Imaging gigahertz zero-group-velocity Lamb waves

Abstract

Zero-group-velocity (ZGV) waves have the peculiarity of being stationary, and thus locally confining energy. Although they are particularly useful in evaluation applications, they have not yet been tracked in two dimensions. Here we image gigahertz zero-group-velocity Lamb waves in the time domain by means of an ultrafast optical technique, revealing their stationary nature and their acoustic energy localization. The acoustic field is imaged to micron resolution on a nanoscale bilayer consisting of a silicon-nitride plate coated with a titanium film. Temporal and spatiotemporal Fourier transforms combined with a technique involving the intensity modulation of the optical pump and probe beams gives access to arbitrary acoustic frequencies, allowing ZGV modes to be isolated. The dispersion curves of the bilayer system are extracted together with the quality factor Q and lifetime of the first ZGV mode. Applications include the testing of bonded nanostructures.

Fig. 1

Sample, surface particle velocity variation, and acoustic frequency spectrum. a Schematic diagram of the sample. DM: dichroic mirror, OL: objective lens. The double-ended arrows represent the probe-beam 2D scanning realized by moving mirrors and a confocal lens pair (see Methods). b Out-of-plane surface particle velocity measured for co-focused pump and probe beams. c Zoom-in of the ZGV Lamb mode amplitude temporal evolution and fit $\propto 1∕\sqrt{t}$. d Frequency spectrum obtained after temporal Fourier transform. Arrows indicate the two modes imaged in Fig. 2. e Evolution of the normalized amplitude in the frequency window 1.688 ≤ f ≤ 1.692 GHz. The solid line is a fit to the experimental data in the form of the square root of a Lorentzian. The dashed line represents $1∕\sqrt{2}$ (i.e., −3 dB) of the maximum amplitude of the fit

Fig. 2

Imaging a ZGV Lamb mode. a, b Normalized images of the measured out-of-plane surface particle velocity at a 0.2391 and b 1.6900 GHz. The x and y axis directions are shown in Fig. 1a. Animations are viewable in the Supplementary Movie 1. c 2D acoustic field at 1.6900 GHz based on a fit to a Bessel function J₀(kr). d Normalized out-of-plane surface particle velocity evolution of the experimental (dots) and fitted (solid line) data as a function of radial distance for a frequency of 1.6900 GHz (dashed lines in b and c)

Fig. 3

Dispersion curves of the bilayer. a, b Experimental (image) and theoretical (blue dashed and green dotted-dashed lines) dispersion curves for a 0 ≤ f ≤ 10 GHz and 0 ≤ k ≤ 4.5 μm⁻¹ b at frequencies near the first ZGV mode for 1.5 ≤ f ≤ 1.9 GHz and −2.5 ≤ k ≤ 2.5 μm⁻¹. The solid navy blue section of the dispersion curves corresponds to the portions where the phase and group velocities are anti-parallel. a The box corresponds to the enlarged region in b for k > 0. a, b Arrows indicate ZGV points. Plots are independently normalized