Time Reversal Differentiation of FDTD for Photonic Inverse Design

Differentiable models enable the efficient computation of parameter gradients for continuous functions, greatly expediting the optimization of high-dimensional systems. This makes them an asset for the design of nanostructured metasurfaces. The adjoint variable method (AVM) is the workhorse for photonic gradient computation but can be challenging to implement with the finite difference time domain (FDTD) electromagnetic simulation method for certain optimization problems. Automatic differentiation (AD) platforms remove the need for manual constructions while retaining favorable computational scaling, but high memory consumption limits their application to small systems. Here, we introduce a method of gradient calculation based on the direct differentiation of the FDTD update equations by leveraging the time-reversible nature of Maxwell's equations. We support open and closed systems by recording the time-dependent fields at lossy boundaries and playing them back during the time-reversed FDTD simulation. The method is generally applicable without the high memory consumption of AD by eliminating redundant memory operations performed at each time step. We demonstrate this architecture in a 3D FDTD simulation. Its computational cost is comparable to the adjoint method, and it reduces memory requirements by 98% compared to an equivalent AD calculation for calculating a 900-element gradient vector. The differentiable simulator is applied to design two systems: a color sorter with frequency-domain behavior and a resonant nanostructure array with time-domain behavior. This approach to differentiate grid-based simulators is applicable to a broad range of physics simulators, thereby broadening the scope of inverse design topology optimization across fields.