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Optimization of photonic crystal nanocavities based on deep learning

Takashi Asano, Susumu Noda

arXiv:1808.05722v1physics.comp-phphysics.app-ph

TL;DR

Optimizing 2D-PC nanocavities with many structural degrees of freedom requires fast gradient evaluation, which direct calculations make difficult. The paper trains a convolutional neural network on simulated cavities to predict Q-factor gradients and obtains a nanocavity with QFDTD = 1.58 × 10^9.

  • Problem

    Optimizing nanocavity structures with large degrees of freedom requires selecting relevant parameters and repeatedly evaluating gradients quickly, which is difficult for direct optimization methods.

  • Method

    The authors train a four-layer neural network with a convolutional layer on 1000 simulated nanocavities to learn the relationship between air-hole displacements and Q factors, then use its predicted gradients for optimization.

  • Results

    QFDTD = 1.58 × 10^9 was obtained, exceeding the manually optimized two-step heterostructure nanocavity by one order of magnitude and the previously reported highest value by more than twice.

  • Takeaways & Limitations

    The method supports high-dimensional 2D-PC nanocavity optimization through rapid gradient evaluation while retaining high Q factors under constrained displacement magnitudes.

Abstract

from arXiv · show

An approach to optimizing the Q factors of two-dimensional photonic crystal (2D-PC) nanocavities based on deep learning is proposed and demonstrated. We prepare a dataset consisting of 1000 nanocavities generated by randomly displacing the positions of many air holes of a base nanocavity and their Q factors calculated by a first-principle method. We train a four-layer neural network including a convolutional layer to recognize the relationship between the air holes' displacements and the Q factors using the prepared dataset. After the training, the neural network becomes able to estimate the Q factors from the air holes' displacements with an error of 13% in standard deviation. Crucially, the trained neural network can estimate the gradient of the Q factor with respect to the air holes' displacements very quickly based on back-propagation. A nanocavity structure with an extremely high Q factor of 1.58 x 10^9 is successfully obtained by optimizing the positions of 50 air holes over ~10^6 iterations, having taken advantage of the very fast evaluation of the gradient in high-dimensional parameter space. The obtained Q factor is more than one order of magnitude higher than that of the base cavity and more than twice that of the highest Q factors reported so far for cavities with similar modal volumes. This approach can optimize 2D-PC structures over a parameter space of a size unfeasibly large for previous optimization methods based solely on direct calculations. We believe this approach is also useful for improving other optical characteristics.

1. INTRODUCTION

The paper addresses the difficulty of optimizing many structural degrees of freedom when fast gradient evaluation is required. It proposes a deep-learning approach that learns the relationship between air-hole displacements and nanocavity Q factors, then uses predicted gradients for repeated optimization.

  • Higher Q factors increase photon storage and light–matter interaction time, while smaller modal volumes strengthen interaction and reduce device footprint.
  • Existing approaches include Gaussian-envelope designs, analytic inverse methods, genetic algorithms, and leaky-position visualization, but unused design freedoms remain difficult to exploit fully.
  • The proposed method trains a neural network on randomly displaced nanocavities and their first-principles Q factors to learn the mapping from air-hole displacement patterns to Q factors.
  • Fast gradient evaluation is required for repeated optimization when structures have many degrees of freedom, whereas direct calculation becomes difficult in such settings.
  • A convolutional neural network is used because convolutional layers can learn spatial features from fixed-size structural inputs.
  • The framework selects structural parameters, generates randomized structures, calculates Q factors in parallel, trains a neural network, and repeatedly updates parameters using predicted gradients.

3. RESULTS

The study uses a manually tuned two-step heterostructure nanocavity as its base structure for further optimization. The base cavity combines a silicon slab, patterned air holes, lattice modulation, and a small modal volume.

  • The base cavity is a two-step heterostructure in a 220 nm silicon slab with 110 nm air holes and a 410 nm lattice constant.
  • Eight central air holes were manually shifted by approximately a/1000 using leaky-position visualization, increasing Q from 50 million to 140 million while leaving Vcav near 1.3 (λ/n)^3.
  • The lattice constants near the cavity center are modulated by 3 nm in two x-direction steps to confine light through the mode-gap effect.
  • The base structure has a theoretical QFDTD of 1.37×10^8 and a modal volume of 1.3 cubic wavelengths in material.

A. Preparation of Dataset for Learning

The dataset contains randomly perturbed versions of the base cavity whose Q factors are calculated by 3D-FDTD. Because the raw Q distribution is highly nonuniform, the study trains on log10(QFDTD).

  • 1000 nanocavities were generated by randomly displacing air-hole positions while preserving structural symmetry.
  • 3D-FDTD calculated the fundamental-mode electromagnetic fields and QFDTD values for every generated structure.
  • QFDTD spans nearly two orders of magnitude, from approximately 10^6 to above 3×10^8, but is concentrated mainly below approximately 3×10^7.
  • The study transforms QFDTD to log10(QFDTD) because the transformed distribution is more uniform and easier for the neural network to learn.
  • The random displacement distribution has a standard deviation of 1/1000 lattice constant.

B. Configuration of Neural Network

The neural network maps spatial air-hole displacement patterns to predicted Q factors. Its convolutional front end extracts local spatial features before fully connected layers produce the output.

  • The input consists of normalized x- and y-displacement vectors for air holes in a rectangular region around the nanocavity center.
  • The first layer is convolutional, applying local filters to displacement inputs to create spatial feature maps.
  • The network uses 50 filters of size 3×5 across two displacement channels, with strides of 1 in x and 2 in y.
  • Fully connected layers with 200 and 50 units use ReLU, affine transformations, and dropout before producing one output unit.
  • The output unit predicts log10(QFDTD) from the air-hole displacement pattern.

