Automatic differentiation accelerated shape optimization approaches to photonic inverse design on rectilinear simulation grids

Shape optimization approaches to inverse design offer low-dimensional, physically-guided parameterizations of structures by representing them as combinations of shape primitives. However, on discretized rectilinear simulation grids, computing the gradient of a user objective via the adjoint variables method requires a sum reduction of the forward/adjoint field solutions and the Jacobian of the simulation material distribution with respect to the structural shape parameters. These shape parameters often perturb large or global parts of the simulation grid resulting in many non-zero Jacobian entries, which are typically computed by finite-difference in practice. Consequently, the gradient calculation can be non-trivial. In this work we propose to accelerate the gradient calculation by invoking automatic differentiation (AutoDiff) in instantiations of structural material distributions. In doing so, we develop extensible differentiable mappings from shape parameters to shape primitives and differentiable effective logic operations (denoted AutoDiffGeo). These AutoDiffGeo definitions may introduce some additional discretization error into the field solutions because they relax notions of sub-pixel smoothing along shape boundaries. However, we show that some mappings (e.g. simple cuboids) can achieve zero error with respect to volumetric averaging strategies. We demonstrate AutoDiff enhanced shape optimization using three integrated photonic examples: a multi-etch blazed grating coupler, a non-adiabatic waveguide transition taper, and a polarization-splitting grating coupler. We find accelerations of the gradient calculation by AutoDiff relative to finite-difference often exceed 50x, resulting in total wall time accelerations of 4x or more on the same hardware with little or no compromise to final device performance. Our code is available open source at https://github.com/smhooten/emopt