Design of optical meta-structures with applications to beam engineering using deep learning

The paper addresses the inverse design of nanophotonic metasurfaces for on-chip beam engineering, a problem characterized by a high-dimensional design space where conventional analytical or numerical optimization can be inefficient and prone to local minima. Metasurfaces composed of sub-wavelength scatterers (meta-atoms) enable precise control over phase, polarization, mode, and spectrum to realize functions such as grating-based coupling, Bragg gratings, and flat metalenses. The authors propose a machine learning approach—specifically deep neural networks—to learn the mapping between metasurface geometric parameters and their electromagnetic (EM) diffraction responses (forward model), and to invert this mapping to directly infer design parameters from a desired diffraction profile (inverse model). The goal is to efficiently design metasurface gratings that engineer out-of-plane beam profiles (Gaussian, focused, collimated) for integrated optofluidic sensing and other on-chip spectroscopic applications, improving scalability, compactness, and potentially enabling high-efficiency grating couplers and LiDAR-like sources.

The Related work section surveys the use of neural networks in photonics and metasurface design: applications of ANNs to RF/microwave design (Zhang et al.), prediction of 2D photonic crystal dispersion using MLPs (Malheiros-Silveira et al.), hybrid EM optimization with ML for permittivity prediction (Ferreira et al.), simulator-based training of generative models for metasurface inverse design (Jiang & Fan), and DNNs for exploring large nanophotonic parameter spaces (Yao et al.). GAN-based metasurface thermal emitter optimization (Kudyshev et al.), DNN-based inverse optimization and equivalent EM solvers for dielectric metasurfaces (Liu et al.) are cited. Additional works include metasurface-enhanced waveguide couplers, out-of-plane waveguide holography, and chip-integrated geometric metasurfaces for directional coupling and polarization sorting. This positions the present work among data-driven inverse design methods, emphasizing a supervised DNN/CNN approach targeted at beam-shaping gratings.

Device and forward model: The metasurface consists of a 5×5 array of meta-scatterers located at the end of a planar silicon nitride (SiN) waveguide on SiO2-on-Si. The waveguide is 400 nm thick and 800 nm wide, tapering to a planar metasurface region via a 3 µm-wide, 5 µm-long taper. The structure supports fundamental TE/TM modes near C/L bands. The metasurface diffracts light out of plane; diffraction is modeled using grating theory and Huygens-Fresnel principle. Space-harmonic fields have propagation constants k_xn = β_0 + 2nπ/λ_x + iα, with diffraction angle θ_n = sin^-1(β_n/k_0). Conditions are given to ensure out-of-plane diffraction and avoid higher orders. Leakage factor α depends on scatterer geometry and a gap factor g via α = α1(w,h,g)(ε_wg−ε_a)^2 sin^2(πg). Design parameters: Primary parameters are grating period λ_x (controls diffraction angle), gap factor g (controls leakage via sin^2 term), etch depth/height h of scatterers (controls leakage behavior), and row-wise scatterer widths C_i in the y-direction (i=1..5) which modulate effective permittivity and local leakage to shape the beam. The scatterer width in x-direction d is also considered. Inverse problem: Define forward operator Ψ(λ_x, g, h, C1..C5, d) → I_p(x,y,z) that maps design parameters to the free-space diffraction profile. The inverse operator Ψ^{-1} maps a target diffraction profile to design parameters. The task is to learn Ψ^{-1} using neural networks. Neural network architectures: Two approaches are evaluated. (1) Feedforward DNN: A fully connected network taking the diffraction image (flattened 2D array) as input and outputting 8 continuous parameters [λ_x, g (or d as presented), h, C1..C5]. A 4-layer architecture with ReLU activations performed best among tested depths. Loss: mean squared error (MSE). Optimizer: stochastic gradient descent (SGD). (2) CNN: A shallow CNN with a single convolutional layer (kernel size 5, stride 1, padding 2), ReLU, max pooling, followed by three fully connected layers to output the 8 parameters. Data generation and simulation: Training data are generated using 3D FDTD simulations (Lumerical FDTD). The platform is a 400 nm SiN waveguide on 3 µm SiO2 with air cladding; refractive indices at λ=1550 nm are n_SiN=1.91 and n_SiO2=1.414. Parameter ranges: λ_x ∈ [0.3, 1.4] µm, d ∈ [50, 1200] nm, h ∈ [100, 400] nm, C_i ∈ [50, 480] nm. Approximately 4000 samples were generated; the input wavelength was fixed at 1.5 µm. Diffraction profiles (top-view) are saved as images labeled with ground-truth parameters. Training procedure and hyperparameters: Implemented in PyTorch. Batch size 125, learning rate 0.06, 1000 iterations. ReLU activations in hidden layers. Loss: MSE. Increasing learning rate caused training jitters; beyond 1000 iterations, losses plateaued. Depth study: 3-layer, 4-layer, and 5-layer DNNs were compared; 4-layer gave lowest test loss, 5-layer showed signs of overfitting. Validation via tandem pipeline: A DNN validator network is formed by cascading the learned inverse model with the FDTD forward model to synthesize beams from predicted parameters and compare with target profiles using correlation coefficients.

