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REVIEW 3 major objections 4 minor 160 references

This thesis argues that generative inverse design—diffusion models and Schrödinger bridges guided by a differentiable surrogate and amplitude constraints—can synthesize silicon metasurfaces whose far-field intensity matches targets at R²≈0.

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2026-08-01 08:28 UTC pith:HQD5JEAE

load-bearing objection A well-executed thesis whose strongest scaling claim rests on a surrogate evaluating itself; the moderate-scale core is credible and worth engaging with. the 3 major comments →

arxiv 2607.21091 v1 pith:HQD5JEAE submitted 2026-07-23 physics.optics math-phmath.MP

Design and Optimization of Metasurfaces for Silicon Photonics: PhD Thesis

Mathys Le Grand (1 , 2)((1) Institut des Nanotechnologies de Lyon , (2) STMicroelectronics) This is my paper
classification physics.optics math-phmath.MP
keywords metasurfacesinverse designelectromagnetic simulationFDTDRCWAsurrogate modelsdiffusion modelsSchrödinger bridges
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The thesis sets out to show that the bottleneck in metasurface design—finding the nanopillar layout that produces a desired light pattern—can be broken by replacing slow full-wave electromagnetic simulation with learned surrogates, and by shifting from iterative optimization to generative modeling. It first argues that FDTD, not RCWA, is trustworthy ground truth: implementations of Li's factorization rules can converge spectrally while silently distorting the material profile, with the worst artifacts in plasmonic structures. It then claims that fully convolutional surrogates trained on FDTD simulations of 23×23 pillar arrays predict the near field almost instantly and apply to much larger apertures. The central result is that diffusion models and Schrödinger bridges, combined with hybrid posterior sampling and amplitude constraints, sustain far-field fidelity of R²≈0.97 on surfaces over 230 times larger than training, while also producing diverse, fabrication-tolerant designs. A database enhancement step lifts every compared approach to roughly the same R²≈0.97 ceiling, suggesting the limit is no longer the optimizer.

Core claim

The paper establishes scale-invariant inverse design of large metasurfaces by pairing a generative model with a differentiable surrogate. RCWA with Li's factorization rules can look spectrally converged while its reconstructed permittivity is distorted—Gibbs oscillations, induced anisotropy, zero-crossings seeding unphysical modes, worst for plasmonic structures—so FDTD is ground truth. A convolutional surrogate trained on 23×23 FDTD arrays predicts near fields in one pass and transfers to larger apertures. In a shared R² benchmark, phase-retrieval-plus-local-model plateaus at 0.925, heuristic-initialized gradient descent at 0.975, and diffusion/Schrödinger-bridge models with hybrid posterio

What carries the argument

The load-bearing object is a differentiable fully convolutional surrogate (a residual U-Net style network) trained on FDTD near fields of 23×23 pillar metasurfaces, used both as a fast forward model and as the guidance signal during generative sampling. Around it sits the generative machinery: diffusion models and diffusion Schrödinger bridges (stochastic-transport generative models), plus amplitude-constrained posterior sampling—spherical, disk, and ring Gaussian constraints with Monte Carlo or stabilized gradient estimators—that keep sampled pillar radii inside the fabrication-allowed range while steering the far field toward the target. The first chapter's machinery is the reconstructed-p

Load-bearing premise

The claim rests on trusting that the neural-network surrogate used to steer the design also stays accurate when predicting light fields on surfaces hundreds of times larger than any it trained on—the paper's large-scale fidelity numbers are computed with that same surrogate rather than with independent full-wave simulation.

