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Wavelength Diversity Expands the Expressive Power of Optical Computing

Shiqi Chen, Che-Yung Shen, Parnian Ghapandar Kashani, Yongkang Cheng, Aydogan Ozcan

arXiv:2610.23070v1physics.opticscs.NEphysics.app-ph

TL;DR

Optical function approximation depends on the computational features encoded in light, yet adding trainable parameters alone does not expand a monochromatic processor’s detector-level frequency support. This paper uses a shared phase encoder under multiple wavelengths to generate scaled Fourier features, and demonstrates improved approximation, including simultaneous experimental approximation of 4,136 nonlinear functions.

  • Problem

    Monochromatic optical processors can remain limited when target functions require detector-level frequency components inadequately represented by their input encoding, and adding trainable phase features does not expand that support.

    Source: Introduction · p. 2

  • Method

    The paper combines multiple wavelengths at a shared phase encoder to generate scaled computational Fourier features and derives approximation-error bounds that account for truncation and PSF synthesis errors.

    Source: Introduction · pp. 2–3

  • Results

    Five-wavelength illumination experimentally approximated 4,136 nonlinear functions; after in-situ learning, MSE was ~4fold lower than for the two-wavelength processor with the same spectral endpoints.

    Source: Introduction · p. 3, Experimental Results · p. 16

  • Takeaways & Limitations

    For nondispersive encoders, illumination wavelengths provide an additional physical resource: spectral endpoints set accessible computational bandwidth, while intermediate wavelengths enrich harmonics within it.

    Source: Discussion · p. 17

    The gains require sufficient trainable optical degrees of freedom to synthesize wavelength-dependent PSF coefficients accurately, and practical accuracy can also depend on dispersion, phase range, throughput, detector response, and noise.

    Source: Wavelength-dependent spatially varying… · p. 7, Discussion · p. 17

Abstract

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Universal function approximation underpins neural networks' capacity to represent nonlinear relationships, but realizing this capability in optical hardware requires engineering the computational features encoded in light. Here, we show that wavelength diversity enhances optical function-approximation expressivity through wavelength-scaled Fourier features generated by a shared phase encoder, without increasing the number of input-encoded harmonics. Multiple wavelengths contribute to the representation of each output function rather than merely serving as independent computational channels. We establish theoretical scaling laws and approximation-error bounds, revealing that, for nondispersive encoders, the spectral endpoints determine the accessible computational bandwidth, while intermediate illumination wavelengths enrich the available harmonics within that bandwidth. Numerical studies demonstrate improved approximation of real- and complex-valued nonlinear functions, with a representative design achieving more than a 100-fold reduction in mean-squared error relative to tested monochromatic optical processors with the same number of trainable phase features. Experimentally, we validate enhanced approximation accuracy and demonstrate the simultaneous approximation of 4,136 distinct nonlinear functions using a compact, single-layer diffractive processor under five-wavelength illumination. These results establish wavelength diversity as an additional physical resource for scaling the expressive power of optical computing and enhancing the accuracy of parallel nonlinear function approximation.

Introduction

Optical function approximation depends on the computational features encoded in light, not only on the processor’s available hardware degrees of freedom. The paper uses multiple wavelengths with a shared phase encoder to expand those features and improve nonlinear function approximation.

Introduction

  • A monochromatic processor’s input encoding sets its computational Fourier frequencies, and square-law detection forms harmonics from pairwise frequency differences.Adding trainable phase features can improve synthesis of the associated PSF coefficients but does not expand detector-level frequency support.Source: Introduction · p. 2
  • Wavelength-dependent phase delays let a shared encoder generate scaled computational Fourier features without increasing the number of input-encoded harmonics.For a nondispersive encoder, spectral endpoints set the accessible bandwidth, while intermediate wavelengths enrich the harmonics within it.Source: Introduction · p. 2
  • The paper develops a framework for wavelength-scaled features, encoder dispersion, wavelength-dependent detector response, and approximation-error bounds.The bounds account for truncation and PSF synthesis errors.Source: Introduction · pp. 2–3
  • Numerical studies show improved approximation of real- and complex-valued nonlinear functions, including comparisons with monochromatic processors having more trainable phase features.The experiments also demonstrate parallel approximation of 4,136 nonlinear functions with a compact, single-layer processor under simultaneous five-wavelength illumination.Source: Introduction · p. 3
  • In situ learning reduces experimental approximation error and narrows the error distribution across output channels, bringing measured performance closer to numerical predictions.The reported experimental illumination spans 500–700 nm.Source: Introduction · p. 3

