High Stability Mechanical Frequency Sensing beyond the Linear Regime

Abstract

Sensing via a mechanical frequency shift is a powerful measurement tool, and, therefore, understanding and mitigating frequency noise affecting mechanical resonators is imperative. Thermomechanical noise fundamentally limits mechanical frequency stability, and its impact can be reduced with increased coherent amplitude of mechanical motion. However, large enough actuation places the resonator in the nonlinear (Duffing) regime, where conversion of amplitude noise (AM) into frequency noise (FM) can worsen sensor performance. Here, we present an experimentally straightforward method to evade this amplitude tradeoff in micromechanical sensors. Combining knowledge of the Duffing coefficients with readily available amplitude measurements, we avoid AM-FM conversion. Our approach uses dual-mechanical-mode operation on a tensioned thin-film resonator to set a baseline thermomechanically-limited stability by eliminating correlated single-mode frequency drifts. Thus, we cleanly observe AM-FM conversion at high drive, and reduce it using our method. The resulting high-stability operation beyond the linear regime contrasts long-standing perspectives in the field.

Figures (6)

• Figure 1: Phase space plot illustrating the Duffing problem and our solution. (a) At a low drive, the resonator amplitude (green lines) and phase inference (purple lines) are dominated by thermomechanical noise. (b) Increasing the amplitude, while remaining below $A_{\text{crit}}$, decreases the relative contribution of $\delta \phi_{\text{th}}$ to the resonator phase inference (angle created by purple lines is smaller in (b) than in (a)). Other noises in addition to thermomechanical noise can be present in the amplitude, leading to a total amplitude noise, $\delta A$. (c) Driving beyond $A_{\text{crit}}$ leads to AM-FM conversion (red) and thus a detected phase noise due to Duffing effects, $\delta \phi_\beta$. Our method removes $\delta f_\beta$, resulting in the blue ellipse.
• Figure 2: Experimental setup and methods. (a) Optical drive and interferometric detection of two modes of a tensioned Si$_3$N$_4$ trampoline (microscope image). Crucially, we record both modes' frequencies and amplitudes, in time. A high-pass optical filter prevents 980 nm light from reaching the detector. (b) Undriven amplitude noise PSD of modes $a$ and $b$ with respective mode shapes above (flat resonator in black, and out-of-plane displacement of each mode in red). Black dashed lines show the imprecision noise floor. (c) Undriven phase space plots for each mode (mode $a$ below, mode $b$ above). The distributions agree with thermal expectations (green lines). (d) Example of two of the four Duffing-coefficient measurements. Detunings of each mode vs. $A_b^2$ during ring up, with arbitrary y-intercepts. Lines (solid light blue for mode $a$ frequency and dashed dark blue for mode $b$ frequency) represent the average of many linear fits of data from each ringdown cycle (circles for mode $a$ and x for mode $b$). e) Conceptual representation of the dual-mode enabled Duffing correction method. Noise contributions of mode $a$ (solid light purple) and of mode $b$ (dotted dark purple) to the relevant frequency signals are shown symbolically as plots versus either time or amplitude as our method is employed.
• Figure 3: Common-mode subtraction data and theory for different drive amplitudes. Solid lines are measured data and dotted lines are the room-temperature thermomechanical limit (Eq \ref{['Eq: Thermomech limit']}) of both modes added in quadrature for each drive. (a) $\sigma_{y_a}$ and $\sigma_{y_b}$ (orange and red) and corresponding $\sigma_{\delta y}$ (solid dark green) for $A_{0,a}=$ 1.4 nm and $A_{0,b} =$ 0.71 nm. Green lines are identical to highest amplitude solid and dotted lines in (b) Measured $\sigma_{\delta y}$ for increasing drive amplitude, where $\frac{A_a}{A_{\text{crit},a}^{\text{SD}}} = \frac{A_b}{A_{\text{crit},b}^{\text{SD}}} =$ 0.14, 0.35, 0.71, 1.06, 1.41, 1.77, 2.12, 2.83. Here, $A_{\text{crit},a}^{SD} \approx 10$ nm, $A_{\text{crit},b}^{SD} \approx 6$ nm.
• Figure 4: Duffing correction results. $\sigma_{\delta y}$ (solid lines) and $\sigma_{\delta y}^{(D)}$ (dashed lines) for varying drive amplitudes with the corresponding thermomechanical limits (dotted lines) as in Fig. \ref{['Fig: Subtracted ADevs with Theory']}. Phase space diagram like in Fig. \ref{['Fig: Duffing subtraction concept']} is shown for an example actuation above the critical amplitude for before (red noise profile) and after (blue noise profile) Duffing correction. Inset: Three example traces taken with white force noise applied to the resonator. The elevated thermomechanical limits (dotted lines) are obtained using a free parameter fit of temperature. A corresponding phase space diagram for actuation in the nonlinear regime is also shown.
• Figure 5: Common-mode subtracted Allan deviations (solid lines) with added white force noise for $\frac{A_a}{A_{\text{crit},a}^{\text{SD}}} = \frac{A_b}{A_{\text{crit},b}^{\text{SD}}} =$0.14, 0.35, 0.71, 1.06, 1.41, 1.77, 2.12, 2.83, as in the main text. Dotted lines are the thermomechanical limit for $T = 27,000$ K (Sec. \ref{['Section: white noise temperature']}) at each drive.