What would settle it

Take a handful of the paper's inverse-designed pillar maps at, say, 200×200 and 500×500 pillars, simulate them with an independent full-wave Maxwell solver, and compute the R² between simulated and target far-field intensity; if R² falls well below 0.97 or systematically degrades with aperture, the scale-invariance claim fails. Separately, for the RCWA artifact claim, reconstruct the real-space permittivity of a dielectric and a plasmonic grating at truncation order N=30: if no induced anisotropy or zero-crossing oscillations appear in the reconstructed profile, the diagnosis is wrong.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Metasurface design no longer needs a full-wave simulation per candidate design: the surrogate plus generative sampler replaces the FDTD-in-the-loop cost, and the convolutional architecture transfers a 23×23-trained model to apertures that are 230× larger.
  • Gradient-descent-based inverse design, when initialized with a physics-informed phase-retrieval/local-model solution, holds R²≈0.975 from small arrays up to the largest tested surrogate-evaluated apertures.
  • Improving the training database (target-driven synthesis rather than uniform sampling) raises every benchmarked method to R²≈0.97, so database quality, not the optimizer, becomes the main lever on performance.
  • Amplitude-constrained posterior sampling produces fabrication-tolerant designs: fidelity stays stable under injected stochasticity until the guidance term contributes less than about one-third of the update.
  • The RCWA analysis warns that spectral convergence is not a sufficient validity check for Fourier-based solvers; reconstructed-permittivity diagnostics should accompany convergence studies, especially for plasmonic or high-contrast geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Because the headline 230× scaling and R²≈0.97 numbers are evaluated with the same surrogate that guides the sampling, the strongest version of the claim—actual millimeter-scale devices—still awaits independent full-wave verification at intermediate apertures; nothing in the paper contradicts that the claim could hold, but the evidence at the extreme scale is surrogate-internal.
  • The symmetry-based database augmentation (quadrupling samples via reflection equivariance) is a transferable trick: any electromagnetic forward operator that respects reflection symmetry could be augmented the same way, which should apply to other pillar-based photonic and acoustic inverse-design pipelines.
  • The convergence of all enhanced methods to R²≈0.97 hints at a shared error floor—probably the surrogate's approximation error or the R² metric's insensitivity to absolute intensity—rather than a fundamental limit of any one inverse-design approach; a natural next test is to swap the loss to a perceptually or radiometrically weighted metric.
  • The diversity/stability trade-off controlled by the stochasticity coefficient suggests a practical manufacturing knob: sweeping that coefficient while measuring fabricated-device yield could let engineers choose designs that are simultaneously high-fidelity and robust to lithography variation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The thesis develops a three-stage pipeline for metasurface inverse design: (i) a critical comparison of RCWA implementations of Li's factorization rules, concluding that spectral convergence does not imply physical fidelity and adopting FDTD as ground truth; (ii) a local phase-approximation model and fully convolutional, FDTD-trained surrogate networks that predict near fields and generalize to larger apertures; and (iii) an inverse-design benchmark comparing Gerchberg–Saxton phase retrieval with the local model (R²≈0.925), heuristic-initialized surrogate-based gradient descent (R²≈0.975), and diffusion-model/Schrödinger-bridge generative frameworks with hybrid posterior sampling and amplitude constraints. The headline claim is that the generative framework restores 'scale-invariant fidelity' on surfaces over 230× larger than the 23×23 training cases, with database enhancement lifting all methods to R²≈0.97.

Significance. If fully substantiated, the work would be a valuable step toward generative inverse design of large-area, fabrication-tolerant silicon metasurfaces without per-device full-wave simulation. The manuscript has real strengths: the moderate-aperture results are validated by FDTD under a common protocol; the surrogate architecture study and symmetry-based database augmentation are carefully documented; the RCWA permittivity-reconstruction analysis provides a useful diagnostic for Fourier factorization artifacts; and the thesis is unusually transparent about where 'direct numerical simulation becomes computationally prohibitive.' These strengths are substantial, but the flagship large-aperture claim currently rests on the same learned surrogate that provides the gradient guidance, so the physical evidence for scale-invariance is incomplete. The paper is publishable after the validation gap is closed or the claims are appropriately bounded.

