HiLAB: A Hybrid Inverse-Design Framework

Abstract

HiLAB (Hybrid inverse-design with Latent-space learning, Adjoint-based partial optimizations, and Bayesian optimization), is a new paradigm for inverse design of nanophotonic structures. Combining early-terminated topological optimization (TO) with a Vision Transformer-based variational autoencoder (VAE) and a Bayesian search, HiLAB addresses multifunctional device design by generating diverse freeform configurations at reduced simulation costs. Shortened adjoint-driven TO runs, coupled with randomized physical parameters, produce robust initial structures. These structures are compressed into a compact latent space by the VAE, enabling Bayesian optimization to co-optimize geometry and physical hyperparameters. Crucially, the trained VAE can be reused for alternative objectives or constraints by adjusting only the acquisition function. Compared to conventional TO pipelines prone to local optima, HiLAB systematically explores near-global optima with considerably fewer electromagnetic simulations. Even after accounting for training overhead, the total number of full electromagnetic simulations decreases by an order of magnitude, accelerating the discovery of fabrication-friendly devices. Demonstrating its efficacy, HiLAB is used to design an achromatic beam deflector for red, green, and blue wavelengths, achieving balanced diffraction efficiencies of ∼25% while mitigating chromatic aberrations, a performance surpassing existing demonstrations. Overall, HiLAB provides a flexible platform for robust, multi-parameter photonic designs and rapid adaptation to next-generation nanophotonic challenges.

Keywords: Bayesian optimization; achromatic beam deflector; deep learning; inverse design; multi‐wavelength optics; topological optimization.

Figure 1

Overview of the demonstrated HiLAB pipeline for multi‐wavelength metasurface optimization. The metasurface consists of a patterned bilayer comprising SiO₂ and TiO₂, fabricated on top of a fused silica substrate. a) Multiple partial TO runs under varied physical parameters, including t ₁ (TiO₂ thickness), t ₂ (SiO₂ thickness), and the unit‐cell period Λ _y . The period along the x‐direction, Λ _x , is fixed at 5 µm to achieve a 41.3° deflection angle. b) Morphological data augmentation expands the library without additional EM simulations. c) VAE‐based dimensionality reduction (using a pre‐trained ViT) learns a compact design manifold. d) BO is performed jointly over the 8‐dimensional geometry latent space and physical design parameters—including (t ₁, t ₂, and Λ _y ) to identify metasurface layouts with optimal broadband performance. Each proposed parameter set is decoded into a structural design using the VAE decoder and evaluated through full‐wave FDTD simulations. The resulting deflection efficiencies at the three design wavelengths (470 nm, 550 nm, and 660 nm) are used to compute a scalar FoM, defined as the worst‐case efficiency across the spectrum: FoM = min {η₄₇₀, η₅₅₀, η₆₆₀}. This conservative formulation ensures robust spectral performance and serves as the objective function value returned to the Bayesian optimizer. The pipeline incorporates post‐decoding smoothing and binarization to ensure that final layouts comply with fabrication constraints such as minimum feature size and pattern co‐registration. A 2D Principal Component Analysis (PCA) projection of the 11‐dimensional sampled design vectors x is shown to visualize optimizer performance and convergence.

Figure 2

Evolution of a metasurface design across partial TO iterations 1, 18, 27, and 35. a) The refractive‐index profile progresses from near‐random initialization (iteration 1) to a predominantly binarized pattern (iteration 35). b) The corresponding deflection efficiencies at red (660 nm), green (550 nm), and blue (470 nm) wavelengths are plotted as functions of outgoing angle. The target (design) deflection angle is 41.3° for all wavelengths, and each efficiency peak shifts progressively closer to this common angle over successive iterations. c) Electric‐field distributions in the xz‐plane for the partially optimized structure (iteration 35) are shown for the three wavelengths. The black arrow highlights the deflected wavefront moving toward the desired output angle.

Figure 3

Validation of the ViT‐based VAE for dimensionality reduction and reconstruction of partially optimized metasurface designs. Columns correspond to five representative device layouts. Row (a) shows the original topologically optimized structures (Original 1–5) used as input. Row (b) displays the learned latent‐space representations (Latent Space 1–5), where each 8‐dimensional vector is projected onto a 2 × 4 grid and visualized as a color map. Row (c) presents the VAE‐decoded outputs (Reconstructed 1–5), which closely match their respective inputs. Row (d) shows the pixel‐wise reconstruction errors (Reconstruction Error 1–5), computed as the absolute difference between the original and reconstructed patterns: |Original − Reconstructed|, are near zero across most regions, with noticeable deviations only along the edges of the freeform structures. The minimal error values and close structural correspondence confirm that the VAE effectively captures key freeform features while reducing the input dimensionality by over 4000‐fold (256 × 128 to 8).

