Wide-field stroboscopic imaging of topologically protected phononic modes

TL;DR

Problem: Imaging spatial mode profiles in phononic circuits over large areas while preserving phase information. Approach: Introduces a frequency-detuned wide-field stroboscopic interferometry technique that uses a collimated probe and CMOS camera to reconstruct the displacement waveform across a complete oscillation cycle in a few seconds via detuning $f_I = f_s - f_m$. Contributions: Applied to Valley-Hall topological phononic waveguides, it images edge-mode profiles over more than 20 mm^2 with ~6 μm resolution and shows that backscattering distorts mode profiles, correlating with measured frequency splittings (e.g., Δ ≈ 2.2 kHz for M=15). Significance: The results align with finite-element simulations for several modes and establish a fast, scalable protocol for characterizing mesoscopic resonators in phononic circuits.

Abstract

Imaging spatial mode profiles is important for understanding the behavior of mechanical resonators. The recent development of phononic circuits has increased the demand for a fast imaging method based on principles of coherent detection. However, it becomes complicated to perform measurements on a large surface area. Here, we present a frequency-detuned collimated-beam interferometry measurement scheme with in-plane spatial resolution of about 6 um, which can provide information about the phase dynamics of the entire mechanical oscillation cycle on a time scale of a few seconds. We employ a stroboscopic pulse probing method to resolve high-frequency vibrational motion with a standard CMOS camera. We use this setup to image megahertz frequency resonant mode profiles present in a Valley-Hall topological triangular cavity, over an area of more than 20 mm2. We relate the obtained data to numerical simulations of the topological edge modes to reveal the relation between backscattering and the mode profile distribution. The presented protocol can become a staple for characterizing mesoscopic mechanical resonators.

Figures (3)

• Figure 1: Experimental protocol. (a) Experimental setup for stroboscopic imaging measurements. Depicted here, AOM: acousto-optic modulator, BS: optical beam splitter with noted R:T values, CMOS: complementary metal–oxide–semiconductor camera, Lock-in: lock-in amplifier, PD: silicon photodetector, Piezo: piezoelectric actuator, SG: signal generator. The dashed characterization box is the setup for measuring the mechanical spectra of a targeted region. (b) Detuned stroboscopic detection scheme: the detecting optical pulses (black line) has a repetition frequency ($f_\textrm{s}$) slightly different from the mechanical excitation (blue line) frequency ($f_\textrm{m}$).
• Figure 2: Topological phononic waveguide. (a) Microscope image of fabricated bulk valley-Hall topological crystals. Dashed lines mark out unit cell of the honeycomb lattice with the basis vectors $\mathbf{a}_1$ and $\mathbf{a}_2$. The right panel shows a single unit cell with a rounded triangular hole, whose orientation is defined by the angle $\theta$ and a size is defined by $r \approx 93µm.$ (b) Microscope image of the topological edge by interfacing two bulk insulators with $\theta = +\pi/6$ and $\theta = -\pi/6$. (c) Simulated band diagram of the edge structure in (b). The black, red, and grey dots respectively represent topological bulk, topological edge, and trivial in-plane mechanical modes. (d) Simulated mode profile of the topological edge state. $u_z$ is the out-of-plane mechanical field component. (e) Microscope image (false color) of fabricated topological triangular cavity. (f) Meaured normalized power spectral density (PSD) from one of the fabricated devices. Inset shows the zoomed-in spectrum for the mode pair with $M = 16$.
• Figure 3: Resonant mode profiles. (a) Experimental mode profiles of $u_z$ from stroboscopic method for the two topological mode pairs 1, 2 and 3, 4 from one device. Mode 1 and 2 have mode number of 15 while mode 3 and 4 have mode number of 16. (b) Simulated modal profile for the mode 1, 2, 3, 4. (c) Zoomed-in comparison for the measured and simulated results around the top triangle corner in (a) and (b).