Lagrangian description of Brillouin scattering and electrostriction in nanoscale optical waveguides

The diffraction of light by sound waves is known as Brillouin scattering. In optical waveguides, Brillouin scattering can arise from both bulk contributions, modelled by photoelasticity, and surface contributions, which are due to the waveguide boundaries being shaken by propagating sound. The reciprocal effect, electrostriction, governs the coherent generation of sound by light. The bulk photoelastic contribution to Brillouin scattering is generally nonlinear but can be limited to a firstorder expansion for small strain. We investigate the moving-interface contribution to Brillouin scattering in optical waveguides and show that it is also inherently nonlinear, leading to multi-phonon processes for large deformations. Limiting the perturbation to first order, we form a Lagrangian describing the interaction of sound and light. The Lagrangian contains both surface and bulk contributions to Brillouin scattering and electrostriction, and allows the derivation of optical and acoustic equations in a single variational formula. A full electrostriction equation is then derived for the phonon distribution, with both bulk and surface effects included. Numerical simulations in the case of a silicon nanowire illustrate the different effects and their respective contributions.