The spatial complexity of optical computing: toward space-efficient design.

Similar to algorithms, which consume time and memory to run, hardware requires resources to function. For devices processing physical waves, implementing operations needs sufficient "space," as dictated by wave physics. How much space is needed to perform a certain function is a fundamental question in optics, with recent research addressing it for given mathematical operations, but not for more general computing tasks, e.g., classification. Inspired by computational complexity theory, we study the "spatial complexity" of optical computing systems in terms of scaling laws-specifically, how their physical dimensions must scale as the dimension of the mathematical operation increases-and propose a new paradigm for designing optical computing systems: space-efficient neuromorphic optics, based on structural sparsity constraints and neural pruning methods motivated by wave physics (notably, the concept of "overlapping nonlocality"). On two mainstream platforms, free-space optics and on-chip integrated photonics, our methods demonstrate substantial size reductions (to 1%-10% the size of conventional designs) with minimal compromise on performance. Our theoretical and computational results reveal a trend of diminishing returns on accuracy as structure dimensions increase, providing a new perspective for interpreting and approaching the ultimate limits of optical computing-a balanced trade-off between device size and accuracy.