Figure 4

Visualization of how individual elements of the latent vectors influence the generated metasurface patterns. a) A randomly selected 8‐dimensional latent vector [2.0, 0.0, 2.5, − 1.5, − 0.1, 0.5, 1.1, − 1.4] is decoded using the trained model to produce a binary pattern. b) Schematic highlighting which latent component (from 1 to 8, red square) is being varied in each row. c) Binarized outputs generated by sweeping the selected latent component from −2.0 to +2.0, while all other seven component remain fixed. Thus, each row corresponds to a unique latent component and each column to a specific value along that component. The distinct structural variations confirm that the learned latent space is interpretable and disentangled, with different elements controlling different pattern characteristics.

Figure 5

2D PCA + Kernel Density Estimation (KDE) visualization of the 11‐dimensional design space x, defined in Equation (11)(composed of 8 latent variables and 3 physical parameters (t ₁, t ₂, and Λ _y )). a) TO results from 70 independent runs, each with 200 iterations. For each run, the physical parameters were randomly sampled to generate diverse initial conditions, and the final structure was encoded using the trained VAE to extract the 8 latent features. These were then combined with the physical parameters to construct the full design vector x for PCA–KDE visualization. The best FoM was 0.188 (minimum efficiency of 18.8% between three wavelengths). b–f) Snapshots of our HiLAB framework at 70‐iteration intervals (0–70, 70–140, 140–210, 210–280, 280–350). Despite requiring only 1400 simulations in total, HiLAB discovers compact, high‐performing clusters with a peak FoM of 0.247(minimum efficiency of 24.7% between three wavelengths). KDE contours in each panel reflect the density of sampled designs in the projected PCA space: while HiLAB progressively concentrates sampling in promising regions, the broader, more diffuse KDE distribution in panel (a) reflects the randomness and lack of convergence in the baseline TO with randomized physical conditions.

Figure 6

a) Projection of all evaluated 11D design vectors x onto a 2D PCA space. Each point represents a specific combination of latent geometry parameters and physical hyperparameters, color‐coded according to the minimum simulated FoM across wavelengths (660, 550, and 470 nm) up to that iteration. The clustering of the FoM data clearly indicates the systematic exploration of the optimizer toward high‐performance regions. A representative optimal design (θ*) is highlighted by the black arrow, with physical parameters: period Λ _y = 500 nm, TiO₂ thickness = 196 nm, and SiO₂ thickness = 400 nm. b) Refractive‐index profile of the optimized metasurface unit cell corresponding to design θ*. c) Simulated deflection efficiencies for red (660 nm), green (550 nm), and blue (470 nm) wavelengths, demonstrating the strong multi‐wavelength achromatic deflection capability of the device at the target 41.3° angle with an unprecedented efficiency and almost equal efficiency among the three colors.

Figure 7

a) Refractive‐index profile of the optimized metasurface unit cell. b) SEM of the fabricated metasurface deflector. The inset (red box) highlights the subwavelength corrugations critical for broadband operation. Scale bars: 5 µm (main image) and 1 µm (inset). c) Experimentally measured deflection efficiency as a function of output angle for three design wavelengths: 470 nm (blue), 550 nm (green), and 660 nm (red). The prominent efficiency peaks near θ = 41.3° confirm the metasurface's broadband and angle‐specific beam deflection performance. The measured efficiencies agree well with theoretical simulations (25.0% for red, 24.2% for green, and 24.1% for blue). d) Schematic of the broadband optical characterization setup. A collimated beam from a broadband source passes through a linear polarizer and imaging optics, illuminates the object and metasurface, and is then focused onto a CCD detector. The system enables angle‐resolved measurements, with the detection angle θ = 41.3° indicated. e) Optical characterization using a USAF 1951 resolution target under illumination at the same three wavelengths and their combinations. Individual and combined RGB (red, green, blue) channel images validate achromatic beam deflection across the visible spectrum.