Fig. 1
From: Method for GHz optical helicity modulation by acoustic drum modes on a chip
Laterally confined Lamb modes. (a) Finite-element (FEM) simulations of the acoustic displacement field (\(\textbf{u}\)) excited by a ring-shaped piezoelectric bulk acoustic wave resonator (BAWR) on a sapphire substrate (see “Methods” for detail). The BAWR consists of a piezoelectric ZnO layer of thickness \(d_\textrm{film}=700\) nm shaped as a disk of radius of \(r_\textrm{film}= 70~\upmu\)m and sandwiched between two thin metal contacts (cf. upper part of panel c). The bottom contact is an Au ring with a radius \(r_\textrm{outer}=27~\upmu\)m while the top contact is a 30 nm-thick Al ring with inner and outer radii of \(r_\textrm{inner}=21.5~\upmu\)m and \(r_\textrm{outer}=27~\upmu\)m, respectively. The double-side polished c-plane sapphire substrate (thickness \(d_\textrm{sub}=425~\upmu\)m) forms Fabry–Pérot acoustic cavities with different lengths in the ZnO coated and uncoated regions, leading to different acoustic resonance frequencies in these two regions. The frequency mismatch laterally confines the acoustic modes in the regions underneath the ZnO-coated region, which is demarked by vertical dashed lines in panels (a), (c) and (d). (b) Calculated frequency (\(f_\textrm{RF}\)) spectrum of the average displacement amplitude \(|\textbf{u}|\) excited by the radio-frequency (rf) voltage (\(V_\textrm{rf}\)) applied to the BAWR within a free spectral range of the acoustic cavity. (c,d) Vertical displacement field (\(u_z\)) for the two resonances indicated by the thick arrows in panel (b), corresponding to modes confined underneath the ZnO layers. (e,f) Radial dependence of the average acoustic amplitude \(u_z\) integrated within the dashed rectangles (blue curves) and at the sample surface (green curves) in panels (c) and (d), respectively. The dashed red lines are Bessel-function fits to the profiles with different radial wavevectors \(k_{r,m_r}\), (\(m_r=1,2\)) as discussed in “Radially confined lamb modes”. The material parameters used in the finite element simulation are listed in Table SMI of the Supplement (SM).
