Fast multi-channel inverse design through augmented partial factorization

Computer-automated design and discovery have led to high-performance nanophotonic devices with diverse functionalities. However, massively multi-channel systems such as metasurfaces controlling many incident angles and photonic-circuit components coupling many waveguide modes still present a challenge. Conventional methods require $M_{\mathrm{i}\mathrm{n}}$ forward simulations and $M_{\mathrm{i}\mathrm{n}}$ adjoint simulations – $2M_{\mathrm{i}\mathrm{n}}$ simulations in total – to compute the objective function and its gradient for a design involving the response to $M_{\mathrm{i}\mathrm{n}}$ input channels. By generalizing the adjoint method and the recently proposed augmented partial factorization method, here we show how to obtain both the objective function and its gradient for a massively multi-channel system in a single simulation, achieving over-two-orders-of-magnitude speedup and reduced memory usage. We use this method to inverse design a metasurface beam splitter that separates the incident light to the target diffraction orders for all incident angles of interest, a key component of the dot projector for 3D sensing. This formalism enables efficient inverse design for a wide range of multi-channel optical systems.