Results

The Results explain how wavelength diversity changes the features available to each output function, rather than merely assigning separate functions to different wavelength channels. A shared encoder generates wavelength-scaled features, with intermediate wavelengths enriching harmonic coverage within the bandwidth set by the spectral endpoints.

Optical forward model of spectral-domain diffractive processors

  • The same physical encoder generates wavelength-scaled computational Fourier features, which an optimized processor weights through wavelength-dependent coherent PSFs before detector intensities are combined.The gains depend on computational bandwidth, harmonic diversity, and accurate synthesis of the wavelength-dependent PSFs.Source: Results · p. 3, Optical forward model of… · pp. 4–5

Results

  • Unlike wavelength multiplexing, which assigns different functions to different channels, wavelength diversity expands the features and improves approximation accuracy of each output function.The computational harmonics are defined over abstract input variables, distinct from spatial frequencies of propagating optical fields.Source: Results · p. 3

Wavelength-scaled computational frequencies of a spectral-domain diffractive processor

  • Adding wavelengths expands the available computational Fourier-feature set without increasing input-encoded harmonics; intermediate wavelengths can enrich feature density even when spectral endpoints stay fixed.The longest wavelength is treated as the reference Fourier feature in the model, and the physical wavelengths are ordered from shortest to longest.Source: Optical forward model of… · pp. 4–5, Wavelength-scaled computational… · p. 6

Wavelength-dependent spatially varying coherent PSFs and output spectral field synthesis

The processor’s wavelength-dependent forward response assigns complex PSF coefficients to wavelength-scaled computational frequencies. Realizing the resulting expressive gains requires sufficient trainable optical degrees of freedom to synthesize those responses accurately.

Wavelength-dependent spatially varying coherent PSFs and output spectral field synthesis

  • The output field model uses wavelength-dependent, spatially varying coherent PSF coefficients to connect encoder inputs to output channels.The forward operator depends on wavelength through optical propagation and diffractive phase modulation.Source: Wavelength-dependent spatially varying… · p. 6
  • Wavelength scaling creates larger computational frequencies, while wavelength-dependent PSFs allow different complex weights to be assigned to those features.The processor’s wavelength-labeled forward map is block diagonal under the stated passive, time-invariant assumption.Source: Wavelength-dependent spatially varying… · pp. 6–7
  • Accurate synthesis of wavelength-dependent PSF coefficients requires sufficient trainable optical degrees of freedom; the proposed parameter-count relation is a design heuristic, not a guarantee.Realized PSF error also depends on architecture, aperture, wavelength span, optimization, dispersion, fabrication tolerances, throughput, and numerical aperture.Source: Wavelength-dependent spatially varying… · p. 7

Nonlinear function approximation via output intensity of a spectral-domain diffractive processor

Wavelength-scaled computational features combine through square-law intensity detection to represent nonlinear functions. The approach improves approximation as illumination wavelengths increase and extends to complex-valued targets.

Nonlinear function approximation via output intensity of a spectral-domain diffractive processor

  • Square-law detection generates harmonics from pairwise differences between wavelength-scaled input frequencies, and the detector-level frequency set determines the available nonlinear representation.For the spectral-domain processor, detected wavelength-specific intensity contributions are summed at each output detector.Source: Nonlinear function approximation via… · p. 8
  • Approximation improves as the number of wavelengths increases across multiple trainable-degree-of-freedom settings.The reported analyses include N=NwNpNf, 2NwNpNf, and 3NwNpNf.Source: Nonlinear function approximation via… · p. 9
  • Direct intensity detection accurately approximates non-negative real functions, while signed or complex targets require a differential or multi-intensity readout convention.The complex-target implementation uses three output detectors.Source: Nonlinear function approximation via… · pp. 9–10
  • Three spatially separated detectors with complex basis coefficients 120° apart represent complex targets using non-negative component functions, without background subtraction.The three real-valued detector signals are combined to reconstruct the complex-valued estimate.Source: Nonlinear function approximation via… · p. 10

