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On the Universality of Microwave Envelope Equations for Narrowband Optoelectronic Oscillators

Narrowband optoelectronic oscillators are time-delayed system that can generate ultrapure microwave signals. In order to study these oscillators, one approach is to derive a time-delayed envelope equation. In this article, we show that this approach is universal and applies to a wide variety of OEO families. By studying different oscillators, it is possible to show that their dynamics is governed by an envelope equation that can be rewritten under a generalized universal form. In this article, this universal microwave envelope equation is proposed and investigated analytically for optoelectronic microwave oscillators, and the bifurcation analysis of the deterministic envelope equation is performed. The universal model is further generalized and written in a stochastic form in order to predict the phase noise performance of the oscillators.