Multiscale phase dynamics and $2\pi$ phase kinks in injection-locked optoelectronic oscillators with large delay

Abstract

Injection locking of optoelectronic oscillators (OEOs) with large delay gives rise to phase dynamics that lie beyond the scope of classical single mode locking theory, including the spontaneous formation of persistent $2\pi$ phase kinks. In this work, a multiscale theoretical framework is developed that explains the origin, structure, and stability of these phase slip phenomena in injection locked (IL) OEOs operating in large-delay regime. Starting from a complex envelope delay differential equation that explicitly incorporates hard-limiting gain saturation and RF BPF dynamics, a reduced phase-only description valid for nearly constant oscillation amplitude is derived. Exploiting the separation between fast round-trip dynamics and slow inter-round-trip evolution, a two-timescale reduction yields a continuum of Adler equations governing phase difference between injected signal and oscillator as a function of round-trip time. Analytical solutions obtained for weak injection and small detuning show how initially smooth phase profiles sharpen into localized $2\pi$ phase kinks, with p kinks appearing per round trip when the injection is tuned near the pth adjacent mode. Time-domain simulations of full complex-envelope model validate the predicted phase kink formation mechanism and reveal essential role of RF resonator dynamics in determining their persistence. While the reduced phase-only model correctly predicts kink sharpening, resonator-induced amplitude excursions associated with steep phase gradients can erase the kinks when the complex-envelope trajectory fails to encircle the origin, restoring conventional phase locking. These results provide a unified physical interpretation of $2\pi$phase kinks in IL-OEOs and delineate the limits of phase-only models in the presence of large, fast phase transients, identifying a regime of frequency locking without phase locking in large delay oscillators.