major comments (3)
  1. [§4.12.7, Figs. 4.26–4.27] The central claim of scale-invariant fidelity on surfaces 'over 230 times larger than training' is evaluated with the surrogate Sφ, and the same Sφ is used as the differentiable guidance model during posterior sampling in §4.11. This makes the reported R²≈0.97 at large apertures a measure of the surrogate's internal consistency rather than physical fidelity. The coupling-range analysis of §3.4.2 establishes that 23×23 is sufficient for the PBC coupling range under a 1% error criterion; it does not establish that error does not accumulate when the learned filterbank is applied at 1200×1200. Please provide full-wave FDTD (or an independent Maxwell solver) validation at intermediate apertures at least, and report the surrogate-vs-FDTD difference on optimized designs, not only on training-distribution samples. Without this, the abstract's scale-invariance statement overstates the evidence.
  2. [§4.11 and §4.12.5, Fig. 4.11] The same conflation affects the gradient-descent scalability results: Fig. 4.11 shows performance up to 1200×1200 pillars using surrogate-predicted far fields, while the FDTD figures in Appendix A.2.1 (Figs. A.21–A.24) appear to cover only moderate apertures. Because the design is optimized against the surrogate, the surrogate-predicted R² can be biased upward relative to the true FDTD R² even at moderate scales. The thesis itself labels §4.12.7 as surrogate-based evaluation and §5.3 acknowledges computational limits; I count this transparency as a strength, but it also identifies precisely the gap that must be filled before the headline claim is supported. A concrete test: evaluate a few large-aperture designs (e.g., 99×99 and 199×199) with FDTD and report the R² gap versus Sφ.
  3. [§4.12.3, Tables 4.2–4.3] The 'fabrication-tolerant' and comparative claims are presented without statistical uncertainty. Diffusion posterior sampling and Schrödinger-bridge sampling are stochastic: different seeds will produce different R² values, yet Tables 4.2 and 4.3 report single numbers and the Hessian spectral analysis in §4.12.3 is performed on the surrogate-guided landscape. To support the claim that 'database enhancement lifts every approach to R²≈0.97,' please report means, standard deviations, and repeated-seed counts for each method and condition. Without these, the observed differences between closely ranked methods (e.g., DSB Robust5 vs. DM Robust5) are not assessable.
minor comments (4)
  1. [Abstract and §4.12.5] The phrase 'over 230 times larger than training' is ambiguous: does it refer to linear dimension, area, or degrees of freedom? The training aperture is 23×23 and the largest reported is 1200×1200; specifying the scaling metric would clarify the claim.
  2. [§4.14?] The text contains typos and grammatical slips, e.g., 'symetries' in §3.4.3, 'peridic' in §3.4.8, and the caption of Fig. 4.14: 'both ... slightly outperform their normalized counterparts when using non-normalized guidance' is logically confusing and should be rephrased.
  3. [§4.4.2] The comparison between raw and hybrid Gerchberg–Saxton variants is stated in terms of R² and oscillation standard deviation, but the manuscript does not specify whether R² is computed on the full far-field image or a masked region. Please state the metric definition explicitly in the validation protocol (§4.3).
  4. [§2.7 and §3.4.5] The 'reordering' reconstruction in Chapter 2 and the FFT downsampling phase correction in Eq. (3.2) are described clearly, but the sign convention of the phase factor in Eq. (3.2) should be checked against the figures: a one-pixel offset in the sign would produce a systematic drift. A short derivation or a numerical consistency check would remove ambiguity.

Circularity Check

2 steps flagged

The 230× scale-invariance claim is evaluated with the same surrogate Sϕ that provides the posterior-sampling gradient guidance, so the headline R² measures the optimizer's own objective rather than independent full-wave fidelity.

specific steps
  1. fitted input called prediction [§4.12.7 (Figs. 4.26–4.27); guidance in §4.11 (Fig. 4.15)]
    "Performance is quantified via the surrogate model Sϕ, enabling a rigorous assessment of generative fidelity at extrapolated scales where direct numerical simulation becomes computationally prohibitive."

    The posterior-sampling guidance used to create the evaluated designs is qt ∝ 1/||c − Sϕ(x̂0|t)||² (Fig. 4.15), i.e., the sampling directly minimizes the surrogate prediction error against the target c. The large-aperture R² reported in §4.12.7 is then computed from the same Sϕ's own far-field output against the same target c. The headline 'scale-invariant fidelity on surfaces over 230 times larger than training' is therefore, by construction, a measure of how well the optimizer minimized its own surrogate objective, not an independent physical-fidelity check. The thesis labels the section 'Extrapolative Scaling and Surrogate-Based Evaluation' and limits FDTD validation to ~100×100; no independent full-wave verification exists at the claimed 230× scale.

  2. fitted input called prediction [§4.5.1, Fig. 4.11]
    "While rigorous full-wave validation via FDTD is computationally limited to metasurfaces of approximately 100×100 unit cells, we leverage the neural network surrogate Sϕ to evaluate the scaling behavior across much larger apertures."

