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Learning qBIC Resonances across Metasurface Families in Dielectric Fourier Space

Shuangteng Lei, Li Yu, Tianxin Li, Wei Lu

arXiv:2610.11500v1physics.opticscs.LGphysics.comp-ph

TL;DR

Geometry-specific parameters make structures from different metasurface families difficult to compare, while low-order Fourier coefficients provide a shared coordinate linked to their mode couplings. The paper uses fixed low-order descriptors to track resonance shifts and field-level calculations to test the Maxwell–Fourier perturbation picture, finding strong within-branch linear fits but less uniform fits for ln Q.

Abstract

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Bound states in the continuum (BIC) metasurfaces are typically described by geometry-specific parameters, hindering cross-geometry comparison, while ultranarrow qBIC features are easily diluted in full-spectrum learning. Here, 2015 samples from seven dielectric metasurface families are mapped to a shared reciprocal-lattice grid, where two frozen low-order Fourier channels capture resonance shifts with mean within-branch $R^2$ values of 0.871-0.999. Field-level analysis of two representative branches further confirms that these shifts are consistent with the Maxwell-Fourier perturbation picture. A five-channel K-space backbone models the broadband spectrum, while a local complex K-space expert parameterizes the qBIC resonance through a differentiable Fano layer. The expert reduces resonance-position mean absolute error (MAE) from 3.2 to 0.95 nm and the resonance-depth error by 14-fold on a geometry-blocked test set. The same coordinate supports spectrum-to-structure reconstruction.

Results

Dielectric Fourier coefficients place geometrically different metasurfaces on a shared reciprocal-lattice grid, where they describe the couplings governing Bloch modes. First-order perturbation motivates low-order descriptors, but its shifts are qualitative rather than quantitative for the structures studied.

Dielectric Fourier coefficients as a shared structural coordinate

  • All 2,015 structures across seven silicon metasurface families are represented on one reciprocal-lattice grid using centred two-dimensional permittivity FFTs.The families share a square lattice, coordinate origin, orientation, film thickness, and normal-incidence excitation, while their real-space parameters are not directly comparable.Source: Dielectric Fourier coefficients as a… · p. 4
  • Permittivity Fourier coefficients couple Bloch harmonics whose wavevectors differ by the corresponding reciprocal-lattice vector, and geometry changes these couplings and shifts the complex eigenfrequency.This is described as the photonic counterpart of the nearly-free-electron model.Source: Dielectric Fourier coefficients as a… · p. 5
  • First-order perturbation links the real-frequency shift to structural wavevectors weighted by modal overlap, motivating scalar low-order descriptors and a local 9 × 9 complex block.For near-Γ qBICs, most modal weight is in low-order harmonics; the complex block retains relative amplitudes and phases.Source: Dielectric Fourier coefficients as a… · p. 5
  • The perturbative shift is not quantitatively exact when high-index-contrast boundaries move, and the same permittivity change can shift different families by different amounts or in opposite directions.The perturbation is used to identify important wavevectors and test trends, not as a quantitative predictor.Source: Dielectric Fourier coefficients as a… · p. 5

Frozen low-order Fourier channels capture the dominant resonance shift

Two fixed low-order Fourier descriptors track resonance position along parameter branches across the seven families, while their relationship to ln Q is less uniform. Phase-sensitive radiation explains why scalar amplitude descriptors do not generally capture the full lineshape.

Frozen low-order Fourier channels capture the dominant resonance shift

  • ExtraTrees importance maps concentrate near Γ and are approximately mirror-symmetric, with directional preferences varying between families, informing the localized and symmetry-aware descriptors.The candidate regions were identified using separate regressors for resonance position and ln Q.Source: Frozen low-order Fourier channels… · p. 6
  • R2 ranges from 0.984–0.999 for AC, AD, CSR, DAN, SR, and TY and is 0.871 for PX for linear fits of selected descriptors to resonance position along 17-point branches.Descriptor choice is fixed across families, scan directions, and slices; CΓ is selected for CSR and PX, and CD for the other five families.Source: Frozen low-order Fourier channels… · p. 6
  • The descriptor fits are in-sample and descriptive, and the affine rescaling that collapses families removes family-specific slopes rather than demonstrating a universal slope.Out-of-sample prediction is evaluated with the neural models.Source: Frozen low-order Fourier channels… · p. 7
  • Within-branch R2 for ln Q exceeds 0.8 for AC, CSR, DAN, and TY, but is 0.670 for AD, 0.765 for PX, and 0.381 for SR.The descriptors describe ln Q less uniformly than resonance position.Source: Frozen low-order Fourier channels… · p. 7
  • ln Q depends on relative Fourier-coefficient phases as well as amplitudes because coherent radiation sums produce cross terms, which amplitude-only scalars cannot generally capture.This motivates the complex local representation for modelling the full lineshape.Source: Frozen low-order Fourier channels… · p. 7

