Transformation-optics simulation method for stimulated Brillouin scattering

We develop an approach to enable the full-wave simulation of stimulated Brillouin scattering and related phenomena in a frequency-domain, finite-element environment. The method uses transformation-optics techniques to implement a time-harmonic coordinate transform that reconciles the different frames of reference used by electromagnetic and mechanical finite-element solvers. We show how this strategy can be successfully applied to bulk and guided systems, comparing the results with the predictions of established theory.

Figure 1

Schematic of the relation between Lagrangian coordinates X, Eulerian coordinates $\tilde{\mathbf{x}}$, and displacement $\tilde{\mathbf{u}}$.

Figure 2

Conceptual schematic of TO method, as applied to the system described in Sec. 4b. Two optical ${\mathrm{TE}}^z$ guided modes (one of which is depicted in the first row) counterpropagate in a dielectric slab waveguide, giving rise to a mechanical potential $\varphi$ and the corresponding force field ${\mathbf{F}}_{\mathrm{v}}$ (second row, color map and arrows, respectively). The force excites one or more elastic modes (third row, warped grid), thus creating a mass density variation field $\mathrm{\Delta }\rho$ (third row, color map). This in turn induces a relative permittivity variation field $\mathrm{\Delta }{\epsilon }_{\rho }$ (fourth row, left column), but there is no effect $\mathrm{\Delta }{\mu }_{\rho }$ on permeability in the case of ordinary nonmagnetic optical materials (fourth row, right column). With our method, we calculate effective anisotropic properties (fifth and sixth rows, left column permittivity, right column permeability) that enable the simulation of SBS coupling while keeping material points fixed (Lagrangian frame). All figures are depicted in reference to a given time $t_0$.

Figure 3

1D backward SBS amplifier: schematic of 2D simulation (inspired by [1]). The lateral boundaries are connected through periodic boundary conditions, making the domain effectively infinite in the transverse direction. Open boundary conditions generate optical fields at one end ($z=0$ for the pump; $z=L_g$ for the signal) and transmit them without reflection at the other. Elastic waves are generated by optical forces, and absorbed at either $z$ boundary by perfectly matched layers. $L_g$ is the characteristic gain length. Elastic waves are computed in the plane strain approximation.

Figure 4

1D backward SBS amplifier at resonance: (top) relative pump intensity (middle) relative signal intensity and (bottom) pressure amplitude, as predicted by theory (blue, dashed line), the TO method (orange, upper continuous line), and a simply coupled simulation (green, lower continuous line).

Figure 5

1D backward SBS amplifier: exponential gain $g$ spectrum predictions: theory (continuous blue line) against simulations run with TO method (orange “x” series) and without (green “o” series).

Figure 6

Dielectric slab SBS amplifier: schematic of the waveguide, of finite thickness $d$. The direction of propagation is $z$, and the problem is translationally invariant in the out-of-plane direction, making it 2D.

Figure 7

Dielectric slab SBS amplifier: schematic of the dispersion diagrams for lossless ${\mathrm{TE}}^z$ electromagnetic and longitudinal elastic waves. In simulations, a fixed optical frequency $\omega$ is picked. Selecting a waveguide thickness $d$, one can read off the corresponding propagation wave number $k_z$ for the desired mode (the lowest order in our case). By phase matching, the corresponding elastic propagation constant $q_z\simeq 2k_z$ is determined, from which one finds the frequency $\mathrm{\Omega }$ of the desired elastic mode.

Figure 8

Dielectric slab SBS amplifier: gain $G$ vs waveguide thickness $d$. Theory (continuous blue line); simulations with TO method (orange “x” series) and without method (green “o” series).