Computational bandwidth expansion and harmonic enrichment through wavelength diversity

Wavelength diversity expands detector-level computational bandwidth and enriches the harmonics available within it. For nondispersive encoders, spectral endpoints set the outer frequency radius, while intermediate wavelengths add distinct, weighted harmonics.

Computational bandwidth expansion and harmonic enrichment through wavelength diversity

  • In the general dispersive case, the wavelength ratio alone does not determine the bandwidth scaling; nonmonotonic encoder dispersion can shift the maximizing wavelength away from the shortest.A dispersion-free encoder, such as a piston-based micro-mirror array in air, permits a simplified scaling expression.Source: Computational bandwidth expansion and… · p. 11
  • For nondispersive encoders, spectral endpoints determine the outer frequency radius, while intermediate wavelengths increase the number and density of distinct detector harmonics and add wavelength-specific coefficient contributions.More generally, the endpoints determine the outer radius when the wavelength scaling factor varies monotonically; in the general dispersive case, the maximizing active wavelength sets it.Source: Computational bandwidth expansion and… · p. 11
  • Wavelength diversity expands computational-frequency support and enriches the harmonic representation within that support; both mechanisms contribute to improved approximation accuracy in the numerical studies.The bandwidth increase depends on synthesis of the required wavelength-dependent spatially varying responses with sufficiently small error.Source: Computational bandwidth expansion and… · p. 11

Approximation-error bound of spectral-domain diffractive processors

The approximation-error framework separates finite-frequency truncation from imperfect synthesis of wavelength-dependent coherent responses. Additional implementation-dependent noise terms can be included in a more complete physical error budget.

Approximation-error bound of spectral-domain diffractive processors

  • The truncation error is defined for a non-negative target function that does not lie exactly in the finite function class.The bound also accounts for wavelength-specific complex-field approximation errors and their weighted sum across illuminated wavelengths.Source: Approximation-error bound of… · p. 12
  • A more complete physical error budget can include additional implementation-dependent noise terms, such as illumination phase error.Source: Approximation-error bound of… · p. 12
  • The error bounds distinguish limitations from the available computational Fourier representation from those due to imperfect synthesis of wavelength-dependent coherent responses.The framework connects wavelength diversity and trainable optical degrees of freedom to achievable approximation accuracy.Source: Approximation-error bound of… · p. 12

Expanding the expressive power of single-layer spectral-domain diffractive optical processors

The single-layer encoder–decoder architecture extends wavelength-diverse function approximation by combining wavelength-specific optical responses and expanding computational frequency support.

Expanding the expressive power of single-layer spectral-domain diffractive optical processors

  • The co-localized E+D architecture places a dynamic input-dependent encoder and an optimized static decoder on the same phase-only plane, where their fields interfere before square-law detection.The coherent mixing term produces a programmable real-valued nonlinear function.Source: Expanding the expressive power of… · pp. 12–13
  • For each wavelength, the detector response includes encoder self-intensity and encoder–decoder mixing terms, with the mixing term providing programmable contributions at encoded computational frequencies.The self-intensity term adds difference-frequency components to the full detector response.Source: Expanding the expressive power of… · pp. 13–14
  • The E+D processor’s frequency endpoint-scaling factor is the same as for the multilayer processor, and the numerical comparison reports improved accuracy as wavelength count rises within fixed endpoints.The improvement also holds against monochromatic E+D references with the same number of phase degrees of freedom.Source: Expanding the expressive power of… · p. 15
  • At αG max = 20, the nine-wavelength E+D processor achieved an MSE of ~3×10-5, more than two orders of magnitude below enlarged monochromatic references with the same trainable phase features.Increasing wavelengths from 2 to 9 within the same spectral endpoints reduced MSE by ~40-fold in this case, without increasing the outer computational-frequency radius.Source: Expanding the expressive power of… · p. 15

Experimental Results

Experiments show that multi-wavelength illumination improves approximation accuracy in a compact, single-layer processor and supports parallel approximation across thousands of functions.

