Beyond the Thin-Layer Limit: Differentiable Volumetric Training for Visible-Range Diffractive Neural Networks

Diffractive deep neural networks (D2NNs) promise miniaturized, power-efficient, light-speed optical front-ends for machine vision, yet the most mature demonstrations remain in the terahertz regime, built from readily fabricated millimeter-scale neurons. Translating D2NNs to the visible range, where nearly all vision pipelines operate, was long blamed on the difficulty of fabricating nanoscale neurons; but even after recent advances removed that barrier, visible-range D2NNs matching their terahertz counterparts remain out of reach. We identify the true obstacle as the thin-layer approximation underlying nearly all D2NN training, which treats each diffractive layer as an infinitely thin mask. It fails not because of the short wavelength, as is commonly assumed, but because the low-refractive-index materials (n approximately 1.3-1.5) used at visible wavelengths require relief structures thick enough that intra-layer diffraction and phase accumulation become significant. To overcome this, we introduce a differentiable beam-propagation ($\partial$BPM) layer that models each element as a finite-thickness volume and propagates light through it during training, keeping the fabrication-compatible height map end-to-end trainable without full-wave simulation in the loop. Across MNIST, Fashion-MNIST, and CIFAR-100 classification and imaging, $\partial$BPM training substantially reduces the design-to-device mismatch, and full-wave FDTD validation raises classification accuracy from 50% to 90% without re-optimization. The $\partial$BPM layer thus offers a scalable, physics-aware bridge between efficient optical neural-network optimization and fabrication-consistent diffractive design.