Simultaneous Inverse Design of Materials and Structures via Deep

Research Article Cite This: ACS Appl. Mater. Interfaces 2019, 11, 24264−24268

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Simultaneous Inverse Design of Materials and Structures via Deep Learning: Demonstration of Dipole Resonance Engineering Using Core−Shell Nanoparticles Sunae So,†,¶ Jungho Mun,‡,¶ and Junsuk Rho*,†,‡ †

Department of Mechanical Engineering and ‡Department of Chemical Engineering, Pohang University of Science and Technology (POSTECH), Pohang 37673, Republic of Korea

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S Supporting Information *

ABSTRACT: Recent introduction of data-driven approaches based on deep-learning technology has revolutionized the field of nanophotonics by allowing efficient inverse design methods. In this paper, a simultaneous inverse design of materials and structure parameters of core−shell nanoparticles is achieved for the first time using deep learning of a neural network. A neural network to learn the correlation between the extinction spectra of electric and magnetic dipoles and core−shell nanoparticle designs, which include material information and shell thicknesses, is developed and trained. We demonstrate deep-learning-assisted inverse design of core−shell nanoparticles for (1) spectral tuning electric dipole resonances, (2) finding spectrally isolated pure magnetic dipole resonances, and (3) finding spectrally overlapped electric dipole and magnetic dipole resonances. Our finding paves the way for the rapid development of nanophotonics by allowing a practical utilization of deep-learning technology for nanophotonic inverse design. KEYWORDS: neural network, nanophotonics, plasmonics, scattering, metamaterials, deep learning

1. INTRODUCTION The optical interaction between subwavelength particles and light has been an important problem in many fields of physics and materials science. However, subwavelength nanoparticles usually undergo rather uninteresting Rayleigh scattering; on the other hand, subwavelength plasmonic nanoparticles have shown sharp and strong extinction peaks due to the localized surface plasmon resonance, which has been used in refractive index sensing.1,2 The subwavelength plasmonic nanoparticles generally have dominant electric dipole (ED) modes, with its resonance determined by the plasmonic materials used. Recently, structured subwavelength meta-atoms with higherorder multipole modes have been realized,3,4 whose contributions have demonstrated many interesting phenomena including directional scattering,5 Fano-like resonance,6 and negative index media.7 Core−shell nanoparticles are metaatom candidates for exhibiting such exotic phenomena,8−14 but their higher degree of freedom makes designing difficult. Also, it is usually considered difficult to independently engineer ED and magnetic dipole (MD) resonances. These design problems due to the higher degree of freedom can be overcome by utilizing data-driven approaches based on deep learning (DL). By training with a vast amount of data, artificial neural networks (NNs) can learn the correlation between various photonic structures and their optical properties and inversely design complex photonic structures.15−19 © 2019 American Chemical Society

Once trained, an NN can be repeatedly employed to design photonic structures from given arbitrary optical properties, and this design process takes only a fraction of time compared to conventional labor-intensive parameter optimization methods. This feature is especially beneficial to computational electromagnetics, which often requires significant computational costs, whereas an NN-based design process can be conducted in a single GPU processor. Nevertheless, to the best of our knowledge, a simultaneous inverse design of structural parameters and material information of photonic structures has not been demonstrated. This difficulty of simultaneous inverse design arises from problem setting where structural parameters with continuous values should be predicted by regression problem while materials be predicted by classification problem; therefore, a joint of regression and classification problems should be solved to inversely design materials and structural parameters simultaneously. In this study, a loss function, which is a function that an NN aims to minimize during a training process, is developed to solve such a multitask problem20 consisting of regression and classification problems. Received: April 9, 2019 Accepted: June 14, 2019 Published: June 14, 2019 24264

DOI: 10.1021/acsami.9b05857 ACS Appl. Mater. Interfaces 2019, 11, 24264−24268

Research Article

ACS Applied Materials & Interfaces

Figure 1. Schematics of (a) a core−shell nanoparticle and (b) the utilized DL model consisting of two NNs (DN and SN). The DN learns mapping from S (input extinction spectra) to D (design parameters), and the SN learns mapping from D to S′.

optical properties. The developed overall loss function of DN (loverall) is defined as the weighted average of design (ldesign) and spectrum losses (lspectrum)

In this work, we propose a DL-assisted method to inversely design structural parameters and material information of core− shell nanoparticles simultaneously from given ED and MD extinction spectra. Details of the developed NNs and loss function are discussed. To discuss the practical usage of a DLassisted inverse design, three cases of dipole resonance engineering are demonstrated: (1) spectral tuning ED resonances, (2) finding spectrally isolated MD resonances, and (3) finding spectrally overlapped ED and MD resonances.

loverall = wlspectrum + (1 − w)ldesign

where lspectrum is calculated by the mean squared error (MSE) of S (target spectrum) and S′ (predicted response by DL). The weight w is chosen by the parametric study to minimize the MSE between S and S″ (calculated spectra using the predicted designs D by DN) for the final test data set (see Supporting Information, Section S1). The simultaneous inverse design of optical materials and structural parameters requires a development of the loss function because classification and regression problems should be solved at the same time. The devised design loss function is the weighted average of material and structural losses