    The surrogate-based gradient descent (Eq. 4.1) minimizes the MSE between the target far-field and Sϕ's predicted intensity, and the large-aperture scaling claim (up to 1200×1200 pillars) is then 'evaluated' with this same Sϕ. Thus the reported R²≈0.975 at large scale is the minimized training loss of the optimization reported as a scaling result: the same learned function appears both as the differentiable design objective and as the evaluator, so the large-scale 'prediction' is forced by construction and lacks independent FDTD confirmation.

full rationale

The circularity is real but partial. The RCWA/Li-rule analysis (Ch. 2) is an independent numerical diagnostic, not circular. The moderate-aperture inverse-design benchmarks (23×23 and ~100×100) are validated with full-wave FDTD and provide genuine independent content. The circular reduction is confined to the extrapolated-scale headline claims: the same convolutional surrogate Sϕ is used (i) as the differentiable guidance model during posterior sampling and gradient descent, minimizing ||c − Sϕ(x)||², and (ii) as the evaluator producing the reported R² at scales where FDTD is said to be 'computationally prohibitive' (§4.12.7, Figs. 4.26–4.27; §4.5.1, Fig. 4.11). The 230× scale-invariance and large-aperture R²≈0.97 claims therefore reduce by construction to the surrogate's internal consistency with its own training objective, exactly the pattern of a fitted input renamed as a prediction. The paper's own section titles ('Extrapolative Scaling and Surrogate-Based Evaluation') concede this. Because substantial FDTD-verified results exist at moderate apertures and the surrogate is a legitimate forward model there, the paper is not wholly circular; but its most prominent novelty—scale-invariant fidelity far beyond training—is self-referential as reported.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The pipeline rests on FDTD as ground truth, a per-pillar near-field representation, Fourier far-field propagation, symmetry augmentation, and surrogate generalization. The paper introduces no new physical entity; its contributions are algorithmic.

free parameters (6)
  • Surrogate architecture hyperparameters (width c, depth N) = c∈{32,64,128}, N varied; final model parameters not fixed in excerpt
    Selected by empirical validation-set performance; no separate hold-out for the scaling claim.
  • Near-field amplitude constraint μ for Gerchberg-Saxton phase retrieval = Mean of transmission amplitude distribution (Fig. 4.2b)
    Fitted to the nanopillar library; the phase-retrieval result depends on this constraint.
  • Posterior-sampling guidance normalization (qt∝1/||·||² vs q=1) = Normalized variant preferred for scaling
    This tuning choice changes R² at large apertures (Figs. 4.14–4.16).
  • Stochasticity/diversity coefficient α in guided sampling = α=1 used in final comparisons
    Controls the magnitude of gradient guidance vs noise injection; hand-set.
  • Consistency loss schedule/weight = Uniform vs scheduled variants; explicit values not stated in excerpt
    Training enhancement chosen to improve R² distribution (Fig. 4.13).
  • Noise variance schedule for DMs/DSBs = Not specified in excerpt
    A required generative-model choice that materially affects sampling fidelity.
axioms (6)
  • domain assumption FDTD is treated as artifact-free ground truth for the entire pipeline.
    Chapter 2 concludes RCWA distorts permittivity and retains FDTD as reference; FDTD's own discretization/PML errors are not independently benchmarked against experiment.
  • domain assumption The near field is adequately represented by one complex value per pillar after FFT downsampling.
    §3.4.4 uses a top-hat Fourier mask; far-field patterns survive, but sub-cell near-field information is discarded.
  • standard math The far field is obtained from the near field via a Fourier transform (Fraunhofer approximation).
    §3.1.3; this is standard scalar/vector diffraction theory, not verified against full 3D propagation in the paper.
  • domain assumption Symmetry equivariance F(s(radius map))=s(F(radius map)) holds for flips under normal incidence.
    Eq. 3.1 is used to quadruple the database; the paper itself notes it breaks for oblique incidence.
  • domain assumption A surrogate trained on 23×23 patches generalizes to apertures of 1,200×1,200 pillars.
    Fully convolutional architecture with periodic padding (§3.4.6) is assumed to preserve coupling physics far outside the training distribution; this surrogate is then used as the evaluator in §4.12.7.
  • domain assumption The staircase discretized geometry is the correct reference for scoring reconstructed RCWA permittivity.
    §2.7 uses R² between the reconstructed permittivity and the staircase profile as the fidelity metric; a physical 'true' profile may differ due to fabrication curvature.

pith-pipeline@v1.3.0-alltime-deepseek · 45916 in / 13137 out tokens · 114527 ms · 2026-08-01T08:28:47.940341+00:00 · methodology