Experimental Results

  • The five-wavelength processor’s MSE fell from ~7.9 × 10−3 zero-shot to ~4.9 × 10−3 after in-situ learning, about fourfold below the two-wavelength processor with the same endpoints.The experiment used five wavelengths spanning 500–700 nm in 50-nm increments.Source: Experimental Results · p. 16
  • Measured outputs after in-situ learning agreed well with the distinct oscillatory profiles of all 25 target functions for the five-wavelength processor.The experiments demonstrate compatibility with massively parallel approximation using a compact, single-layer processor.Source: Experimental Results · p. 16
  • In-situ learning reduced median experimental MSE and narrowed the interquartile range across 4,136 functions, bringing measured errors closer to numerical predictions.After learning, mean MSE was lower in central detector regions and relatively higher near the detector-array boundary.Source: Experimental Results · p. 16

Discussion

The discussion frames illumination spectrum as a physical resource for optical expressivity while noting that accuracy depends on response synthesis and remains experimentally scoped.

Discussion

  • A shared phase encoder uses multiple wavelengths to generate more computational Fourier features without increasing the number of input-encoded harmonics.For nondispersive encoders, spectral endpoints determine maximum accessible bandwidth, while intermediate wavelengths enrich harmonic coverage and wavelength-dependent coefficient contributions.Source: Discussion · p. 17
  • Wavelength diversity complements rather than replaces trainable hardware degrees of freedom, and its benefits depend on accurately synthesizing wavelength-dependent coherent responses.Encoder dispersion, finite phase-modulation range, optical throughput, detector response, and experimental noise can influence achievable accuracy.Source: Discussion · p. 17
  • Experiments focus on real-valued single-input functions, while the theoretical framework extends to higher-dimensional inputs and numerical studies include complex-valued functions.The authors identify multivariable mappings, experimentally learned functions, and integrated photonic implementations as possible future investigations.Source: Discussion · p. 17

Supplementary Information includes:

The supplementary information lists materials on the optical model, approximation and training, optimization, experimental setup, and figures.

Supplementary Information includes:

Figures

The figures compare monochromatic and multi-wavelength diffractive processors on real- and complex-valued function approximation, including experimental parallel implementations. At higher target bandwidths, the supplied comparisons report that monochromatic designs do not match multi-wavelength accuracy, even with substantially more trainable features.

Figures

  • At higher target bandwidths, monochromatic E+D processors do not match multi-wavelength approximation accuracy, even with substantially more trainable features.The comparisons cover multiple feature-count settings and plot MSE against target maximum frequency.Source: Figures · p. 24
  • Examples show two target nonlinear functions and their corresponding diffractive-processor outputs for αG max = 16, 20, 24, and 28, comparing single- and multi-wavelength illumination.Single-wavelength cases use 600 nm and 400 nm; multi-wavelength cases use Nw = 2, 3, 5, or 9 wavelengths spanning 400–600 nm.Source: Figures · pp. 25, 26, 28
  • At higher target bandwidths, monochromatic diffractive systems likewise cannot match multi-wavelength accuracy despite substantially increased degrees of freedom.The figure compares spectral illumination configurations across target-frequency values and feature-count settings.Source: Figures · p. 27
  • A separate experimental figure presents all 25 target functions and their corresponding processor outputs implemented in parallel.The processor uses five wavelengths and shows measurements after in-situ training.Source: Figures · p. 32
  • The five-wavelength processor is shown approximating 4,136 target functions in parallel, with representative outputs measured after in-situ training.The figure also presents distributions of MSE across the functions and spatial averages across detector subregions.Source: Figure 10 · p. 33, Figures · p. 34

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