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Machine learning for nanophotonics

Published online by Cambridge University Press:  11 March 2020

Itzik Malkiel
Affiliation:
School of Computer Science, Tel Aviv University, Israel; itzik.malkiel@gmail.com
Michael Mrejen
Affiliation:
Femto-Nano Laboratory, Department of Condensed Matter Physics, School of Physics and Astronomy, Tel Aviv University, Israel; michael.mrejen@berkeley.edu
Lior Wolf
Affiliation:
School of Computer Science, Tel Aviv University, Israel; wolf@cs.tau.ac.il
Haim Suchowski
Affiliation:
Department of Condensed Matter Physics, School of Physics and Astronomy, Tel Aviv University, Israel; haimsu@tauex.tau.ac.il

Abstract

The past decade has witnessed the advent of nanophotonics, where light–matter interaction is shaped, almost at will, with human-made designed nanostructures. However, the design process for these nanostructures has remained complex, often relying on the intuition and expertise of the designer, ultimately limiting the reach and penetration of this groundbreaking approach. Recently, there has been an increasing number of studies in applying machine learning techniques for the design of nanostructures. Most of these studies engage deep learning techniques, which entail training a deep neural network (DNN) to approximate the highly nonlinear function of the underlying physical process of the interaction between light and the nanostructures. At the end of the training, the DNN allows for on-demand design of nanostructures (i.e., the model can infer nanostructure geometries for desired light spectra). In this article, we review previous studies for designing nanostructures, including recent advances where a DNN is trained to generate a two-dimensional image of the designed nanostructure, which is not limited to a closed set of nanostructure shapes, and can be trained for the design of any geometry. This allows for better generalization, with higher applicability for real-world design problems.

Information

Type
Metasurfaces for Flat Optics
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Materials Research Society 2020
Figure 0

Figure 1. Deep learning nanophotonics. (a) Interaction of light with plasmonic nanostructures. Incoming electromagnetic radiation interacts with human-made subwavelength structures in a resonant manner, leading to an effective optical response where the optical properties for both horizontal and vertical polarizations of the designed metamaterial are dictated by the geometry at the nanoscale rather than the chemical composition. (b) To date, the approach enabled by computational tools allows only for “direct” modeling (predicting the optical response in both polarizations (H = horizontal and V = vertical) of a nanostructure based on its geometry, constituent and surrounding media). However, the inverse problem, where the tool outputs a nanostructure for an input desired optical response, is much more relevant from a designer point of view and is currently unachievable in a time efficient way. Note: nanofab, nanofabrication. (c) The plot shows that if a more complex optical response is desired, the solution of the inverse problem becomes increasingly unattainable. A deep learning approach bridges this gap and unlocks the possibility to design, at the single nanoparticle level, complex optical responses with multiple resonances and for both polarizations. (d) The different categories of generalization as explained in the main text. In Category 1, a model is capable of designing nanostructures from the same shape and material it was trained on, but with different properties, such as sizes, angles, and host material. In Category 2, a model is able to generalize and design geometries with shapes that differ from the training set shapes but are still considered to be in the same family. In Category 3, a model is able to design any geometry, with any shape, achieving what is called generalization capability. Note: FEM, finite element method; NP, nanoparticle.

Figure 1

Figure 2. Design of thin-film multilayer filters for on-demand spectral response using neural networks.2 (a) A thin film composed of m layers of SiO2 and Si3N4. The design parameters of the thin film are the thicknesses of the layers di (i = 1, 2, ..., m), and the device response is the transmission spectrum. (b) The forward neural network takes D = [d1, d2, ..., dm] as inputs and discretized transmission spectrum R = [r1, r2, ..., rm] as output. A tandem network is composed of (left) an inverse design network connected to a (right, dashed box) forward modeling network. The forward modeling network is trained in advance. In the training process, weights in the pretrained forward modeling network are fixed and the weights in the inverse network are adjusted to reduce the cost (i.e., error between the predicted response and the target response). Outputs by the intermediate layer M (labeled in blue) are designs D. (c, d) Example test results for two different target designs queried with the tandem network method showing successful retrieval compared to the target design. Note: c, speed of light; a, maximum allowed thickness of each layer.