0 comments
read the original abstract

Metasurfaces, two-dimensional arrangements of subwavelength nanopillars, provide local control over the phase of light, enabling flat optical functions beyond conventional refractive components. Their inverse design, finding the pillar distribution producing a target response, faces two obstacles: the immense dimensionality of the design space and the prohibitive cost of rigorous electromagnetic simulations, precluding exhaustive exploration at device scale. This thesis addresses the challenge in three stages. The first assesses the reliability of rigorous Maxwell solvers: comparing three implementations of Li's factorization rules for RCWA shows that spectral convergence does not guarantee physical fidelity, as these rules implicitly distort the simulated permittivity, most severely in the plasmonic regime. FDTD, immune to such artifacts, is retained as ground truth throughout. The second stage removes the computational bottleneck: a local phase-approximation model, then fully convolutional surrogates trained on FDTD simulations of large pillar metasurfaces, predict the near field almost instantaneously. Exploiting problem symmetries quadruples the training database, and the surrogates generalize to much larger apertures while remaining differentiable. The third stage benchmarks three strategies under a common FDTD protocol ($R^2$ between realized and target far fields): Gerchberg-Saxton retrieval and Local Model ($R^2\approx0.925$), surrogate-based and heuristic-initialized gradient descent ($R^2\approx0.975$), and a generative framework based on diffusion models and Schr\"odinger bridges. Hybrid posterior sampling and amplitude constraints restore scale-invariant fidelity on surfaces over 230 times larger than training, with diverse, fabrication-tolerant designs. Database enhancement lifts every approach to $R^2\approx0.97$.

Figures

Figures reproduced from arXiv: 2607.21091 by 2)((1) Institut des Nanotechnologies de Lyon, (2) STMicroelectronics), Mathys Le Grand (1.

