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Computer-automated design and discovery have led to high-performancenanophotonic devices with diverse functionalities. However, massivelymulti-channel systems such as metasurfaces controlling many incident angles andphotonic-circuit components coupling many waveguide modes still present achallenge. Conventional methods require $M_{\rm in}$ forward simulations and$M_{\rm in}$ adjoint simulations -- $2M_{\rm in}$ simulations in total -- tocompute the objective function and its gradient for a design involving theresponse to $M_{\rm in}$ input channels. By generalizing the adjoint method andthe recently proposed augmented partial factorization method, here we show howto obtain both the objective function and its gradient for a massivelymulti-channel system in a single simulation, achievingover-two-orders-of-magnitude speedup and reduced memory usage. We use thismethod to inverse design a metasurface beam splitter that separates theincident light to the target diffraction orders for all incident angles ofinterest, a key component of the dot projector for 3D sensing. This formalismenables efficient inverse design for a wide range of multi-channel opticalsystems.

Fast multi-channel inverse design through augmented partial factorization