Passive Phase Noise Cancellation Scheme

We introduce a new method for reducing phase noise in oscillators, thereby improving their frequency precision. The noise reduction is realized by a passive device consisting of a pair of coupled nonlinear resonating elements that are driven parametrically by the output of a conventional oscillator at a frequency close to the sum of the linear mode frequencies. Above the threshold for parametric instability, the coupled resonators exhibit self-oscillations which arise as a response to the parametric driving, rather than by application of active feedback. We find operating points of the device for which this periodic signal is immune to frequency noise in the driving oscillator, providing a way to clean its phase noise. We present results for the effect of thermal noise to advance a broader understanding of the overall noise sensitivity and the fundamental operating limits.

Figure 1

An illustration of the phase noise cancellation scheme. An oscillator produces a signal with a noisy frequency around ${\omega }_1+{\omega }_2$. This signal parametrically drives a pair of coupled NEMS or MEMS beams with a relative phase of 180°. The output signal at a frequency around ${\omega }_2-{\omega }_1$ is given by squaring and filtering.

Figure 2

The periodic solutions of Eqs. (2). (a) The squared mode amplitudes (the in-phase mode has the larger amplitude); (b) Twice the frequency of the periodic solutions. Solid and dashed lines are stable and unstable solutions, respectively. The parameters are $\Delta =7$, $g=10$, and $\eta =1$.

Figure 3

Phase diffusion of the output signal induced by frequency noise of unit strength for different values $\eta$ of the nonlinear damping. For each curve, ${\Omega }_p^{\max }$ is the value of ${\Omega }_p$ for which the diffusion is zero. Note that $D_{{\Omega }_p}$ is also smaller than the phase diffusion of the driving oscillator along most of the curve, and that the solid curve is the squared derivative of the solid curve in Fig. 2b.

Figure 4

Thermal noise limit of the phase diffusion of the output signal, for $F_{\mathrm{th}}=1$ and the same parameters as in Fig. 2. The asterisk indicates the value at ${\Omega }_p^{\max }$.