Figure 2.1
Figure 2.1. Figure 2.1: Normal Vector Field built for a square structure with the algorithm presented in [ [PITH_FULL_IMAGE:figures/full_fig_p033_2_1.png] view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Dielectric structure in the patterned layer. [PITH_FULL_IMAGE:figures/full_fig_p036_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Plasmonic structure in the patterned layer. [PITH_FULL_IMAGE:figures/full_fig_p037_2_3.png] view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Transmission spectrum of the dielectric structure over the wavelength range [PITH_FULL_IMAGE:figures/full_fig_p038_2_4.png] view at source ↗
Figure 2.5
Figure 2.5. Figure 2.5: Transmission spectrum of the dielectric structure over the wavelength range [PITH_FULL_IMAGE:figures/full_fig_p038_2_5.png] view at source ↗
Figure 2.6
Figure 2.6. Figure 2.6: Transmission spectrum of the dielectric structure over the wavelength range [PITH_FULL_IMAGE:figures/full_fig_p039_2_6.png] view at source ↗
Figure 2.7
Figure 2.7. Figure 2.7: Convergence analysis of the transmission for the plasmonic structure at [PITH_FULL_IMAGE:figures/full_fig_p040_2_7.png] view at source ↗
Figure 2.8
Figure 2.8. Figure 2.8: Convergence analysis of the transmission for the plasmonic structure at [PITH_FULL_IMAGE:figures/full_fig_p040_2_8.png] view at source ↗
Figure 2.9
Figure 2.9. Figure 2.9: Transmission spectrum of the plasmonic structure with [PITH_FULL_IMAGE:figures/full_fig_p041_2_9.png] view at source ↗
Figure 2.10
Figure 2.10. Figure 2.10: Transmission spectrum of the plasmonic structure with [PITH_FULL_IMAGE:figures/full_fig_p042_2_10.png] view at source ↗
Figure 2.11
Figure 2.11. Figure 2.11: Convergence analysis of the transmission for the plasmonic structure at [PITH_FULL_IMAGE:figures/full_fig_p043_2_11.png] view at source ↗
Figure 2.12
Figure 2.12. Figure 2.12: Reconstructed permittivity profiles of the dielectric structure at a truncation order [PITH_FULL_IMAGE:figures/full_fig_p045_2_12.png] view at source ↗
Figure 2.13
Figure 2.13. Figure 2.13: Reconstructed permittivity profiles of the dielectric structure at a truncation order [PITH_FULL_IMAGE:figures/full_fig_p046_2_13.png] view at source ↗
Figure 2.14
Figure 2.14. Figure 2.14: Reconstructed permittivity profiles of the dielectric structure at a truncation order [PITH_FULL_IMAGE:figures/full_fig_p046_2_14.png] view at source ↗
Figure 2.15
Figure 2.15. Figure 2.15: Reconstructed permittivity profiles of the dielectric structure anti diagonal term [PITH_FULL_IMAGE:figures/full_fig_p047_2_15.png] view at source ↗
Figure 2.16
Figure 2.16. Figure 2.16: Reconstructed y-axis permittivity profiles of the dielectric structure for truncation [PITH_FULL_IMAGE:figures/full_fig_p048_2_16.png] view at source ↗
Figure 2.17
Figure 2.17. Figure 2.17: Evolution of the R2 coefficient of determination as a function of the truncation order N, illustrating the correlation between the reference staircase profile and the reconstructed permittivity. The observed fluctuations in numerical precision correlate strongly with the Gibbs phe￾nomenon. This relationship is demonstrated using a simplified 1D model in [PITH_FULL_IMAGE:figures/full_fig_p049_2_17.png] view at source ↗
Figure 2.18
Figure 2.18. Figure 2.18: 1D permittivity distribution corresponding to a cross-section of the two-dimensional [PITH_FULL_IMAGE:figures/full_fig_p050_2_18.png] view at source ↗
Figure 2.19
Figure 2.19. Figure 2.19: Evolution of the R2 coefficient of determination as a function of the truncation order N, illustrating the correlation between the reference staircase profile and the reconstructed permittivity [PITH_FULL_IMAGE:figures/full_fig_p050_2_19.png] view at source ↗
Figure 2.20
Figure 2.20. Figure 2.20: Permittivity profile reconstruction for various truncation orders [PITH_FULL_IMAGE:figures/full_fig_p051_2_20.png] view at source ↗
Figure 2.21
Figure 2.21. Figure 2.21: One-dimensional permittivity distribution corresponding to a transverse cross [PITH_FULL_IMAGE:figures/full_fig_p051_2_21.png] view at source ↗
Figure 2.22
Figure 2.22. Figure 2.22: Evolution of the R2 coefficient of determination as a function of the truncation order N, illustrating the structural correlation between the reference staircase geometry and the reconstructed permittivity for the narrow plateau architecture [PITH_FULL_IMAGE:figures/full_fig_p052_2_22.png] view at source ↗
Figure 2.23
Figure 2.23. Figure 2.23: Reconstructed permittivity profiles across the narrow plateau region for various [PITH_FULL_IMAGE:figures/full_fig_p052_2_23.png] view at source ↗
Figure 2.24
Figure 2.24. Figure 2.24: Reconstructed y-axis real-part permittivity profiles of the plasmonic structure for truncation orders N ∈ {10, 20, 30}, comparing various Fourier factorization implementations based on Li’s rules [PITH_FULL_IMAGE:figures/full_fig_p053_2_24.png] view at source ↗
Figure 2.25
