Efficient mapping and tracking the properties of micromechanical resonators using phase-lock loops with closely-spaced frequencies

TL;DR

This work presents a fast, robust method to map and track the properties of micromechanical resonators by employing three closely spaced drive tones locked to phases $\phi_1=-90^\circ$, $\phi_2=-45^\circ$, and $\phi_3=-135^\circ$ with phase-locked loops. By analyzing the PLL frequencies and phases, the resonance frequency $f_0$, linewidth $w$, and Duffing nonlinearity $\alpha$ are extracted without full frequency sweeps, enabling rapid imaging and real-time monitoring. The approach is demonstrated on a high-stress SiN membrane, showing accurate tracking during large temperature ramps and high-resolution spatial maps, both with and without active feedback. In the nonlinear regime, pump–probe measurements quantify the Duffing nonlinearity and determine a critical pump power $P_c$ at which nonlinear effects become significant, validating the method as a versatile tool for MEMS/NEMS characterization and real-time sensing.

Abstract

Studying the dynamical behavior of micro- and nano-mechanical systems (MEMS and NEMS) is essential in various fields from nonlinear dynamics to quantum technologies. Hence, it is important to be able to precisely monitor the mechanical properties of MEMS and NEMS devices. In this work, we show how to track and spatially map various properties of a mechanical resonator, such as frequency shift, linewidth, and nonlinearity, by aptly choosing three closely-spaced drive frequencies and using phase-locked loops. This technique tracks changes in the system faster and more efficiently, without the need for repeated frequency sweeps of the oscillator response, simply by employing three phase-locked tones.

Figures (14)

• Figure 1: (a) Illustration of the concept showing the frequency response of a driven harmonic oscillator with phase (top, in deg.) and magnitude (bottom) for every panel. (i) PLLs can be used to lock to specific phases (filled symbols), resulting in different driving frequencies and amplitudes (open symbols). Now, (ii) a shift in resonance frequency, (iii) damping rate, and (iv) nonlinearity all result in distinguishable shifts in $f_i$. The original response with $w/f_0 = 0.01$ and $\alpha = 0$ is shown in gray. (b) Schematic of the setup with a lock-in amplifier and digital implementation of the PLLs; for details, see Ref. hoch_MM_mode_mapping
• Figure 2: (a) Driven response of a hexagonal SiN membrane with resonances near the simulated eigenfrequencies (dashed lines). The corresponding mode shapes are shown as insets. (b) Magnitude and (c) phase of the fundamental mode. The setpoints are indicated as symbols.
• Figure 3: Tracking the (a) Frequency shift and (b) Linewidth during a large temperature ramp. Black data points indicate calibration NWA measurements. The inset in (a) is a zoom in the reverse point of the temperature showing a small offset.
• Figure 4: Maps in the presence of feedback ($g_\mathrm{FB} = 250 \un{V/V}$, $\theta_\mathrm{FB} = -45 \un{^o}$). (a) reflected laser light, (b) signal amplitude, i.e. the product of the mode shape and transduction (see Ref. hoch_MM_mode_mapping for details on the meaning of $\mathrm{Re}~S$). (c) the frequency shift and (d) the estimated linewidth. The black line is a contour of the reflectivity and outlines the edge of the membrane. White in (c) and (d) are excluded regions where the signal is too low for the PLL to update the frequencies, or the error for the estimator is too large.
• Figure 5: Operation in the nonlinear regime. (a) Driven responses for varying excitation power. The dashed lines show the fit of the phase. (b) Response of a weak (-30 dBm) probe for different pump powers. The pump at 1692555 Hz is grayed out. (c) Calculated probe response phase for different probe frequencies and strength of the nonlinearity. Contour lines at the indicated values are shown in black; the blue line shows the frequency of pump when locked to $-90\degree$. (d) Evolution of the PLL frequencies for increasing pump strength. The pump is in blue and the two probes are gold and orange. For clarity, the markers at -40 dBm have been enlarged. The solid lines are the scaled contours from (c).