Generation of Acoustic Phonons in Nanoscale Photonic Waveguides and Cavities
Résumé
Interaction of light and sound in tiny optical cavities and waveguides is a vibrant topic nowadays, as for instance manifested in cavity optomechanics, but also in Brillouin scattering measurements in micro-wires and tapered optical fibers. Indeed, both optical and elastic wave fields are tightly confined to a very small volume and surface effects can become significant with the size reduction. Recent progress has shown that opto-acoustic interactions can benefit from the combination of the photoelastic and of the moving-interface effects. While the photoelastic effect is classical in bulk acousto-optics, the moving-interface effect is specifically driven by the vibrations of surfaces of a resonator or a waveguide. Assuming a photonic mode and a phononic mode are known beforehand, the diffraction efficiency for the creation of new photons can be estimated by overlap integrals. Reciprocally, the confined optical fields exert mechanical forces on the material composing the cavity. In order to describe this effect, we propose a variational principle describing the acousto-optical interaction. We construct a 3-wave Lagrangian describing the interaction of the original optical wave, the Doppler-shifted optical wave, and the acoustic phonons. The three waves are further phased-matched in waveguides. The Lagrangian exhibits both volume and surface contributions to the interaction energy. First, electrostriction results in a bulk force governed by the photoelastic tensor. Second, coupling of the electromagnetic field with the mechanical motion of the cavity further results in an effective surface force. We then solve the resulting elastodynamic equation subject to bulk and surface optical forces. A finite element model is specifically derived from the variational formulation. It is applied to acousto-optical (or optomechanical) interactions in a nanoscale silicon photonic cavity and in a silica micro-wire. The simulation results show that acoustic resonances can be excited all-optically in the multi-gigahertz range from infrared light. Significantly, the excited acoustic phonons are not necessarily unique normal modes of the structure, but their distribution is found to be governed by the spatial distribution of the optical forces. The finite element model is further applied to tiny optical waveguides and compared to experiments performed with silica micro-fibers, revealing the generation of surface acoustic waves propagation at the waveguide boundaries.