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Functional Superoscillations in a Wave Processor

Yuntian Wang, Md Sadman Sakib Rahman, Aydogan Ozcan

arXiv:2610.34111v1physics.opticscs.NEphysics.app-ph

TL;DR

Conventional optical superoscillation work generates predetermined fields, while this paper studies continuously queried function evaluation. It encodes an argument onto a wavefront and uses a fixed diffractive processor to approximate superoscillatory outputs, including visible-light reconstruction beyond the encoder’s harmonic content.

  • Problem

    Conventional optical superoscillation implementations generate predetermined spatial fields, whereas this work addresses continuously queried function mappings over an input coordinate.

    Source: Discussion · pp. 13–14

  • Method

    A fixed diffractive processor encodes the scalar argument onto a wavefront and uses coherent interference and spatially allocated output channels to approximate superoscillatory functions.

    Source: Discussion · pp. 12–14

  • Results

    With Nsup=9 target harmonics from Nbase=4 input harmonics (ρh=2.25), visible-light experiments reconstructed a superoscillatory function beyond the encoder’s native content.

    Source: Discussion · p. 12

  • Takeaways & Limitations

    Functional superoscillation provides a principle for analog optical processor design, with domain partitioning and spatial multiplexing extending approximation to functions whose global spectral content exceeds the input encoding bandwidth.

    Source: Discussion · pp. 13–14

    Precise destructive interference required for superoscillation is harder to establish than gross amplitude modulation.

    Source: Discussion · p. 13

Abstract

from arXiv · show

Physical wave processors are constrained by diffraction, but the bandwidth of the mathematical functions they implement need not be set by it. Here, we demonstrate continuous and multiplexed optical processing of superoscillatory functions, in which a scalar argument is encoded onto a wavefront and a fixed diffractive wave processor synthesizes optical responses that approximate the corresponding function outputs in parallel. Unlike conventional superoscillatory optics that generates predetermined fields, superoscillation here is expressed in the input-output dependence on the encoded argument. Its frequencies are therefore distinct from the transverse spatial frequencies of propagating light and are not subject to the same diffraction-imposed cutoff. We demonstrate complex-valued superoscillatory function synthesis and introduce a multiplexed architecture that simultaneously evaluates distinct target superoscillatory functions whose harmonic content exceeds that of the optical input encoding. Visible-light experiments further validate this beyond-encoding superoscillatory regime. Together, these results establish functional superoscillation as a design principle for wave-based computation and show how interference, diffractive depth and spatial channel allocation extend the functional reach of passive wave processors. More broadly, these analyses establish a separation between functional bandwidth and physical wave bandwidth in analog optical computation.

Results

The results show that a fixed diffractive processor can approximate complex-valued superoscillatory functions, including locally rapid variations beyond the encoded harmonic band. Increasing diffractive depth improves fidelity, especially for stronger superoscillations.

Results

  • The scalar argument t is encoded as pixel phases k_jt, and coherent propagation creates position-dependent harmonic superpositions measured through intensity detection.Differential detector pairs recover the real and imaginary target components; the diffractive response is optimized over continuously sampled inputs.Source: Results · pp. 3–5
  • The processor uses four phase-only layers to approximate 100 complex-valued Berry–Popescu functions with a_i∈[1,5], recovering real and imaginary components through differential detector pairs.The target and input encoding share the same harmonic count in this demonstration.Source: Results · p. 5
  • For a_i=1.5, 2.5, and 3.5, reconstructed local oscillation rates approach a_i near the origin, beyond the constituent bandlimit k_max=1.The processor reproduces both target profiles and locally superoscillatory behavior; stronger targets require rapidly growing expansion coefficients and strong cancellation.Source: Results · pp. 5–6
  • For a_i>2, local spectral features exceed the quadratic difference-frequency bound |k_p−k_q|≤2, indicating local superoscillatory variation rather than a new global Fourier component.The finite-window spectra show increasing residual error as a_i rises.Source: Results · p. 6
  • At fixed feature ratio, increasing depth from L=2 to L=8 reduces median MSE by approximately an order of magnitude, while gains from increasing r saturate beyond r≈2.The depth comparison redistributes the same total number of trainable features.Source: Results · p. 7
  • At a_i≈5, median errors range from ∼1 × 10^-3 for L=2 to ∼7 × 10^-5 for L=8, and stronger superoscillations are generally harder to approximate.Reconstructed local oscillation rates follow k_local,i(0)=a_i across the ensemble, with larger deviations at higher a_i.Source: Results · p. 7

Piecewise optical synthesis of superoscillations beyond the encoding basis

Spatial multiplexing partitions a target function into locally rescaled intervals, assigning each interval to an output channel. This enables approximation beyond the encoder’s harmonic content, with fidelity shaped by interval width, encoding scale, and channel resources.