Field-level Maxwell–Fourier validation

Field-overlap analysis on two qBIC branches supports the Maxwell–Fourier account of resonance shifts, while a local complex-Fourier expert improves resonance recovery alongside a broadband model.

Field-level Maxwell–Fourier validation

  • The central 7 × 7 block contains 93.3% of TY and 93.2% of CSR modal weight, and that concentration persists along both branches.The validation uses a reference state from each branch: TY represents C_D and CSR represents C_Γ.Source: Field-level Maxwell–Fourier validation · pp. 7–8
  • Descriptor-based first-order shifts are linear with resonance shift across the 17 states, with R2 = 0.996 for TY and 0.995 for CSR.For TY, the four CD wavevectors give a tighter linear relation than the full sum; the absolute slope remains affected by approximations in Eq. (4).Source: Field-level Maxwell–Fourier validation · p. 8

A local complex Fourier-space resonance expert reconstructs sharp qBIC resonances

  • A five-channel Fourier backbone predicts broadband transmission while a local complex-Fourier expert predicts four resonance parameters for a differentiable Fano layer.The expert uses the central 5 × 9 × 9 tensor block; the Fano layer lets spectral error update the resonance parameters directly.Source: A local complex Fourier-space… · p. 9
  • Resonance-position MAE falls from 3.000 to 0.998 nm on the real-space backbone and from 3.223 to 0.947 nm on the Fourier backbone; minimum-transmission MAE drops 91% and 93%, respectively.These test-set results are averaged over five random seeds.Source: A local complex Fourier-space… · p. 10
  • ln Q MAE falls from 0.542 to 0.289 on the real-space backbone and from 0.673 to 0.295 on the Fourier backbone with the expert.Source: A local complex Fourier-space… · p. 10
  • The expert's gain is large for AD, CSR, DAN, PX and SR, but it adds no improvement for AC and TY, whose broader resonances are easier for background models to learn.Source: A local complex Fourier-space… · p. 10
  • The Fourier backbone reaches training loss 0.06 in 69.3% fewer epochs than the real-space backbone, but its test errors are not lower.The background stage of expert models likewise uses 69.7% fewer epochs with Fourier input.Source: A local complex Fourier-space… · p. 10
  • The local 9 × 9 complex Fourier block lowers position MAE from 3.00 to 1.00 nm versus a background-only model, retaining low-order amplitude and phase information from the whole unit cell.The two scalar descriptors capture branchwise position shifts but discard phase and directional information needed for lineshape.Source: A local complex Fourier-space… · p. 11

Fourier input as the output of inverse reconstruction

The shared Fourier coordinate supports spectrum-to-structure reconstruction and provides a common structural representation for qBIC learning. The discussion connects the model design to Maxwell–Bloch coupling while noting its known-design-space scope.

Fourier input as the output of inverse reconstruction

  • A spectrum encoder predicts complex Fourier coefficients, which an inverse FFT converts to a real-space unit cell; family labels supervise training but are not inference inputs.The comparison includes Fourier decoding, direct real-space mask decoding, and joint supervision.Source: Fourier input as the output of inverse… · p. 11
  • 0.944 mean foreground IoU for Fourier reconstruction compares with 0.853 for direct real-space decoding and 0.926 for joint supervision on 221 test samples.On 153 masks without a pixel-identical training counterpart, the corresponding values are 0.927, 0.816 and 0.920; Fourier and joint routes are statistically indistinguishable, and inverse comparisons use one seed.Source: Fourier input as the output of inverse… · p. 11

