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Machine-Learning-Assisted Metasurface Design for High-Efficiency Thermal Emitter Optimization

Zhaxylyk A. Kudyshev, Alexander V. Kildishev, Vladimir M. Shalaev, Alexandra Boltasseva

arXiv:1910.12741v2physics.opticsphysics.comp-ph

TL;DR

The paper addresses the computational cost and limited scalability of topology optimization for highly constrained, high-dimensional nanophotonic design. It couples topology optimization with adversarial autoencoders to search a compact latent representation while controlling its distribution. The resulting method reports approximately 4900-times faster optimization searches and thermal-emitter reshaping efficiencies up to 98%.

  • Problem

    Topology optimization requires substantial computational resources, limiting its applicability to highly constrained problems in high-dimensional parametric spaces.

  • Method

    The approach couples a topology-optimization engine with adversarial autoencoders to perform optimization in a compressed latent design space.

  • Results

    Thermal-emitter designs achieve thermal reshaping efficiencies up to 98%, while the AAE+VGGnet approach demonstrates approximately 4900-times faster optimization search than conventional direct topology optimization.

  • Takeaways & Limitations

    Compact hyperparametric representations with controlled latent-space configurations support efficient optimization over high-dimensional parametric landscapes.

Abstract

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With the emergence of new photonic and plasmonic materials with optimized properties as well as advanced nanofabrication techniques, nanophotonic devices are now capable of providing solutions to global challenges in energy conversion, information technologies, chemical/biological sensing, space exploration, quantum computing, and secure communication. Addressing grand challenges poses inherently complex, multi-disciplinary problems with a manifold of stringent constraints in conjunction with the required system's performance. Conventional optimization techniques have long been utilized as powerful tools to address multi-constrained design tasks. One example is so-called topology optimization that has emerged as a highly successful architect for the advanced design of non-intuitive photonic structures. Despite many advantages, this technique requires substantial computational resources and thus has very limited applicability to highly constrained optimization problems within high-dimensions parametric space. In our approach, we merge the topology optimization method with machine learning algorithms such as adversarial autoencoders and show substantial improvement of the optimization process by providing unparalleled control of the compact design space representations. By enabling efficient, global optimization searches within complex landscapes, the proposed compact hyperparametric representations could become crucial for multi-constrained problems. The proposed approach could enable a much broader scope of the optimal designs and data-driven materials synthesis that goes beyond photonic and optoelectronic applications.

I. INTRODUCTION

Practical nanophotonic design requires searching high-dimensional spaces under multiple stringent constraints, but conventional topology optimization is computationally expensive and sensitive to initialization. The paper combines topology optimization with adversarial autoencoders to compress the design space, accelerate searches, and optimize a thermal emitter.

  • Practical optical structures must satisfy multiple stringent constraints spanning materials, scalability, experimental tolerances, and optical performance.
  • High-dimensional, multi-objective design searches are resource-heavy, while human intuition limits perception of multivariate optimization models.
  • Topology optimization locally depends on the initial material distribution, so inadequate initial geometries require multiple runs to identify strong solutions.
  • The computational expense of topology optimization limits its use for multi-constrained problems requiring larger parameter domains with mechanical, chemical, and optical properties.
  • The proposed adversarial-autoencoder and topology-optimization approach compresses the design space, controls latent-space distributions, and supports rapid nanophotonic optimization.
  • For a thermophotovoltaic metasurface thermal emitter, the method achieves 98% efficiency versus 92% for an adjoint-based topology-optimized design and provides three times speedup.

II. ADVERSARIAL AUTOENCODERS FOR DESIGN OPTIMIZATION

The paper couples topology optimization with adversarial autoencoders to create a compact, continuous design space for generating and refining nanophotonic structures. It demonstrates the approach on a TiN/Si3N4 thermal emitter for thermophotovoltaic applications, where conventional designs face spectral and optimization constraints.

