Imaging gigahertz zero-group-velocity Lamb waves

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.


SUPPLEMENTARY NOTE 1: THEORETICAL MODEL
The geometry is shown in Supplementary Fig. 1. We assume that the two layers are isotropic, homogeneous and infinite, with mass density ρ i , longitudinal and transverse velocities v Li and v T i , and thicknesses h i , where i = 1, 2 indicates the layer number. The coupling between the layers is taken to be perfect, i.e., by assuming continuity of the displacement and stress components at the interface (z = 0). The ω (angular frequency) -k (wavenumber) relation is solved using the scalar potential φ and the vector potential ψ, where the latter is reduced to a scalar as the problem is two-dimensional. The tangential and normal displacements are derived from these potentials as follows: and the stresses are given by where λ, µ are the Lamé coefficients 1 . The potentials in the layers can be expressed as where p and q are the z-components of the longitudinal and transverse wave vectors, respectively. The wavenumbers k Li = ω/v Li and k T i = ω/v T i satisfy dispersion relations for bulk waves k Li 2 = k 2 +p i 2 and k T i 2 = k 2 +q i 2 . A iL , B iL are the amplitudes of longitudinal components and A iT , B iT are the amplitudes of shear components.
At the free boundaries (z = −h 1 and h 2 ), the stresses normal to the surface (σ xz and σ zz ) vanish, whereas at the interface (z = 0), the continuity of displacement and stresses is applied. It follows that From Supplementary Eqs. (S1-S5), the problem can be rewritten in matrix form, M · U = [0]: Non-trivial solutions are found when the determinant of the 8 × 8 matrix M vanishes, i.e., det(M) = 0. In order to avoid (unwanted) bulk waves propagating at velocities v Li (p i = 0) and v T i (q i = 0), the terms p 1 , q 1 , p 2 and q 2 can be factorized in the 2 nd , 4 th 6 th and 8 th rows, respectively. The dispersion curves of the bilayer structure is then estimated by determining the zeros of the secular equation. As the structure is spatially asym- metric, modes cannot be classified exactly as symmetric and antisymmetric. For a given mode, the group velocity is extracted using v g = ∂ω/∂k. A solution (ω, k) is identified as a ZGV mode if v g = 0 with k = 0. Furthermore, normal and tangential displacements-u z and u x , respectively-can be estimated from the dispersion curves. For a solution (ω, k), the equations representing the boundary conditions can be be solved once a component common to U is fixed (e.g., A 1L = 1). This gives access to the relative displacements u x,z .