Figure 2.25. Figure 2.25: Reconstructed y-axis imaginary-part permittivity profiles of the plasmonic structure for truncation orders N ∈ {10, 20, 30}, comparing various Fourier factorization implementations based on Li’s rules. In summary, for RCWA, the choice of Fourier factorization rule is a critical determinant of both numerical stability and geometric fidelity in RCWA simulations. While Li’s rules and their derivatives (NV … view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Comparison of Local Phase Approximation (LPA) look-up tables generated via three [PITH_FULL_IMAGE:figures/full_fig_p058_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Phase shift distribution as a function of pillar diameter for a [PITH_FULL_IMAGE:figures/full_fig_p058_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Near-field R2 coefficient of determination for the three LUT methodologies as a func￾tion of increasing metasurface aperture. The plot illustrates the scaling of predictive accuracy relative to the ground-truth FDTD simulations [PITH_FULL_IMAGE:figures/full_fig_p059_3_3.png] view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Relative error of near field phase and amplitude with two kind of boundary conditions [PITH_FULL_IMAGE:figures/full_fig_p061_3_4.png] view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Radius map of the metasurface after different studied symetries operation for [PITH_FULL_IMAGE:figures/full_fig_p062_3_5.png] view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Near-field phase distribution of the metasurface, obtained via FDTD simulation un [PITH_FULL_IMAGE:figures/full_fig_p063_3_6.png] view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Downsampled near-field phase distribution of the metasurface, obtained via FDTD [PITH_FULL_IMAGE:figures/full_fig_p063_3_7.png] view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Near-field phase distribution of the metasurface, obtained via FDTD simulation [PITH_FULL_IMAGE:figures/full_fig_p064_3_8.png] view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Downsampled near-field phase distribution of the metasurface, obtained via FDTD [PITH_FULL_IMAGE:figures/full_fig_p064_3_9.png] view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Comparative analysis of near-field and far-field distributions across different spatial [PITH_FULL_IMAGE:figures/full_fig_p066_3_10.png] view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Spatial reconstruction of a cross-shaped geometry highlighting the necessity of [PITH_FULL_IMAGE:figures/full_fig_p067_3_11.png] view at source ↗
Figure 3.12
Figure 3.12. Figure 3.12: Comparative schematic of the developed deep learning surrogate models. (a) A [PITH_FULL_IMAGE:figures/full_fig_p068_3_12.png] view at source ↗
Figure 3.13
Figure 3.13. Figure 3.13: Architectural overview of the ResUnet surrogate model. (a) Global topology of the [PITH_FULL_IMAGE:figures/full_fig_p069_3_13.png] view at source ↗
Figure 3.14
Figure 3.14. Figure 3.14: Comparative performance of diverse neural network architectures as a function of [PITH_FULL_IMAGE:figures/full_fig_p070_3_14.png] view at source ↗
Figure 3.15
Figure 3.15. Figure 3.15: Comparative performance (R2 ) of surrogate architectures across varying structural dimensions. The x-axis denotes the feature channel width (c ∈ {32, 64, 128}), while the different series represent increasing network depths (N) as indicated in the legend. The plot illustrates the scaling behavior of the near-field prediction accuracy as a function of the total parameter number of the models. 3.4.9 Archi… view at source ↗
Figure 3.16
Figure 3.16. Figure 3.16: relative error of near field phase and amplitude for a increasing coupling range [PITH_FULL_IMAGE:figures/full_fig_p072_3_16.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Schematic of the inverse design validation workflow. The pipeline initiates with a [PITH_FULL_IMAGE:figures/full_fig_p075_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Numerical characterization of the nanopillar library via FDTD. (a) Mapping of [PITH_FULL_IMAGE:figures/full_fig_p076_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Phase retrieval results for three distinct far-field binary targets (a, e, j). The Ger [PITH_FULL_IMAGE:figures/full_fig_p077_4_3.png] view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: R2 coefficient of determination quantifying the fidelity between the target far-field intensity and the pattern reconstructed via the phase retrieval algorithm. Comparisons are provided for two distinct aperture scales: (a) a 28 × 28 nanopillar array and (b) an expanded 98 × 98 metasurface, illustrating the scalability of the iterative convergence. 4.4.3 Selection of the Optimal Near-Field Phase Distribu… view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: standard deviation quantifying the oscillation of the target far-field intensity re [PITH_FULL_IMAGE:figures/full_fig_p079_4_5.png] view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Benchmarking the inverse design precision of the Local Model combined with Phase [PITH_FULL_IMAGE:figures/full_fig_p080_4_6.png] view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: FDTD simulated far-field intensity distributions for two representative metasurface [PITH_FULL_IMAGE:figures/full_fig_p081_4_7.png] view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Benchmarking the reconstruction precision ( [PITH_FULL_IMAGE:figures/full_fig_p082_4_8.png] view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: FDTD-simulated far-field intensity for two representative designs across increasing [PITH_FULL_IMAGE:figures/full_fig_p082_4_9.png] view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Simulated |far field| performance of two representative designs, contrasting the [PITH_FULL_IMAGE:figures/full_fig_p083_4_10.png] view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Surrogate-predicted far-field scaling for heuristic-initialized Gradient Descent. Sim [PITH_FULL_IMAGE:figures/full_fig_p083_4_11.png] view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Final performance metrics are compared with and without score conditioning on [PITH_FULL_IMAGE:figures/full_fig_p090_4_12.png] view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Statistical distribution of reconstruction metrics for varying