1. Loss function

The neural network is trained with a loss function combining prediction error and weight decay, while generalization is evaluated on held-out data. The prediction-error metric expresses the typical multiplicative deviation between predicted and calculated Q factors.

  • Loss function: 900 structures are used for training and 100 are held out as test data to evaluate generalization and reduce overfitting risk.The test set checks whether the network predicts meaningful answers for unseen inputs.
  • Loss function: The loss combines deviation from log10(QFDTD) with a weight-penalty term, using λ = 0.001 to discourage large connection weights.The weight-decay term is optimized through stochastic gradient descent with back-propagation.
  • Loss function: EQ measures the average multiplicative prediction error, with 68.27% of QNN values statistically lying within (1 ± EQ) × QFDTD.The metric is derived from the standard deviation in log10(Q) between predictions and true values.

2. Example of training

Training accuracy depends strongly on the selected input-hole area. For (Nx, Ny) = (13, 5), the network reaches approximately 13% test error and high correlation, whereas larger inputs can add noise.

  • 2. Example of training: For (Nx, Ny) = (10, 5), test error falls to ~16% within 2×10^5 iterations, with a test correlation coefficient of 0.92.Training and test errors initially exceed 80% and decrease below 20% during learning.
  • 3. Dependence on input area size: The minimum test prediction error is ~13% for (Nx, Ny) = (13, 5), with test and training correlation coefficients of 0.96 and 0.99.The minimum is obtained during 1 million learning iterations.
  • 3. Dependence on input area size: Prediction error decreases as the input area expands from small sizes but increases again for larger areas when weakly correlated displacements introduce learning noise.This behavior is used to select the (13, 5) air-hole region for optimization.
  • 2. Example of training: The training results are summarized by a learning curve and by QFDTD-versus-QNN correlations for test and training datasets.Figure 5 uses an input air-hole area of (Nx, Ny) = (10, 5).

1. Loss function

The optimization uses a loss combining Q-factor targeting with a displacement penalty, then updates air-hole positions using NN-derived gradients and momentum. Optimization behavior depends on the displacement constraint parameter λ′ and initial structure.

  • Loss function: The gradient of L′ with respect to air-hole displacements is calculated by back-propagation through the trained NN for high-speed structural optimization.The method leverages the same NN framework used during training.
  • Loss function: The loss includes an artificial penalty that constrains air holes to remain near the parameter space learned by the NN, where prediction error is small.The penalty discourages large displacements from the original structure.
  • Loss function: The structure is updated step by step with the Momentum method using a momentum damping factor γ=0.9 and optimization rate 1 × 10^-5 over 10^6 iterations.The displacement and momentum variables are updated iteratively for each air hole.
  • Loss function: Starting from a structure with QNN≈1×10^7, optimization with λ′=0.05 increased QNN to 4.84×10^8 and produced a result nearly identical to that from the base initial structure.The final QNN and optimized structures were nearly the same for the two different initial structures.
  • Loss function: The optimized structures are evaluated by their Q factors, electric-field distributions Ey, and cavity modal volumes Vcav.The validation results are summarized for the optimized structures.

4. DISCUSSION

The discussion shows that NN prediction bias at high Q does not prevent useful gradient directions, while moderate displacement constraints produce the strongest validated Q factors. Results from different starting structures also support optimization within the learned parameter space.

  • 4. DISCUSSION: QNN was less than one-third of QFDTD for the highest-Q structure because high-Q samples were rare in the 1000-sample dataset.Only 40 samples had QFDTD>1×10^8, causing the NN to predict high-Q structures conservatively.
  • 4. DISCUSSION: Despite the prediction bias, the optimized structure had larger QFDTD than the initial structure, indicating that the NN evaluated the Q-factor gradient direction properly.The result supports the usefulness of the learned gradient for optimization even when absolute Q prediction is inaccurate.
  • 4. DISCUSSION: QFDTD increased from 4.48×10^8 to 1.58×10^9 as λ′ increased from 0.01 to 0.1 while air-hole displacements decreased.For λ′>0.5, displacements were overly constrained, but QFDTD values above 1.39×10^9 were still realized.
  • 4. DISCUSSION: Different initial structures produced almost identical optimized structures and QFDTD values, indicating global optimization within and near the learned parameter space.The comparison used cases with substantially different initial QNN values.

5. CONCLUSION

The paper demonstrates deep-learning-based optimization of 2D-PC nanocavities using a convolutional neural network trained on 1000 generated structures. The method achieved a theoretical Q factor of 1.58 × 10^9 while exploring a parameter space impractical for direct-calculation methods.

  • 5. CONCLUSION: A neural network with one convolutional layer and three fully connected layers was trained on 1000 randomly generated nanocavities and their Q factors.The network learned the relationship between air-hole displacement patterns and Q factors.
  • 5. CONCLUSION: The trained network predicted Q factors from air-hole displacement patterns with a standard-deviation error of 13%.The prediction model was then used to estimate gradients for structural optimization.
  • 5. CONCLUSION: A nanocavity with theoretical Q=1.58 × 10^9 was obtained, 10 times larger than the manually optimized base structure and more than twice the highest previously reported value for similar 2D-PC cavities.The result was achieved by optimizing air-hole displacements using gradients estimated by the trained network.
  • 5. CONCLUSION: The authors attribute the high Q factor to optimization over a parameter space too large for previous methods based solely on direct calculations.The conclusion identifies the accessible parameter-space size as the method’s distinguishing capability.
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