Nonlinear and Stochastic Dynamics of Optoelectronic Oscillators

Abstract

Optoelectronic oscillators (OEOs) are nonlinear, time-delayed and self-sustained microwave photonic systems capable of generating ultrapure radiofrequency (RF) signals with extensive frequency tunabilities. Their hybrid architectures, comprising both optical and electronic paths, underscore their merits. One of the most notable points of OEOs can be unprecedentedly high quality factors, achieved by storing optical energies for RF signal generations. Thanks to their low phase noise and broad frequency tunabilities, OEOs have found diverse applications including chaos cryptography, reservoir computing, radar communications, parametric oscillator, clock recovery, and frequency comb generation. This thesis pursues two primary objectives. Firstly, we delve into the nonlinear dynamics of various OEO configurations, elucidating their universal behaviors by deriving corresponding envelope equations. Secondly, we present a stochastic equation delineating the dynamics of phases and explore the intricacies of the phase dynamics.

The outputs of OEOs are defined in their RF ports, with our primary focus directed towards understanding the dynamics of these RF signals. Regardless of their structural complexities, we employ a consistent framework to explore these dynamics, relying on the same underlying principles that determine the oscillation frequencies of OEOs. To comprehend behaviors of OEOs, we analyze the dynamics of a variety of OEOs. For simpler systems, we can utilize the dynamic equations of bandpass filters, whereas more complex physics are required for expressing microwave photonic filtering. Utilizing an envelope approach, which characterizes the dynamics of OEOs in terms of complex envelopes of their RF signals, has proven to be an effective method for studying them. Consequently, we derive envelope equations of these systems and research nonlinear behaviors through analyses such as investigating bifurcations, stability evaluations, and numerical simulations. Comparing the envelope equations of different models reveals similarities in their dynamic equations, suggesting that their dynamics can be governed by a generalized universal form. Thus, we introduce the universal equation, which we refer to as the universal microwave envelope equation and conduct analytical investigations to further understand its implications.

While the deterministic universal equation offers a comprehensive tool for simultaneous exploration of various OEO dynamics, it falls short in describing the stochastic phase dynamics. Our secondary focus lies in investigating phase dynamics through the implementation of a stochastic approach, enabling us to optimize and comprehend phase noise performance effectively. We transform the deterministic universal envelope equation into a stochastic delay differential form, effectively describing the phase dynamics. In our analysis of the oscillators, we categorize noise sources into two types: additive noise contribution, due to random environmental and internal fluctuations, and multiplicative noise contribution, arising from noisy loop gains. The existence of the additive noise is independent of oscillation existence, while the multiplicative noise is intertwined with the noisy loop gains, nonlinearly mixing with signals above the threshold. Therefore, we investigate both sub- and above-threshold regimes separately, where the multiplicative noise can be characterized as white noise and colored noise in respective regimes. For the above-threshold regime, we present the stochastic phase equation and derive an equation for describing phase noise spectra. We conduct thorough investigations into this equation and validate our approaches through experimental verification.

In the sub-threshold regime, we introduce frameworks to experimentally quantify the noise contributions discussed in the above-threshold part. Since no signal is present here and the oscillator is solely driven by the stochastic noise, it becomes feasible to reverse-engineer the noise powers using a Fourier transform formalism. Here, we introduce a stochastic expression written in terms of the real-valued RF signals, not the envelopes, and the transformation facilitates the expressions of additive and multiplicative noise contributions as functions of noisy RF output powers. The additive noise can be defined by deactivating the laser source or operating the intensity modulator at the minimum transmission point, given its independence from the loop gains. Conversely, the expression for the multiplicative noise indicates a dependence on the gain, however, experimental observations suggest that its magnitude may remain relatively constant beyond the threshold.