Discussion

  • Dielectric Fourier space serves as a shared input coordinate for branch resonance shifts, a complex local input for resonance lineshapes, and an output coordinate for inverse reconstruction.Its role is tied to permittivity coefficients coupling Bloch harmonics in the Maxwell–Bloch eigenproblem.Source: Discussion · p. 12
  • The resonance expert learns a mapping from coupling matrix elements to pole-based lineshape parameters, with a Fano output layer; ablations support joint improvement from that output and local complex Fourier input.The Fourier backbone mainly accelerates optimization and does not improve test accuracy at the present data size.Source: Discussion · p. 12
  • All seven families appear in training, so the results establish learning within a known design space; transfer to unseen families remains untested and is proposed for larger datasets and LOFO evaluation.The shared family conditions include lattice, material, film thickness and normal-incidence excitation; the perturbative analysis assumes weak leakage and small perturbations, with field tests on two branches.Source: Discussion · pp. 12–13

Full-wave simulation and dataset construction

Seven silicon metasurface families were simulated under shared conditions and divided into geometry-blocked train, validation and test sets. Fixed low-order Fourier descriptors and a Fourier-space resonance expert were then specified for analysis and learning.

Full-wave simulation and dataset construction

  • 2,023 structures were simulated across seven two-parameter families on nominal 17 × 17 grids, with common lattice, silicon thickness, substrate, excitation, boundaries and monitors; eight structures were excluded, leaving 2,015.Transmission was sampled at 2,001 wavelengths, and the index monitor supplied real-space and Fourier inputs.Source: Full-wave simulation and dataset… · p. 23
  • The strict geometry-hash and principal-scan-block split assigned 1,556 samples to training, 238 to validation and 221 to testing, and was fixed across models.A continuous target qBIC branch was selected per family and its position, minimum transmission, linewidth and Q were extracted.Source: Full-wave simulation and dataset… · p. 23

Low-order dielectric Fourier-channel analysis

  • ExtraTrees importance maps guided two descriptors, CΓ and CD, whose definitions were fixed across families, scan directions and slices; ordinary least squares fits were evaluated within each 17-point slice.Reported R² is averaged over the 17 slices in each family.Source: Low-order dielectric Fourier-channel… · p. 24

Physics-guided spectral learning

  • The backbone uses either a real-space map or five-channel Fourier tensor to predict 32 PCA spectrum coefficients; an expert adds a central complex-Fourier branch and differentiable Fano layer for resonance parameters.The expert predicts λ0, ln Q, depth and asymmetry; models were validation-selected and tested with five seeds, using full-window, local, core, resonance-position, minimum-transmission and spectral-correlation metrics.Source: Physics-guided spectral learning · pp. 24–25

Spectrum-driven Fourier-space inverse reconstruction

The inverse model maps a target BIC-band spectrum and normalized wavelength to family probabilities and complex Fourier coefficients, then converts predicted coefficients into a foreground-probability map. The supplementary setup describes 2,015 valid samples and a fixed strict split.

Spectrum-driven Fourier-space inverse reconstruction

  • The inverse model uses a one-dimensional convolutional network to map the target BIC-band spectrum and normalized wavelength to family probabilities and complex Fourier coefficients.Family labels provide auxiliary supervision during training, while inference uses only predicted probabilities.Source: Spectrum-driven Fourier-space inverse… · p. 25
  • Inverse-transformed coefficients pass through a sigmoid to produce a 67 × 67 foreground-probability map, with Fourier-only, joint Fourier/real-space, and direct real-space supervision compared.The Fourier target is an orthonormal real FFT of binary-mask logits standardized with training-set statistics.Source: Spectrum-driven Fourier-space inverse… · p. 25

Supplementary Information

Supplementary Note 1 | Dataset, simulation settings and continuous spectral branches

  • The dataset contains 2,015 valid samples after excluding eight structures whose target resonance could not be identified.The exclusions were seven CSR samples and DAN231; samples were simulated on nominal 17 × 17 parameter grids.Source: Supplementary Note 1 | Dataset,… · p. 27

Supplementary Note 2 | Compact Maxwell–Fourier derivation

The derivation connects dielectric Fourier coefficients to resonance shifts through Maxwell perturbation theory and complex field-overlap terms. Its trend interpretation is conditional on weak leakage and other stated approximations.