Piecewise optical synthesis of superoscillations beyond the encoding basis

  • Each interval is mapped onto the native encoding range, so narrower intervals present slower functional variation to each channel while preserving the complete target through parallel responses.The prescribed interval map selects both the encoding coordinate and detector for a query.Source: Piecewise optical synthesis of… · p. 8
  • With 10 simultaneous targets, even the worst-case function achieves MSE on the order of 10^-6, with uniform quality across intervals.The interval error distribution shows no significant degradation across output channels.Source: Piecewise optical synthesis of… · pp. 8–9
  • All ten randomly generated targets exhibit local oscillation rates beyond their constituent bandlimit, and synthesized outputs reproduce the broad rate structure and superoscillatory peaks.Rate deviations are concentrated near zero-amplitude regions, where the local rate diverges and is sensitive to small approximation errors.Source: Piecewise optical synthesis of… · p. 9
  • Local oscillation rates reach approximately 226, more than six times the encoder bandwidth of approximately 33.9.The piecewise architecture achieves low MSE despite the target rates exceeding the native encoder bandwidth.Source: Piecewise optical synthesis of… · p. 9
  • Unequal-width partitions remain accurate, with wider tiles only modestly harder to approximate and no systematic error dependence on tile position.This robustness leaves room for adaptive, target-specific tiling.Source: Piecewise optical synthesis of… · p. 9
  • At N_sup = 16, R ≈ 0.4 achieves mean MSE below 10^-6, while optimal error progressively worsens as N_sup rises to 25, 36, 49, and 64.The optimal encoding scale balances insufficient phase diversity at small R against ambiguity from larger phase excursions.Source: Piecewise optical synthesis of… · p. 10
  • Mean MSE is minimized at N_Δ ≈ 8–10 and rises by approximately 3–4 orders of magnitude on either side.Too few intervals leave each detector a wide, difficult target region; too many divide a fixed feature budget and approach the processor’s spatial-resolution limit.Source: Piecewise optical synthesis of… · p. 10

Experimental validation

A visible-light SLM experiment tests spatially multiplexed synthesis with fewer input harmonics than target harmonics. In situ refinement reconstructs the target across nine intervals and reproduces superoscillatory rate structure, with deviations near sharp peaks and low-amplitude regions.

Experimental validation

  • The experimental phase pattern sums an argument-dependent encoder with an input-independent decoder and reference field on the same SLM plane.The reference field coherently interferes with encoded components at the detector, introducing a zero-frequency component.Source: Experimental validation · pp. 10–11
  • With a 520 nm laser, four input harmonics reconstruct a nine-harmonic target across nine intervals, reaching final MSE on the order of 10^-3 after approximately 580 in situ epochs.At epoch 580, the measured output closely follows the target across t ∈ [−0.5, 0.5].Source: Experimental validation · pp. 10–11
  • Optimization proceeds from an angular-spectrum numerical design to model-free PPO refinement using measured responses, with the coarse envelope captured before fine structure.The phase profile is transferred to the physical SLM for in situ refinement without gradients through the physical system.Source: Experimental validation · p. 11
  • The measured output reaches a peak local oscillation rate of approximately 226, beyond the target bandlimit of 18π, and reproduces its broad rate structure.Deviations occur near sharp peaks and regions of low analytic-signal amplitude.Source: Experimental validation · p. 11
  • A second experiment reaches a measured local oscillation rate of approximately 75, nearly twice its target bandlimit, using four input harmonics.The experiment uses four intervals and a native encoder bandwidth of 4.8π before interval rescaling.Source: Experimental validation · p. 11

Discussion

The discussion frames superoscillation as a function evaluated by a wave processor, rather than a predetermined optical field. It connects local interference and spatial multiplexing to approximation beyond encoded harmonic content, while identifying design trade-offs and future extensions.

Discussion

  • The work connects function evaluation with optical computation by combining complex-valued superoscillatory approximation and spatial allocation of locally tractable target regions.The conclusion presents functional superoscillation as a principle for designing analog optical processors.Source: Discussion · pp. 12, 14
  • At fixed degrees-of-freedom ratio, increasing layer count from 2 to 8 reduces MSE by an order of magnitude; increasing r beyond approximately 2 yields diminishing returns in tested architectures.The analysis suggests allocating resources toward architectural depth rather than further lateral resolution.Source: Discussion · p. 12
  • A visible-light system reconstructs a nine-harmonic target from four input harmonics, a target-to-encoding ratio of 2.25.The coarse envelope appears early during in situ optimization, while fine superoscillatory structure requires extended training.Source: Discussion · p. 12
  • Precise destructive interference required for superoscillation is harder to establish than gross amplitude modulation.The discussion points to adaptive partitioning and multidimensional or wavelength-dependent channel allocation as possible extensions, with scaling requiring coordinated encoder, feature, and detector design.Source: Discussion · p. 13
  • Spatial multiplexing assigns locally rescaled target portions to separate output channels, extending representation to targets whose global spectral content exceeds the input encoding bandwidth.The framework relies on restricting the approximation domain rather than generating additional Fourier components through square-law mixing.Source: Discussion · p. 13
  • The approach separates functional bandwidth from physical wave bandwidth: propagating optical fields remain diffraction-governed while queried outputs can vary rapidly with the input argument.Local variations beyond global Fourier support arise through superoscillatory interference over a restricted argument interval.Source: Discussion · pp. 12–13

