in magnitude and 180 degrees out of phase - experimentally achieved with a 180 degree hybrid coupler. To extract analytical predictions, we apply a linearized slowly varying envelope approach to derive a set of time-delayed linear rate equations. The OEO's inherent time delay results in a transcendental characteristic equation. In distinct operating regimes, we find pairs of analytical eigenvalue solutions. Analytical solutions allow us to identify the boundaries that separate operation regimes. At the first bifurcation boundary, where the eigenvalues cross the imaginary axis, the OEO starts to self-oscillate in both experiment and simulation. An exceptional boundary is indicated by a coalescence of the eigenvalues and eigenfunctions along with a transition from purely real to complex eigenvalues. Where the bifurcation and exceptional boundaries intersect, experiment and simulation show that the coupled OEO has an enhanced output power sensitivity and marks a clear frequency splitting. This splitting experimentally manifests as two distinct single-frequency states, which correspond to the imaginary components of the linearized eigenvalues. This frequency splitting gives novel control over the operating frequency of the coupled OEO system via the coupling strengths. Finally, we discuss the relevance of the time scale separation, which allows phase control of the complex couplings.