Analysis of Phase-Locking in Optoelectronic Microwave Oscillators Due to Small RF Signal Injection

In this paper, we analyze the phase-locking phenomenon in a single-loop optoelectronic microwave oscillator, when subjected to the influence of small radio-frequency (RF) signal. We derive the differential equations for the amplitude and phase variations in the oscillator. Using quasi-linear approximation, analytical expressions for the lock range and phase-shift after phase-locking are presented. In addition, beat frequency of the unlocked-driven optoelectronic oscillator (OEO) is obtained and the phase-locking dynamics of the driven oscillator is discussed. Also, the spectrum components of the pulled OEO is derived as a function of the frequency detuning, lock range, beat-frequency, and frequency-shift induced by the phase perturbation of the injection signal. It is shown that all the analytical closed-form expressions clearly demonstrate the phase-locking mechanism starting from the fast-beat state through the quasi-locked state to the locked state of the pulled OEO. Finally, the simulation results are given to validate the analytical results.