Stimulated Brillouin scattering in integrated photonic waveguides: Forces, scattering mechanisms, and coupled-mode analysis

Recent theoretical studies of stimulated Brillouin scattering (SBS) in nanoscale devices have led to intense research effort dedicated to the demonstration and application of this nonlinearity in on-chip systems. The key feature of SBS in integrated photonic waveguides is that small, high-contrast waveguides are predicted to experience powerful optical forces on the waveguide boundaries, which are predicted to further boost the SBS gain that is already expected to grow dramatically in such structures because of the higher mode confinement alone. In all recent treatments, the effect of radiation pressure is included separately from the scattering action that the acoustic field exerts on the optical field. In contrast to this, we show here that the effects of radiation pressure and motion of the waveguide boundaries are inextricably linked. Central to this insight is a new formulation of the SBS interaction that unifies the treatment of light and sound, incorporating all relevant interaction mechanisms—radiation pressure, waveguide boundary motion, electrostriction, and photoelasticity—from a rigorous thermodynamic perspective. Our approach also clarifies important points of ambiguity in the literature, such as the nature of edge effects with regard to electrostriction and of body forces with respect to radiation pressure. This new perspective on Brillouin processes leads to physical insight with implications for the design and fabrication of SBS-based nanoscale devices.

Figure 1

Schematic of the forward (Stokes field represented by the red arrow) and backward (Stokes field in blue) SBS interactions in a waveguide aligned along the $z$ axis. Although a specific waveguide cross section is shown here, the results apply to waveguides of arbitrary cross section and composition.

Figure 2

The two leftmost columns show the electric field distributions of an optical eigenmode in an rectangular waveguide (white outline) before and after a deformation. The other two columns show the deformation-related perturbations of the electric field and the electric induction. The waveguide deformation in this figure is grossly exaggerated for reasons of illustration. The color scale is linear and symmetric.

Figure 3

Sketch of the boundary displacement effect. A waveguide with dielectric permittivity ${\varepsilon }_a$ is embedded in a background material with permittivity ${\varepsilon }_b$. Please note that the deformation is grossly exaggerated in the left sketch. For surface displacements of realistic magnitude, the quasiparallel shift of the interface by the distance $\widehat{n}·\mathbf{U}$ is the dominant effect and the second-order effects related to a change in the normal vector $\widehat{n}$ are negligible. This situation is shown in the close-up on the right side.