Large-scale nonlinear optical computing with incoherent light via linear diffractive systems

We demonstrate that nonlinear computing can be achieved with a single linear diffractive surface under coherent illumination. We introduce a compact encoder-decoder co-localization (E+D) architecture in which an input-dependent dynamic encoder and a static optimized decoder are integrated within the same phase-only diffractive plane. Following free-space propagation, coherent interference between the encoder and decoder fields, combined with intensity detection, generates programmable nonlinear input-output mappings without requiring nonlinear optical materials or multiple diffractive layers. We prove that the proposed E+D optical processor is a universal approximator for arbitrary real-valued band-limited nonlinear functions and identify the physical factors governing its approximation fidelity, including the decoder degrees-of-freedom, detector aperture, and axial propagation distance. Crucially, we demonstrate that introducing a trained, frozen phase bias to the encoder region systematically enhances functional expressivity, providing robustness against coarse phase quantization on spatial light modulators. Using this framework, we accurately synthesize diverse nonlinear functions, including commonly used neural network activation functions and complex-valued nonlinear functions. Finally, we experimentally validate the proposed approach using a visible-light optical set-up trained through in situ learning, demonstrating the parallel approximation of 9 nonlinear functions in a single optical forward pass. By collapsing nonlinear optical computation into a single diffractive surface, the E+D architecture substantially reduces hardware and alignment complexity while preserving powerful function-approximation capabilities, providing a compact and scalable framework for analog information processing.

We present a unified theoretical framework connecting classical universal approximation theory, Fourier-feature approximation, and diffractive optical processors. We show that phase-encoded diffractive processors implement finite Fourier-feature expansions whose mathematical completeness follows from Fourier/Stone-Weierstrass arguments, while their physical realizability is governed by finite coefficient synthesis through optimized spatially varying coherent point-spread functions (PSFs). Our analyses derive approximation-error bounds that separate Fourier truncation, PSF-synthesis, input phase error, optical hardware, readout, and noise contributions; establish scaling relationships linking approximation complexity to optical degrees of freedom and input/output space-bandwidth products; derive photon-budget and throughput limits imposed by photon statistics; formulate finite-class statistical learnability bounds for phase-quantized diffractive function approximators; and analyze the impact of spatially incoherent illumination. We further analyze coherent optical cascadability and show that quadratic feature expansion through coherent mixing and optical readout provides a mechanism for enhanced representation while remaining fundamentally distinct from the depth-separation results established for digital neural networks. Our analyses provide a rigorous theoretical foundation for diffractive nonlinear function approximation and establish quantitative relationships among mathematical expressivity, optical hardware resources, statistical learning, and physical performance limits, thereby offering general design principles for large-scale analog optical computing systems.

Diffractive optical neural networks (DONNs) have demonstrated unparalleled energy efficiency and parallelism by processing information directly in the optical domain. However, their computational expressivity is constrained by static, passive diffractive phase masks that lack efficient nonlinear responses and reprogrammability. To address these limitations, we introduce the Recurrent Diffractive Optical Neural Processor (ReDON), a novel architecture featuring reconfigurable, recurrent self-modulated nonlinearity. This mechanism enables dynamic, input-dependent optical transmission through in-situ electro-optic self-modulation, providing a highly efficient and reprogrammable approach to optical computation. Inspired by the gated linear unit (GLU) used in large language models, ReDON senses a fraction of the propagating optical field and modulates its phase or intensity via a lightweight parametric function, enabling effective nonlinearity with minimal inference overhead. As a non-von Neumann architecture in which the primary weighting elements (metasurfaces) remain fixed, ReDON substantially extends the nonlinear representational capacity and task adaptability of conventional DONNs through recurrent optical hardware reuse and dynamically tunable nonlinearity. We systematically investigate various self-modulation configurations to characterize the trade-offs between hardware efficiency and computational expressivity. On image recognition and segmentation benchmarks, ReDON improves test accuracy and mean intersection-over-union (mIoU) by up to 20% compared with prior DONNs employing either optical or digital nonlinearities at comparable model complexity and negligible additional power consumption. This work establishes a new paradigm for reconfigurable nonlinear optical computing, uniting recurrence and self-modulation within non-von Neumann analog processors.