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

Abstract: Nonlinear computation is essential for various information processing tasks. Optical implementations are attractive because passive light propagation can manipulate high-dimensional signals with extreme throughput and parallelism; yet realizing nonlinear mappings in optical hardware remains challenging due to the weak nonlinearity of optical materials and the large intensities required to induce nonlinear interactions. This challenge is further amplified in many systems that operate with incoherent illumination, motivating a coherence-aware framework for scalable optical nonlinear processing. Here, we show that linear optical systems, in particular, optimized diffractive processors comprising passive surfaces, can perform large-scale nonlinear function approximation under spatially incoherent or partially coherent illumination, when preceded by intensity-only input encoding. We quantify how the accuracy of the nonlinear function approximation varies with the degree of parallelism, the number of diffractive layers, and the number of trainable diffractive features. Numerical results demonstrate snapshot computation of up to one million distinct nonlinear functions in a single forward pass through a diffractive processor, with the function outputs spatially multiplexed and read out using densely packed detectors at the output. We further provide a proof-of-concept experimental demonstration under incoherent illumination from a liquid crystal display (LCD), enabled by a model-free in situ learning strategy that jointly optimizes the diffractive profile and detector readout geometry in the presence of hardware imperfections and misalignments. Our findings establish diffractive processors as a massively parallel universal function approximator for both spatially incoherent and partially coherent illumination.

• The paper demonstrates that optimized phase-only diffractive systems can approximate arbitrary nonlinear functions using intensity-only encoding under incoherent illumination.
• It achieves massive parallelism, computing up to 10^6 distinct functions in one pass with significant error reduction from multilayer architectures.
• The work introduces differential detection to emulate negative weights, enabling robust function approximation and establishing scaling laws for trainable features.

Large-Scale Nonlinear Optical Computing with Incoherent Light via Linear Diffractive Systems

Introduction

This paper introduces an optical framework that enables massively parallel nonlinear function approximation under spatially incoherent or partially coherent illumination using fully passive, linear diffractive systems. Unlike previous work, which predominantly focused on coherent sources or required nonlinear optical materials, the approach leverages linear diffractive modulators optimized in silico and intensity-only input encoding. Snapshot computation of up to $10^6$ distinct nonlinear functions is demonstrated numerically, supplemented by a hardware proof-of-concept using an LCD source and phase light modulator. These results position linear diffractive processors as universal function approximators for analog optical computing in regimes beyond fully coherent illumination.

Technical Contributions

The main innovation is the demonstration that phase-only diffractive optical processors, when appropriately optimized and paired with an intensity-encoded input representation, can realize arbitrary sets of nonlinear mappings under incoherent or partially coherent illumination. This is achieved through:

• Intensity-Only Input Encoding: The argument $a$ of nonlinear target functions is mapped into a vector of nonnegative intensity values on the input plane, corresponding to harmonics in a truncated Fourier expansion. Both cosine and sine channels are spatially multiplexed.
• Linear Diffractive Processing: The input pattern propagates through a series of $K$ trainable diffractive layers, each modulating the optical phase. Under incoherent light, the optical system implements a linear transformation of the input intensity pattern, effectively learning a set of spatially varying intensity point spread functions.
• Differential Detection for Negative Weights: Each output function is read out from a dedicated $2 \times 2$ detector tile. A differential combination of the four responses per tile achieves the effect of signed Fourier coefficients despite the nonnegativity of physical intensities.
• Massive Parallelism: The architecture simultaneously approximates $N_{\rm f}$ nonlinear functions using a single forward pass, with all outputs multiplexed on a detector array.

The approach generalizes via a statistical forward model to account for partial spatial coherence, simulating or experimentally validating robustness to source coherence variations.

Numerical and Experimental Results

The framework achieves highly accurate nonlinear function approximations at scale:

• With $N_{\mathrm{f}}=10^4$ and $10^5$, median per-function MSE under incoherent illumination is $O(10^{-4})$; at $N_{\mathrm{f}}=10^6$, the MSE rises to $O(10^{-3})$, but most functions remain accurately approximated.
• The representational capacity improves dramatically with multilayer ($a$0) architectures versus single-layer designs, evidenced by a two-order-of-magnitude MSE reduction for fixed total parameters.
• Increasing the number of harmonics ($a$1) in the input encoding enhances approximation accuracy for high-bandwidth target functions, including standard deep learning activations.
• The relationship between total trainable features $a$2 and performance shows diminishing returns for $a$3.
• Under partially coherent illumination, the system retains low MSE and uniform error distributions across the output field of view for $a$4.
• Experimental realization using a PLM and CMOS sensor under incoherent LCD illumination (with $a$5) achieves average MSE $a$6, using a model-free in situ learning strategy that jointly optimizes phase profiles and output detector placements.

Theoretical and Practical Implications

This work demonstrates that nonlinear optical processing does not fundamentally require optical nonlinearities or coherent illumination. Instead, nonlinearity can be transferred to encoded input statistics and appropriate output readout. The differential detection mechanism extends the function class to those requiring negative weights, overcoming the non-negativity restriction of intensity measurements. The analysis establishes scaling laws: the required trainable feature count grows linearly in the number of functions and input harmonics for satisfactory approximation. Optical architectural depth, achieved via multilayer diffractive systems, confers substantial gains in effective degrees of freedom and error robustness.

From a practical standpoint, this architecture's throughput is fundamentally limited only by the spatial bandwidth products of the input and detector planes and the fabrication constraints of large-area diffractive elements and sensor arrays. The one-shot massively parallel nature is particularly advantageous for analog preprocessing in computational imaging, phase retrieval, or machine vision domains where the signal of interest is inherently optical.

The demonstration of an in situ, model-free training methodology mitigates alignment and modeling errors in real hardware, enabling robust operation despite physical imperfections, and supports scalability toward higher-function-count deployments.

Future Directions

Several extensions are identified:

• Scaling Up: Realization of massively parallel ($a$7) systems, subject to advances in high-resolution diffractive optics, 3D alignment, and sensor integration.
• Generalization to Multi-dimensional Inputs: Extending encoding schemes and differential readouts to vector-valued function domains.
• Customized Feature Bases: Exploring non-Fourier bases customized for target function families (e.g., wavelets, Hermite functions).
• Co-Optimization for Task Objectives: End-to-end training of diffractive systems for domain-specific analog computation (e.g., feature extraction, dimensionality reduction) under realistic illumination.
• Expressive Modulation Schemes: Examining amplitude-and-phase or multi-state modulators for higher representational efficiency.

Conclusion

This paper establishes that linear, multilayer diffractive systems equipped with input intensity encoding and differential spatially multiplexed readout are universal, massively parallel nonlinear function approximators under incoherent and partially coherent illumination. The approach holds significant promise for energy-efficient, high-throughput analog optical processing, marking a substantive advance toward optical analog computation platforms that obviate the need for optical nonlinearities or coherent sources (2603.29131).