  • AAE-assisted optimization: The encoder compresses nanoantenna patterns into a compact, continuous latent space, while the discriminator enforces a predefined latent distribution.The decoder reconstructs designs from latent coordinates, enabling the latent representation to serve as a generative design space.
  • AAE-assisted optimization: AAE-assisted optimization combines data generation with adjoint topology optimization, AAE training, and subsequent structure refinement.The trained decoder acts as a generator that maps latent vectors to candidate designs, which are then filtered for instability and low efficiency.
  • AAE-assisted optimization: Continuous latent representations support interpolation across high-dimensional parametric spaces and generate broader design variation.The AAE also permits flexible control of latent-space configuration and applies adversarial learning to compressed representations rather than directly to patterns.
  • Thermal-emitter application: Thermophotovoltaic emitters must maximize radiation within the GaSb working band while minimizing out-of-band radiation that heats the PV cell.The ideal emissivity profile is approximately a step function over the photovoltaic working band and zero elsewhere.
  • Thermal-emitter application: The TiN/Si3N4 platform is selected because transition-metal nitrides retain plasmonic properties while remaining stable at very high temperatures.The study uses TiN and Si3N4 for metasurface thermal-emitter designs in TPV applications.
  • Thermal-emitter application: Simple cylindrical gap-plasmon emitters offer intuitive fabrication but restrict optimization degrees of freedom and reach only 84% mean in-band emissivity.The cylindrical reference uses a back reflector, dielectric spacer, and top resonator array, illustrating the trade-off between simplicity and spectral performance.

IV. TO-GENERATED TRAINING SET AND LATENT SPACE ENGINEERING

The training set is generated with fabrication-aware topology optimization, then used to engineer and sample a latent design space. The resulting nontrivial TO structures improve in-band thermal-emitter performance over the cylindrical reference while preserving out-of-band suppression.

  • Training-set generation: Adjoint topology optimization uses forward and adjoint full-wave simulations to update the material distribution toward higher figure of merit.The resulting heat map determines dielectric-function perturbations, and repeated iterations converge the optimization region to a binary air/TiN structure.
  • Training-set generation: The training structures use a TiN back reflector, Si3N4 spacer, and patterned TiN layer in a 280×280 nm^2 unit cell.The top TiN layer is the optimization region, and the domain is discretized into 60 × 60 optimization elements.
  • Training-set generation: The training set retains designs with more than 85% average in-band absorption and less than 40% mean out-of-band absorption.The figure of merit weights in-band absorption and out-of-band reflectivity relative to an ideal emitter spectrum.
  • Performance comparison: The best TO emitter reaches 92% normalized efficiency versus 83% for the cylindrical emitter.The TO designs provide more uniform, higher in-band absorption while maintaining a rapidly decaying out-of-band spectral tail.
  • Performance comparison: At 1800 °C, out-of-band emission equals 32% of blackbody radiation for the TO design and 29% for the cylindrical emitter.The TO design’s denser population of in-band modes enables higher in-band emission, while both designs substantially suppress out-of-band emission.

V. AAE-OPTIMIZED THERMAL EMITTER AND LATENT SPACE ENGINEERING

The AAE compresses topology-optimized thermal-emitter designs into a 15-dimensional latent space, generates new structures, and supports TO refinement or VGGnet filtering. For the thermal-emitter task, AAE-assisted designs improve efficiency while substantially reducing optimization time and enabling controlled latent-space sampling.

  • AAE design generation: The encoder compresses 64 × 64 topology-optimized patterns into a 15-dimensional latent-space vector, which the decoder uses to generate designs.The training set is expanded to 8400 samples using lateral translations and 90-degree rotations before AAE training.
  • AAE design generation: TO refinement binarizes generated structures, removes sub-30 nm features, and preserves the direct-TO efficiency constraints.The constraints require at least 85% of mean in-band absorption and less than 40% out-of-band absorption.
  • Thermal-emitter optimization: 90% mean efficiency was obtained for 200 AAE designs, compared with 82% for 200 direct-TO designs; the best designs reached 98% and 92%, respectively.The best AAE design exhibits near-unit in-band absorption and a comparable decaying out-of-band tail.
  • Thermal-emitter optimization: The AAE design enabled 98% in-band emission while suppressing out-of-band radiation to 30% of blackbody radiation at 1800 °C.The comparison includes direct-TO and trivial emitters, whose out-of-band absorption spectra have similar decaying tails.
  • Optimization search efficiency: Generating 100 designs required 164 hours with direct TO, 54 hours with AAE+TO, and 2 minutes with AAE+VGGnet, making the latter more than 4900 times faster than direct TO.Average optimization time was 1.64 hours per direct-TO design and 31 minutes per AAE-based design.
  • Latent space engineering: Latent-space engineering maps predefined distributions, including a 3 Gaussian mixture and Swiss roll, into the compact design representation.The paper presents this controlled distribution mapping as a route toward global optimization in high-dimensional spaces.

VI. CONCLUSION

The conclusion presents a topology-optimization and adversarial-autoencoder combination for faster searches in high-dimensional, multiconstrained photonic design spaces. In a thermal-emitter metasurface, AAE+TO reaches up to 98% efficiency, while AAE+VGGnet provides further acceleration.