Figure 2

Figure 3. Multilayer shell nanoparticle inverse design using neural networks (NNs).1 (a) The NN architecture has as its inputs the thickness (xi) of each shell of the nanoparticle and as its output the scattering cross section at different wavelengths of the scattering spectrum (yi). (b) NN versus numerical nonlinear optimization. The legend gives the dimensions of the particle, and the blue is the desired spectrum. The NN is seen to solve the inverse design much more accurately.

Figure 3

Table I. Performance comparison with respect to different parameters between different computational approaches (shallow neural network, deep neural network, and genetic algorithms).

Figure 4

Figure 4. Our previous work5,7,8 introduced a bilateral deep learning (DL) network able to predict the response (in two polarizations) of any of the structures defined in the “H” family (see main text). It also allows the design of “H” structures based on the required response. (a) The “H” geometry is parametrized to allow easy vector representation where the presence of Legs 1–5 is binarily coded (1 = leg present, 0 = leg absent). Note: L1 = length of Legs 1, 2, 4, and 5; L0 = Leg 3 length; φ = Leg 1 angle. (b) The network architecture allows the input of the horizontal and vertical spectrum vectors (sampled at 43 wavelength points each) as well as a material’s properties vector representation (43 parameters). This input is then fed into the first three fully connected 100 neuron (described by the solid black circles) layers followed by eight fully connected layers. The DL is given a (c) measured horizontal polarization (horizontal red double-headed arrow) and (d) vertical polarization (vertical blue double-headed arrow). (e) The predicted geometry, which is in good agreement compared to the geometry measured in scanning electron microscope (inset [c]). Comparison between the fed spectra and the predicted ones are found in (c, d). Note: DNN, deep neural network.

Figure 5

Figure 5. Bidirectional deep learning network applied to the design of chiral metamaterials.4 (a) Schematic of the designed chiral metamaterial. The inset is the zoomed-in structure of a single meta-atom. (b) A bidirectional deep neural network is designed to retrieve the chiral metamaterial geometry from the reflections (σ+-input-σ+-output [blue curve], σ-input-σ-output [green curve]), and the cross-polarization term σ+-input-σ-output (red curve) and CD, and vice versa. (c, d) CD spectra predicted by the deep neural network (blue dots), which are in good agreement with the simulations (red curve). Note: CD, circular dichroism; σ+, σ right-handed and left-handed circular polarization, respectively.

Figure 6

Figure 6. Spectrum prediction for one polarization based on the geometry represented as a 2D map of pixels using convolutional neural networks. (a) Convolutional neural networks are used to extract spatial features from an image of a structure by extracting data from smaller parts of the image. (b) (Right) The solid blue lines show the absorption curves obtained from the simulation package (SIM) for (left) a random geometry, and the dotted orange lines show the absorption curves predicted by the deep learning (DL) model. These curves show a one-to-one comparison of predicted absorption value versus real absorption value for each frequency. Adapted with permission from Reference 3. © 2019 Nature Publishing Group. Note: relu, rectified linear unit.

Figure 7

Figure 7. Generative adversarial networks-based inverse design of metasurfaces.6 (a) Three networks, the generator, the simulator, and the critic constitute the complete architecture. The generator accepts the spectra T and noise z and produces possible patterns. The simulator is a pretrained network that approximates the transmittance spectrum for a given pattern at its input, and the critic evaluates the distance of the distributions between the geometric data and the patterns from the generator. While training the generator, the produced patterns vary according to the feedback obtained from S and D. Valid patterns are documented during the training process, and are smoothed to qualify as candidate structures. (b) Test patterns are depicted in the top row and the corresponding generated patterns are listed in the bottom row. Each shape provides a sample of the different classes of geometric data input to the critic network. Note: S, simulator network; D, critic network.

Figure 8

Figure 8. Six samples from the transform data set of spectra2pix.14,15 We see the wide variety of (left) spectra and (right) geometries spanned by the family of the “H” shapes. The spectra correspond to horizontal and vertical polarizations in transmission. In all experiments, the structures are made of gold, each with a different host dielectric, with dielectric values varying in range (1.0, 3.0) (not shown in the figure). Each image is composed of 64 × 64 pixels, with a pixel size of 15.625 × 15.625 nm.

Figure 9

Figure 9. Results of queries with three designs from the test set (“L” family) to spectra2pix after the learning phase.14 (a) Input spectra are presented; (b) the predicted geometry by spectra2pix, and (c) the ground truth geometry is depicted. Each image is composed of 64 × 64 pixels, with a pixel size of 15.625 × 15.625 nm.