Frequency modulated enhancement of microwave resonator sensing

TL;DR

The study applies the Pound-Drever-Hall technique to a microwave-frequency SAW resonator sensor built on Y-cut LiNbO$_3$ to characterize center-frequency stability $f_0$ and linewidth across multiple resonant modes. Compared with a conventional Phase-Locked Loop, PDH is insensitive to parasitic phase errors and achieves a substantially lower Allan deviation for $f_0$, demonstrating enhanced frequency stability. A fully digital, FPGA-based lock-in generates a three-tone FM spectrum with carrier $f$ and sidebands at $f\pm f_m$, demodulates each tone to form the PDH observable $y_{PDH}$, and allows direct suppression of spurious resonances via sideband tuning. The results show robust long-term stability and effective suppression of unwanted acoustic modes, highlighting PDH as a practical, wide-applicable method for high-precision microwave-resonator sensing and a pathway toward integration in hybrid quantum systems.

Abstract

We use the Pound-Drever-Hall (PDH) technique to characterize the frequency stability of a microwave-frequency surface acoustic wave (SAW) resonator-based sensor. The multi-mode acoustic resonator is integrated in a notch geometry with a transmission line, all fabricated on Y-cut lithium niobate. We measure the amplitude and phase of the resonator transfer function and the PDH signal across the resonator full spectral range. We use these measurements to emphasize the differences between the PDH measurement and a standard Phase-Locked Loop (PLL) technique. As compared to a PLL, we demonstrate that PDH is insensitive to phase error and exhibits a reduced Allan deviation of the center frequency measurement, in each case by up to an order of magnitude. The method rejects spurious effects and background frequency drift, demonstrating the enhancements possible with PDH-based measurements, which can be realized in a wide range of microwave-frequency resonator-based sensors and devices.

Figures (5)

• Figure 1: (a) Time-domain representation of a frequency-modulated (FM) microwave input (red), detuned from the resonator center frequency, $f_0$, interacting with a resonator and resulting in amplitude modulation, (blue). Demodulation of the signal can be performed through a square-law detector, at $f_m$, or via the modulation analysis (MOD) option of a Zurich Instrument lock-in.(b) Frequency-domain representation of the input FM signal (red) interacting with a resonator (black), resulting in an amplitude-modulated output (blue). (c) PDH observable, $y_{PDH}$, versus frequency offset $f-f_0$, which can be used as an error signal in a feedback loop for resonance tracking. (d) Functional diagram of the PDH feedback loop using a Zurich Instruments Microwave Lock-in: MOD extracts the error signal, PID minimizes it by adjusting the carrier frequency, and the updated FM drive is fed back to MOD.
• Figure 2: (a) SAW device enclosed in a copper housing with two SMA connectors. The device package includes a custom PCB featuring a central transmission line and surrounding ground planes. The transmission line on the PCB is SMA containerized on both sides, forming the input and output ports. A central rectangular slot in the PCB securely holds the SAW resonator chip. (b) Zoomed-in microscope image of the device, showing the SAW resonator galvanically connected between the center trace and one of the ground planes of a CPW. (c) Zoomed-in false colored microscopic image of the SAW resonator showing the central IDT (yellow) and two acoustic Bragg mirrors (brown), each $110~\mu$m from the IDT. All structures are aluminum on Y-cut LiNbO$_3$.
• Figure 3: (a) Transmission spectrum ($S_{12}$) of the SAW resonator using a lock-in amplifier, showing resonant dips (dashed vertical lines). Inset: Corresponding phase response. (b) PDH observable of the SAW resonator, recorded at $f_m$ = 200 kHz, with zero crossings at the SAW resonances. The PDH measurement also reveals fine 719 kHz oscillations not apparent in the $S_{21}$ amplitude or phase response. (c) PDH signals with ($f_m$) set to integer multiples of 719 kHz, demonstrating suppression of the spurious oscillatory features.
• Figure 4: PLL versus PDH response to microphonic noise induced externally by mechanically tapping the connected coaxial cables. (a) PLL estimate $f_{0,PLL}$ exhibits large frequency jumps ($\approx 100$ kHz) and slow drifts, (b) PDH estimate $f_{0,PDH}$ shows smaller spikes ($\lesssim50$ kHz), rapid recovery, and negligible drift.
• Figure 5: PLL versus PDH frequency measurement stability. (a) Time traces of the tracked resonance frequency by PLL (blue) and PDH (orange). (b) Allan deviation (AD) of the same data verses averaging time $\tau$, illustrating that PDH is over an order of magnitude quieter than PLL for $\tau>0.02$ s.