Supplementary Note 2 | Compact Maxwell–Fourier derivation

  • The Bloch-field Maxwell equation couples Fourier harmonics through a dielectric matrix whose elements are ε_G−G′.The reciprocal-lattice vector is q = m b1 + n b2, and L_G contains the slab's z-dependent differential operator.Source: Supplementary Note 2 | Compact… · p. 29
  • At fixed A, first-order variation of the generalized eigenproblem relates perturbations to resonance shifts, with the common unit-cell area factor canceling under a consistent Fourier convention.For a perturbation uniform through the patterned thickness, the z integral defines W_q; weak leakage permits E_L ≈ E_R.Source: Supplementary Note 2 | Compact… · p. 29
  • Coherent summation over outgoing channels creates cross terms, so radiative Q depends on channel amplitudes rather than the scalar structure descriptors alone.The expression is ln Q_rad ≈ C − ln Σ_μ |d(μ,0) + Σ_q R(μq)Δε_q|^2.Source: Supplementary Note 2 | Compact… · pp. 30–31
  • The perturbative trend relation assumes negligible absorption and other losses, while linewidth labels are not direct complex-eigenfrequency measurements.Strong leakage, dispersion, moving discontinuous boundaries, or modal hybridization require generalized normalization, boundary perturbation, or multimode treatment.Source: Supplementary Note 2 | Compact… · pp. 30–31

Supplementary Note 3 | Frozen channel assignment and branchwise validation

The supplement distinguishes descriptive branchwise channel fits from held-out spectral-network predictions and reports the frozen-channel assignment and TY statistics. The reported TY R2 values summarize separate branch fits, not a pooled regression.

Supplementary Note 3 | Frozen channel assignment and branchwise validation

  • Channel selection ranks median within-branch R2 before mean R2, minimum R2, and normalized weighted MAE, with ordinary least squares fit separately within each branch.These are descriptive in-dataset fits rather than held-out spectral-network predictions.Source: Supplementary Note 3 | Frozen channel… · p. 32
  • Supplementary Table S3 reports per-family Fourier-channel assignments and branch statistics for resonance position and ln Q.The peak assignment and ln Q channel are selected and reported separately.Source: Table S3 · p. 32, Supplementary Note 3 | Frozen channel… · p. 32
  • For TY, the selected CD channel has mean R2 0.999, compared with 0.827 for CΓ across 17 separate branch fits.The reported means are arithmetic averages in the selected direction, not pooled-point R2 or the median used for channel selection.Source: Supplementary Note 3 | Frozen channel… · p. 32

Supplementary Note 4 | Field-level validation and reference-state dependence

Driven-field analyses measure modal Fourier weight and evaluate complex field-overlap responses along representative TY and CSR branches. Their linear-trend results support resonance-position trends, but do not validate radiative Q.

Supplementary Note 4 | Field-level validation and reference-state dependence

  • Driven fields at six TY and CSR reference resonances yield modal weight by summing vector-component |E_G|^2 over an x–y FFT and integrating over z.This field quantity differs from the dielectric coefficient F_mn; Supplementary Fig. S1 shows low-order localization across the six reference states.Source: Supplementary Note 4 | Field-level… · p. 33, Figure S1 · p. 33
  • The branchwise responses retain complex Δε_q and W_q weights, with complete, low-order, and directional responses evaluated for TY and CSR.TY uses S_CD as its directional response, while CSR uses S_CΓ.Source: Supplementary Note 4 | Field-level… · p. 34, Figure S2 · p. 34
  • TY branch R2 is 0.986–0.996 for S_CD, while CSR branch R2 is 0.995–0.997 for S_CΓ.Across the tested references, TY S_all yields 0.959–0.968 and CSR S_all yields 0.996–0.997; R2 measures linear trend agreement, not captured frequency-shift energy.Source: Supplementary Note 4 | Field-level… · p. 34
  • The driven-field tests support resonance-position trends but provide no normalized outgoing-channel projection to validate radiative Q.The results are reference-state and branch-specific.Source: Supplementary Note 4 | Field-level… · p. 34

Supplementary Note 5 | Forward learning, split boundaries and expert-input controls

The forward model combines a Fourier-space backbone for broadband response with a local Fourier expert for the qBIC resonance, and evaluates alternatives under a geometry-blocked split. Results vary across families, and the expert’s resonance-position performance depends on its input representation.