  • VI. CONCLUSION: The method merges adjoint-based topology optimization with an adversarial autoencoder to accelerate searches and control latent-space configuration.The authors identify latent-space control as important for optimization over high-dimensional parametric landscapes.
  • VI. CONCLUSION: The approach targets multiconstrained, multifunctional photonic devices and is demonstrated on a TiN/Si3N4 thermal-emitter metasurface for high-efficiency emission reshaping.The demonstrated device is framed as a thermal-emitter design problem.
  • VI. CONCLUSION: AAE+TO emitter designs achieve thermal-emission reshaping efficiencies up to 98%.The reported result concerns the thermal-emitter optimization demonstration.
  • VI. CONCLUSION: AAE-assisted optimization provides approximately 4900-times-faster searches than conventional direct topology optimization.The conclusion reports the speedup as part of the method's optimization-search improvement.
  • VI. CONCLUSION: The proposed method is presented as applicable to optical design and data-driven materials synthesis across photonics, optoelectronics, MEMS, and biomedical applications.The stated scope extends beyond the demonstrated thermal-emitter example.

SUPPLEMENTARY MATERIAL

The supplementary material contains technical details supporting the paper’s optimization and machine-learning methods, material models, training procedures, and additional designs.

  • SUPPLEMENTARY MATERIAL: Supplementary information covers dielectric permittivity functions for TiN and Si3N4.These material functions support the thermal-emitter optimization setup.
  • SUPPLEMENTARY MATERIAL: The supplementary material documents the topology-optimization framework and adversarial-autoencoder structure.These details concern the principal computational design methods.
  • SUPPLEMENTARY MATERIAL: Additional supplementary data describe the training process, data augmentation procedure, and AAE-optimized designs.These materials support the dataset and generated-design components of the workflow.
  • SUPPLEMENTARY MATERIAL: The structure of the VGGnet is provided in the supplementary material.This supports the CNN-based filtering approach.

S1. DIELECTRIC PERMITTIVITY FUNCTIONS OF TiN AND Si3N4

The supplementary material describes temperature-dependent dielectric-permittivity retrieval for TiN and Si3N4, whose functions are used in the optimization and correspond to a 1000 C temperature response.

  • S1. DIELECTRIC PERMITTIVITY FUNCTIONS OF TiN AND Si3N4: Dielectric permittivity functions were obtained with a custom-built heating stage integrated into a spectroscopic ellipsometer setup.Further details on the experimental setup and permittivity retrieval process are referenced separately.
  • S1. DIELECTRIC PERMITTIVITY FUNCTIONS OF TiN AND Si3N4: The optimization uses dielectric functions for TiN and Si3N4 corresponding to a 1000 C temperature response.The associated material-permittivity plots are shown in Figure S1.

S2. TOPOLOGY OPTIMIZATION FRAMEWORK

The framework combines adjoint-based topology optimization with filtering, spectrally weighted FOM gradients, robustness projection, and iterative material updates for TiN/air structures.

  • Optimization workflow: Adjoint-based topology optimization is coupled to a commercial FDTD solver for photonic-structure design.The implementation uses Matlab as an interface to Lumerical FDTD.
  • Optimization workflow: The optimization loop consists of filtering, FOM-gradient calculation, and material-distribution updating.These steps are repeated for 50 iterations, with filtering applied at every tenth iteration.
  • Material representation: The TiN/air material distribution is represented by a continuous nonlinear interpolation between air and TiN permittivities.The material component is selected through the index j in the interpolation scheme.
  • Objective and gradients: The FOM is a weighted average of in-band absorption and out-of-band reflectivity, spectrally weighted against an ideal step-function emitter.Adjoint fields provide the wavelength-dependent FOM gradient used for updates.
  • Robustness control: Robustness is controlled with dilated, intermediate, and eroded threshold projections that progressively drive the material distribution toward binary composition.The threshold sharpness parameter β is increased during optimization.
  • Objective and gradients: Forward and adjoint simulations generate gradient profiles for the three perturbed design types, which are combined into a total gradient before updating the base pattern.The final design is saved after meeting the main optimization constraints.

S3. EFFICIENCY CALCULATION

Emitter efficiency is calculated from spectral emissivity and black-body radiance over the photovoltaic cell’s wavelength band, while accounting for corrected out-of-band performance.