SUPPLEMENTARY NOTE 2: SAMPLE AND EXPERIMENTAL PARAMETERS
The sample consists of a silicon-nitride membrane provided by NTT Advanced Technology Corporation (MEM-N0302) with a nominal thickness of 2.0 ± 0.2 µm. It is mostly composed of Si 3 N 4 , but is not a pure crystal (the composition ratio Si:N is between 3:4 and 1:1). Nevertheless, it is hereafter denoted as Si 3 N 4 . The membrane is supported on its edges by a Si frame, providing a 3 × 3 mm 2 area with free surfaces, necessary to generate ZGV Lamb modes. The membrane is coated with a ∼650 nm sputtered polycrystalline titanium film. To calculate the dispersion curves, the elastic constants and density are taken from Ref. In order to accurately determine the thicknesses, an experiment measuring the surface particle velocity in the time domain is carried out using an interferometric pulse-echo method with focused pulsed-laser beams (∼1.5 µm 1/e 2 diameter) incident from the top side of the sample and with picosecond time resolution. The pump beam is modulated at f p = 1 MHz, and we monitor the in-phase output of the lock-in amplifier. The result is shown in Supplementary Fig. 2a. The first minimum in the variation at t 0 = 0 is related to the temperature rise and deformation caused by the laser pulse. The echo at t 1 corresponds to the acoustic pulse reflected from the Si 3 N 4 /Ti interface, whereas the second echo at t 2 corresponds to the acoustic pulse reflected from the rear surface of the membrane. The weak reflection from the interface (at t 1 ) indicates good adhesion (as our model assumes). The corresponding time intervals are ∆t 1 = 215 ± 1 ps and ∆t 2 = 560 ± 1 ps, allowing us to evaluate the thicknesses of 659 ± 3 and 1830 ± 10 nm for the Ti and the Si 3 N 4 layers, respectively, from the known v L values. (Errors correspond to those arising from the time resolution of the apparatus.). For Si 3 N 4 the thickness agrees within the 10% uncertainty given by the supplier.
The corresponding predicted dispersion curves are shown in Supplementary Fig. 2d. The mode classification follows the one suggested by Mindlin 3 , where the integers correspond to the number of antinodes of the mechanical displacement. This integer can be negative in case of negative group velocity. The 'q' denomination relates to the term quasi-in the appellations quasisymmetric and quasi-antisymmetric, related to the sample spatial asymmetry. For the zero wave-vector modes (i.e., for k = 0 µm −1 ), qA 2n , qS 2n+1 have an out-of-plane displacement whereas qA 2n+1 , qS 2n have an in-plane displacement. Therefore, the former are more likely to be observed in our experiments. Three ZGV Lamb modes are predicted below 10 GHz. They are then referred as qS 1 , qA 3 and qA 7 , and are labelled 1, 2, 3, respectively, for simplicity (see Supplementary Table I). Their frequencies and associated wavenumbers are displayed in Table I. With the arbitrary-frequency method (see Methods in the Main text), these frequencies are accessible by modulating the pump beam at the frequency f p = 36.8, f p = 27.3, and f p = 34.9 MHz, for the first, second and third ZGV Lamb modes, respectively. We also present the normal and tangential displacements of these three ZGV modes in Supplementary Figs 2e-g. At the top free surface, i.e., where the excitation and detection occur, the tangential displacement is significant for the three modes. Conversely, the normal displacement is different for these modes: it is predominant for the lowest ZGV mode at f th 1 = 1.7248 GHz ( Supplementary Fig. 2e), still significant for the second one at f th 2 = 3.0014 GHz (Supplementary Fig. 2f) and relatively weak for the third one at f th 3 = 6.9476 GHz ( Supplementary Fig. 2g).
Finally, the pump beam radius should be carefully chosen to enhance ZGV Lamb mode generation. For a single isotropic plate, Bruno et al. demonstrated that, for a Gaussian beam, the optimum response is reached when the 1/e 2 radius is 2 √ 2/k 4 . Extending this result for our bilayer system leads to an ideal pump radius of ∼4.6 µm for the first ZGV mode. In our set-up, detection sensitivity is inversely proportional to the probe beam radius. As both pump and probe beams are focused with the same objective lens (see Fig. 1(a) in the main text), it is difficult to achieve the ideal case. A good compromise is found with the pump and probe 1/e 2 radii, measured by knife-edge technique, set to be 4.2 and 2.8 µm, respectively. This facilitates the generation of propagating modes with wavenumber k = 0.67 µm −1 , but modes in the range 0.3 k 1.4 µm −1 should also be generated. In the case of the line pump spot with a 1/e 2 intensity half-width of 1.5 µm and a length of 5 µm (used for the dispersion relation measurement), modes with wavenumbers in the range 0.8 k 3.9 µm −1 are expected to be generated, as observed in experiment.

SUPPLEMENTARY NOTE 3: TEMPERATURE RISE EVOLUTION
The steady state temperature rise T of the sample at the centre of the optical pump spot is estimated by considering a finite-sized effectively 2D circular plate with its circumference held at constant temperature and approximating the laser intensity profile to a top hat distribution. Under such assumptions, the solution of the heat diffusion equation gives where P is the power absorbed by the sample (P = P 0 T 0 (1 − R 0 ) with P 0 = 6 mW the measured incident power before the objective lens, T 0 =0.83 the optical transmittance of the objective lens at 415 nm-the pump wavelength-and R 0 =0.444 the optical reflection coefficient of Ti at 415 nm), h the bilayer thickness (with h Ti = 660, h Si3N4 = 1830 nm, see Supplementary Note 2), κ = 27.8 W.m −1 .K −1 the thermal conductivity estimated by weighting the values for each layer by their thickness (κ Ti = 21.9, κ Si3N4 = 30 W.m −1 .K −1 ), a = 5.64 mm the plate radius (the circular plate being chosen to have the same area as the square sample plate surface 10 × 10 mm 2 ) and w = 4.2 µm the 1/e 2 intensity radius of the pump beam. Reflection coefficients and thermal conductivities are taken from Supplementary Ref. 5. It follows that T =49 K.