Forward-model design

  • The backbone predicts broadband response, while a central 5 × 9 × 9 complex-Fourier expert predicts λ0, ln Q, depth, and Fano asymmetry for differentiable Fano synthesis.All configurations use the complete 2,001-point spectra; preprocessing statistics and the PCA basis are fitted on training data only.Source: Supplementary Note 5 | Forward… · p. 35

Split boundaries

  • The strict split separates source-grid geometry hashes and principal-scan parameter blocks, but 68 of 221 test masks duplicate training masks at 67 × 67 resolution.The remaining 153 masks are nonduplicated and still belong to the seven known families and current scan range.Source: Supplementary Note 5 | Forward… · p. 36, Figure S3 · p. 36

Family-resolved results

Evaluation

  • Complete-spectrum, local, core, minimum-transmission, resonance-position, and median spectral R metrics are reported, with five-seed means and standard deviations.Local and core errors evaluate windows around the labeled center; they are evaluation masks, not inference inputs.Source: Figure S5 · p. 38, Supplementary Note 5 | Forward… · p. 38

Expert-input controls

  • 0.996 nm (central complex Fourier), 2.049 nm (full real-space), 1.979 nm (full Fourier), 2.691 nm (central real-space crop), and 2.548 nm (CD, CΓ) are the resonance-position MAEs.The ablation fixes the real-space backbone, split, and seed; it distinguishes input representations but does not statistically rank them.Source: Supplementary Note 5 | Forward… · p. 39

Surrogate-model controls

  • Additional controls compare real-space ResNet, a 2D Fourier neural operator, and an adapted offline two-stage model on the same strict split and target spectra.The offline two-stage model is a Ma-style adaptation, not an exact reproduction of RIDL; representative offset spectra are shown for the controls.Source: Supplementary Note 5 | Forward… · pp. 40–41, Figure S8 · p. 41

Compute cost

  • 2,015 FDTD simulations required 146,984.4 CPU·h total, while prepared-input batch-one inference for 67 × 67 inputs took 1.593–3.403 ms per sample.Inference timings exclude geometry construction, FFT, disk I/O, and model loading; CPU·h and GPU·h are not converted into a direct speedup.Source: Supplementary Note 5 | Forward… · p. 42, Figure S9 · p. 42

Supplementary Note 6 | Inverse reconstruction and its evaluation boundary

The inverse routes reconstruct binary masks from spectra using either Fourier coefficients, joint real-space and Fourier prediction, or direct real-space prediction. Fourier-only has the highest reported IoU on both test sets, while reconstruction remains conditional on paired data and known topology classes.

Reconstruction method

  • The inverse encoder maps the 2,001-point spectrum and normalized wavelength to a 67 × 67 binary mask, with family labels used as auxiliary training supervision.Fourier routes predict coefficients of mask logits and reconstruct maps through inverse rFFT, sigmoid, and thresholding; these are not the physical dielectric coefficients used in channel analysis.Source: Supplementary Note 6 | Inverse… · p. 43

Test results

  • Full-test probability MAE is 0.030 (Fourier only), 0.015 (joint), and 0.027 (direct real), complementing IoU and exact-mask accuracy.Probability MAE evaluates the unthresholded map, whereas IoU evaluates the thresholded contour and exact-mask accuracy requires every pixel to match.Source: Supplementary Note 6 | Inverse… · p. 43, Figure S10 · p. 44
  • 0.944 IoU (Fourier only), 0.926 (joint), and 0.853 (direct real) on all 221 test samples; on 153 no-training-repeat masks, IoU is 0.927, 0.920, and 0.816, respectively.The no-training-repeat subset still consists of masks from the seven known families.Source: Supplementary Note 6 | Inverse… · p. 43, Figure S10 · p. 44

Evaluation boundary

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