Physical forward model and wave processor architecture

The forward model represents the processor as trainable phase-only layers separated by free-space propagation. Propagation is modeled with the angular spectrum method, and each mask’s phase is constrained to a physically realizable range.

Physical forward model and wave processor architecture

  • The processor has L trainable phase-only layers and L + 1 propagation segments, with numerical studies using 300 nm pixel pitch and 5.5 μm inter-layer spacing unless noted otherwise.The operating wavelength is 550 nm.Source: Physical forward model and wave… · p. 14
  • Free-space propagation between layers is modeled with the angular spectrum method, using a transfer function for the optical field and a two-dimensional Fourier-transform propagation calculation.For propagating waves in the non-evanescent spatial-frequency band, the transfer function has unit magnitude.Source: Physical forward model and wave… · p. 14
  • Each layer uses a trainable phase-only mask whose latent real-valued parameter maps to phase in [0, 2π], yielding unit-magnitude transmission.The bounded phase profile is described as physically realizable with phase-only SLMs or fabricated diffractive elements.Source: Physical forward model and wave… · p. 15

Superoscillatory target function generation

The targets are constructed from harmonic sums whose local oscillation rate can exceed the constituent bandlimit within a region of interest. For beyond-bandwidth synthesis, the target uses more harmonics than the input encoding and is evaluated piecewise across sub-intervals.

Superoscillatory target function generation

  • Berry–Popescu targets combine harmonics within [−1, +1] while their local oscillation rate at t = 0 equals the superoscillation parameter a_i, sampled from.The approximation is considered over [−t_roi, t_roi], with t_roi = 2 unless otherwise specified.Source: Superoscillatory target function… · pp. 15–16
  • Beyond-bandwidth targets use N_sup harmonics exceeding the N_base input harmonics, with ρ_h = N_sup/N_base > 1.The target harmonics have wavenumbers κ_j distinct from the input encoding wavenumbers k_j.Source: Superoscillatory target function… · p. 16
  • Each beyond-bandwidth target is restricted to [−t_roi, t_roi] and divided into N_Δ non-overlapping, function-dependent sub-intervals.The sub-interval widths sum to 2t_roi.Source: Superoscillatory target function… · p. 16

Figures

The figures show harmonic-sum targets and their optical approximations, then examine approximation accuracy across processor depth, piecewise synthesis, and a visible-light experiment. The later figures focus on targets whose harmonic content exceeds the input encoding and on nonuniform domain partitions.

Figures

  • The Fig. 1 schematic combines weighted harmonic basis functions into a band-limited target and encodes the scalar argument in the input wavefront for optical approximation.Real and imaginary components are recovered with differential intensity detectors.Source: Figure 1 · p. 20, Figures · p. 20
  • Increasing diffractive depth from 2 to 8 layers reduces median MSE by approximately an order of magnitude at fixed feature ratio across 100 target functions.The reported ensemble contains complex-valued Berry–Popescu targets.Source: Figures · p. 24
  • Fig. 4 depicts beyond-bandwidth synthesis by assigning sub-intervals of the target domain to dedicated detectors and concatenating their readouts.The example uses N_base = 9 and N_sup = 36, so the target harmonic count exceeds the encoded count.Source: Figure 4 · p. 25, Figures · p. 26
  • Fig. 5 compares target and processor-output local oscillation rates against the target constituent bandlimit.The caption identifies black solid and red dashed curves for target and processor results, respectively, and a blue dashed bandlimit.Source: Figure 5 · p. 27, Figures · pp. 27–28
  • With nonuniform partitioning, wider sub-intervals are systematically, though modestly, harder to approximate.Fig. 6B plots per-interval MSE against sub-interval width; its least-squares fit has R2 = 0.21.Source: Figure 6 · p. 29
  • The visible-light experiment uses a phase-only SLM and compares measured optical outputs with the target at training epochs 0, 300, and 580.The figure also shows the target and measured-output local oscillation rates at epoch 580.Source: Figures · p. 31
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