  • Radiance calculation: The emitter’s radiance is obtained by multiplying spectral emissivity by black-body spectral radiance at temperature T.The black-body term uses frequency, Planck’s constant, Boltzmann’s constant, light speed, and temperature.
  • Radiance calculation: Radiance within the wavelength interval [λ_min, λ_max] is integrated over wavelength to quantify emitted power.The interval is defined by the lower and upper bounds of the PV cell’s spectral response.
  • Efficiency metric: The ideal target is a step-function emissivity spectrum, with in-band performance shaped by the top-layer antenna’s resonant response.The back reflector controls the long-wavelength response of the gap-plasmon structure.
  • Efficiency metric: Thermal-emitter efficiency is defined as the product of in-band efficiency and corrected out-of-band efficiency.The correction accounts for the long-wavelength response determined by the back-reflector material.

S4. ADVERSARIAL AUTOENCODER FOR DESIGN PRODUCTION

The adversarial autoencoder generates compact resonant-design representations, while symmetry- and translation-based augmentation expands the training data and its variance can affect generated designs.

  • AAE architecture: The AAE contains coupled encoder, decoder/generator, and discriminator networks that learn latent representations and generate design patterns.The encoder maps 64 × 64 binary patterns to a 15-dimensional latent vector, and the decoder reconstructs 4096-element outputs.
  • Training-data variance: With 10% variance, the AAE generated only 10 robust designs above 80% efficiency within one hour instead of the targeted 200.This condition is the clearest failure to reach the prescribed robust, high-efficiency sample count.
  • Data augmentation: Up to 8400 resonant patterns are produced by applying 20 random translations to each design together with its rotated counterpart.The rotation augmentation relies on freedom in choosing the primary polarization direction.
  • Training-data variance: Lower training-set variance narrows generated-efficiency distributions and biases designs toward particular shapes and efficiencies.Because the source designs have efficiencies between 80% and 90%, generated efficiencies localize around that region; the 10% case favors two shapes.
  • AAE training: AAE training first minimizes encoder-decoder reconstruction error, then adversarially matches encoded latent vectors to a predefined distribution.The discriminator classifies encoded and random inputs, and its feedback updates the encoder.

S5. AAE-OPTIMIZED DESIGNS OF HIGH-EFFICIENCY THERMAL EMITTERS

AAE-based generation is evaluated against GAN generation for topology-optimized thermal-emitter designs, using VGGnet to assess robustness and efficiency.

  • AAE-optimized designs: The top 55 AAE-optimized resonant patterns use unit cells of 280 × 280 nm^2.These patterns are presented as design profiles in the supplementary figure.
  • AAE rationale: AAE selection is motivated by dense latent-space distribution and control over latent-space configuration.These properties are presented as advantages over alternative generative architectures including VAE and GAN.
  • AAE–GAN comparison: The comparison evaluates GAN and AAE networks trained on the topology-optimized design set used in the main text.Both generated designs are coupled with VGGnet-based robustness and efficiency assessment.
  • AAE–GAN comparison: 94% efficiency is achieved by the best AAE+VGGnet generated design, compared with 92% for the best GAN design and 92% for the best training design.The corresponding emissivity spectra are shown for the best GAN and AAE designs.
  • AAE–GAN comparison: 4 min is required for AAE to generate 200 designs above 80% efficiency and predefined robustness, versus 29 min for GAN.The reported comparison supports better AAE performance in best-design efficiency and generation time.
  • VGGnet assessment: AAE and GAN outputs are assessed with VGGnet models that classify robustness and regress efficiency from 64 × 64 design images.The two models share a convolutional architecture but use different final-layer activation and loss functions.

S9. AAE OPTIMIZATION FOR DIFFERENT UNIT CELL DIMENSIONS

The AAE+VGGnet approach was evaluated for 250 nm and 300 nm unit cells using augmented topology-optimization datasets and generated-design filtering. It produced higher-efficiency designs and generated 200 designs faster than direct topology optimization.

  • Experimental setup: The AAE was trained on topology-optimization datasets for 250 nm and 300 nm unit-cell sizes, with 150 optimized designs per size.The same optimization scheme and parameters were used for each unit-cell size.
  • Experimental setup: Each unit-cell dataset was augmented to 8400 designs, and a pretrained VGGnet filtered robust designs with efficiency above 80%.
  • Results: For both unit-cell sizes, AAE+VGGnet generated design sets with higher mean efficiency and better best-design performance than direct topology optimization.
  • Results: 95.75% versus 94.5% was achieved for the best 250 nm designs using AAE+VGGnet and direct topology optimization, respectively.
  • Results: 95% versus 92% was achieved for the best 300 nm designs using AAE+VGGnet and direct topology optimization, respectively.
  • Results: 200 highly efficient designs were generated in 4 min with AAE+VGGnet, whereas direct topology optimization required 328 min.
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