- The 4-layer DNN achieved the best inverse-design performance with a test loss of ~0.012 per sample after 1000 iterations, outperforming the 3-layer DNN (~0.030) and the 5-layer DNN (~0.021). - The best CNN (1 conv + 3 FC) attained a higher test error (~0.0170 per sample) than the DNN, indicating inferior parameter estimation for this dataset/architecture. - Per-parameter errors show the inverse model predicts λ_x, d (or g), and h with low error (<0.005), while row-wise scatterer widths C_i exhibit higher errors (~0.025), reflecting the ill-posedness (multiple C_i combinations can yield similar diffraction profiles). - Tandem validation (inverse DNN + FDTD) reproduces target beams with high correlation: Gaussian ~0.986; focused ~0.925; collimated reported as 0.890 in the figure and 0.925 in text; random ~0.996. The highest correlation reached 0.996. - 3 dB operation bandwidths (example designs): collimated ~30 nm, focused ~24 nm, Gaussian ~65 nm, random ~70 nm. Beam profiles with more extreme spatial distributions tend to have smaller bandwidths. - The ML inverse design rapidly estimates metasurface parameters from desired diffraction profiles, offering a time-efficient alternative to conventional iterative optimization methods.

The study demonstrates that supervised deep learning can effectively learn an inverse mapping from desired diffraction profiles to metasurface design parameters for beam engineering on an integrated SiN platform. The superior performance of the 4-layer DNN over deeper (5-layer) and CNN architectures suggests that a moderately deep, fully connected model generalizes well for the given dataset size and parameterization, whereas excessive depth risks overfitting. Low errors for λ_x, d/g, and h indicate these parameters predominantly control the diffraction characteristics, while higher errors for C_i confirm the inverse problem’s ill-posedness—many scatterer-width configurations can realize similar beam shapes. The tandem validator corroborates that predicted parameters yield beam profiles closely matching targets (up to 0.996 correlation), though focused and collimated beams are somewhat less accurate than Gaussian or random profiles. Bandwidth analysis reveals a trade-off between beam profile extremity and operational bandwidth, and suggests that adding design degrees of freedom (e.g., grating apodization) could improve bandwidth. Overall, the approach addresses the research goal of accelerating metasurface inverse design, providing accurate, rapid parameter estimation compared to traditional iterative EM optimization.

The work presents an inverse-design framework for dielectric metasurface gratings that engineer out-of-plane beams (Gaussian, focused, collimated) on a SiN photonic platform. Using FDTD-generated data and deep learning models, the authors show that a 4-layer DNN accurately infers grating period, gap/width, height, and row-wise scatterer sizes from diffraction images, outperforming a CNN baseline. The learned inverse model, validated via an FDTD tandem pipeline, reproduces target beams with high correlation and reveals bandwidth trends across beam types. The approach is general and can be extended to more complex, higher-dimensional photonic structures and to apodized gratings for improved coupler efficiency. Future directions include enlarging the metasurface (more scatterers in x and y), expanding training data, exploring richer parameterizations (e.g., apodized periods), and leveraging advanced architectures or physics-informed learning to further enhance accuracy and bandwidth.

- Inverse prediction of row-wise scatterer widths C_i is less accurate due to the ill-posed nature of the problem (non-unique solutions producing similar diffraction profiles). - The best-performing DNN shows signs of overfitting when depth increases (5-layer), indicating sensitivity to model capacity vs. dataset size (~4000 samples). - CNN architectures underperform DNN for this dataset; performance may be constrained by the shallow CNN used. - The metasurface is limited to a 5×5 scatterer grid, which may restrict beam-shaping fidelity; authors note performance could improve with more scatterers. - Operational bandwidth is smaller for certain engineered beams (e.g., focused, collimated), reflecting design trade-offs. - Training and validation use a fixed illumination wavelength (1.5 µm) and specific material indices; generalization across broader spectral/material variations was not demonstrated. - The approach relies on FDTD-generated data; accuracy and speed depend on simulation fidelity and coverage of the parameter space.