consistency strategies. The histogram compares the final performance of models trained without con￾sistency loss, with uniform consistency loss, and with scheduled consistency loss. The data demonstrates a significant concentration of results toward the optimal metric value (R2 → 1) when physical constraints are applied, highlighting improve… view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: Simulated far-field performance via Posterior Sampling. Comparison of intensity distributions for metasurface designs generated over 1,000 sampling steps. Normalized guidance (qt ∝ 1/∥ · ∥2) is contrasted with non-normalized guidance (qt = 1). While both ap￾proaches outperform ancestral sampling, the normalized variant demonstrates superior scaling for high-dimensional metasurface optimization. However,… view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: Simulated far-field intensity profiles optimized via Monte Carlo Posterior Sampling (MC5). Responses correspond to metasurface geometries inversely designed over 1, 000 diffusion steps using an ensemble of N = 5. The panels contrast normalized guidance (qt ∝ 1/∥c − Sϕ(ˆx0|t )∥2) against non-normalized guidance (qt = 1). 4.11.4 Robust Posterior Sampling Building upon the MC posterior sampling framework [… view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Simulated far-field intensity profiles optimized via PS Robust5. Responses correspond to metasurface geometries inversely designed over 1, 000 diffusion steps using an ensemble of N = 5. The panels contrast normalized guidance (qt ∝ 1/∥c − Sϕ(ˆx0|t )∥2) against non-normalized guidance (qt = 1). Computational Complexity of Robust Guidance Like its Monte Carlo counterpart, Robust Posterior Sampling introd… view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: Impact of stochasticity on reconstruction fidelity. The R2 metric, evaluated after full-wave simulation of the SG-constrained designs, is plotted against the diversity coeffi￾cient α. The performance exhibits high resilience to noise injection, maintaining stability until the gradient-based guidance contributes less than one-third of the total update magnitude. Disk Gaussian Constraint Posterior Samplin… view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: Simulated far-field magnitude obtained from inverse-designed metasurface parame [PITH_FULL_IMAGE:figures/full_fig_p103_4_18.png] view at source ↗
Figure 4.19
Figure 4.19. Figure 4.19: R2 values for DSBs and DMs using amplitude-constrained posterior sampling [PITH_FULL_IMAGE:figures/full_fig_p104_4_19.png] view at source ↗
Figure 4.20
Figure 4.20. Figure 4.20: a) R2 values for DSBs using direction posterior sampling without normlized guid￾ance a) and with guidance b). 4.12.2 Performance evaluation under fixed sampling budgets This section evaluates the generative performance of DMs and DSBs under a standardized computational budget of 1000 sampling steps. As summarized in [PITH_FULL_IMAGE:figures/full_fig_p104_4_20.png] view at source ↗
Figure 4.21
Figure 4.21. Figure 4.21: Spectral analysis of the Hessian for various guidance computation schemes (Raw, [PITH_FULL_IMAGE:figures/full_fig_p107_4_21.png] view at source ↗
Figure 4.22
Figure 4.22. Figure 4.22: Hessian spectral analysis across various amplitude-constrained posterior sampling [PITH_FULL_IMAGE:figures/full_fig_p108_4_22.png] view at source ↗
Figure 4.23
Figure 4.23. Figure 4.23: Hessian spectral characterization for hybrid sampling configurations, integrating [PITH_FULL_IMAGE:figures/full_fig_p109_4_23.png] view at source ↗
Figure 4.24
Figure 4.24. Figure 4.24: Comparison of far-field magnitude R2 performance metrics for DM and DSB frame￾works across various directional guidance posterior sampling schemes. (a) Evaluates performance utilizing amplitude-normalized guidance, while (b) presents the corresponding results obtained without normalization, highlighting the sensitivity of the reconstruction fidelity to gradient scal￾ing [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 4.25
Figure 4.25. Figure 4.25: Comparative analysis of far-field magnitude [PITH_FULL_IMAGE:figures/full_fig_p110_4_25.png] view at source ↗
Figure 4.26
Figure 4.26. Figure 4.26: Comparative analysis of far-field magnitude [PITH_FULL_IMAGE:figures/full_fig_p111_4_26.png] view at source ↗
Figure 4.27
Figure 4.27. Figure 4.27: Comparative analysis of far-field magnitude [PITH_FULL_IMAGE:figures/full_fig_p112_4_27.png] view at source ↗
Figure 4.28
Figure 4.28. Figure 4.28: Representative randomly selected samples ( [PITH_FULL_IMAGE:figures/full_fig_p113_4_28.png] view at source ↗
Figure 4.29
Figure 4.29. Figure 4.29: Representative randomly selected samples ( [PITH_FULL_IMAGE:figures/full_fig_p114_4_29.png] view at source ↗
Figure 4.30
Figure 4.30. Figure 4.30: Comparison of ensemble-averaged far-field amplitude profiles. (a) Mean response [PITH_FULL_IMAGE:figures/full_fig_p114_4_30.png] view at source ↗
Figure 4.31
Figure 4.31. Figure 4.31: Far-field amplitude for two designs using the DM and DSB methods, trained on [PITH_FULL_IMAGE:figures/full_fig_p115_4_31.png] view at source ↗
Figure 4.32
Figure 4.32. Figure 4.32: Comparison of R2 values for the far-field magnitude across increasingly large meta￾surfaces, using DM and DSB trained on either the initial uniform or the enhanced database This chapter completed the arc of the manuscript by turning the validated forward models of the preceding chapters into a full inverse design pipeline. Three methodologies were benchmarked under a common FDTD-validated protocol: the … view at source ↗

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