2. RESULTS AND DISCUSSION 2.1. Deep Learning. In this work, spherical three-layered core−shell nanoparticles are used with a parameterized material type and thickness for each layer (Figure 1a). The outer radius of the core−shell is constrained between 10 and 150 nm. The extinction for stratified spherical particles (data set for DL) was obtained using the T-matrix formalism at plane-wave incidence. The extinction spectra are normalized by maximum values for simplicity of inverse design (see the Supporting Information, Section S7). Each entry of the data set consists of six design parameters (three materials and three thicknesses) and 100 spectral points of each of ED and MD extinction spectra at a wavelength range from 300 to 1000 nm. Optical properties of meta-atoms significantly depend on their materials. However, the refractive index should not be directly used for DL because an arbitrarily returned refractive index cannot generally be realized with real materials. In addition, plasmonic and many dielectric materials are highly dispersive and cannot easily be treated in DL. Therefore, we have listed the seven possible materials (Ag, Au, Al, Cu, TiO2, SiO2, and Si) for design and indexed them through numbering (Table 1).

ldesign = ρlstructure + (1 − ρ)lmaterial

1

2

3

4

5

6

7

none

Ag

Au

Al

Cu

TiO2

SiO2

Si

(2)

where ρ is the weight of the structural error, which is also added as a hyperparameter to be adjusted. lstructure was evaluated by the mean absolute error, and lmaterial was evaluated by binary cross entropy with logits loss (see the Supporting Information, Section S1 for details on DL networks used for training and parametric study of the devised loss function). The total data containing more than 18,000 sets are divided into training (∼80%), validating (∼10%), and test data sets (∼10%). NNs are trained using the training data set, and the trained networks are evaluated by the validation data set on every epoch to avoid the over-fitting problem. After 20,000 epochs, the trained network is finally evaluated by the test data set that has never been used in previous training or validation steps. Four examples of randomly chosen test results (Figure 2) show that the DN is well trained to provide appropriate design parameters of materials and thickness for given input ED and MD responses. The ED and MD responses obtained from predicted designs show good agreement with target input spectra. The average MSE between predicted and target responses of 2313 test data sets is ∼9.43 × 10−3, quantitatively showing that DNs indeed provide appropriate designs that have desired optical properties. DL-based inverse design is practically utilized to find a structure, which reconstructs an input spectrum with specific purposes. In the following sections, we demonstrate the capability of our network to inversely design spherical core− shell nanoparticles using user-drawn spectra. We employed Lorentzian shape resonance to engineer spectral dipole

Table 1. Indexed Materials 0

(1)

We now express the central idea of this work: our DL network architecture and newly devised loss function, which enables simultaneous inverse design of material and structural parameters. The DL model used for training consists of two networks: a design network (DN) and a spectrum network (SN) (Figure 1b). The DN learns mapping from optical properties (EDs and MDs) to design parameters (materials and thicknesses), and the SN learns from design parameters to 24265


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nanoparticles.21 However, it has been difficult to observe a spectrally isolated MD mode from subwavelength nanoparticles.22 Only recently, Feng et al.22 have found that a pure MD mode at a certain wavelength where MD modes are overlapped with a nonradiated mode of the anapole mode. Such a nanoparticle with spectrally isolated MD may allow us to study light−matter interactions involving the MD mode at the absence of the ED mode. Such spectrally isolated MD would enable controlled experiments on optomechanical effects of MD,23,24 optical interactions between MDs,25 and optical interactions between MD and quantum resonators (i.e., molecules) and their modified emission properties.26−29 We searched for spectrally isolated MD mode by using input spectra of Lorentzian MD resonance and flat-zero ED spectra (Figure 4). We could not find a condition where modes other

Figure 2. Test examples of the inverse design using test data sets. Spectra of target input (solid lines) and predicted responses (open circles) obtained from the provided design parameters. ED (red) and MD (black) extinction cross sections are independently expressed. Calculated MSEs between predicted and target spectra are (a) 2.64 × 10−3, (b) 1.10 × 10−4, (c) 1.36 × 10−4, and (d) 2.42 × 10−3. Design parameters of the results are summarized in Supporting Information, Section S3.

resonances. Additional examples not included in the following sections and inverse design using arbitrary input spectra can be found in the Supporting Information (Sections S4 and S5). 2.2. Spectral Tuning of Electric Dipole Resonances. Sharp extinction resonance peaks of plasmonic nanoparticles have enabled highly sensitive sensing and detection, but the resonance wavelength of a single subwavelength nanoparticle is limited by the plasmon resonance wavelength of the plasmonic material.1,2 Our approach allows multimaterial nanoparticles and thereby achieves various target resonance wavelengths (Figure 3). We believe that sensing applications would benefit

Figure 4. Spectrally isolated MD resonance. (a, d) Spectra of userdrawn inputs (solid lines) and predicted responses (open circles); ED (red) and MD (black). Insets are three-dimensional radiation patterns. (b, e) Magnetic field distribution together with the polarization (white arrow) and (c, f) scattering radiation patterns at resonant wavelengths of (b, c) 425 and (e, f) 660 nm.

than MD are completely eliminated over a broad range of wavelengths, but we could still find a condition where the MD mode is dominant at a certain wavelength (∼660 nm in Figure 4d). This mode is almost purely an MD mode as can be seen from its radiation pattern (Figure 4f). It should be noted that it is not always possible to find structures with physically constrained optical properties possibly because of limited design conditions (see Supporting Information, Section S5). 2.4. Finding Spectrally Overlapped ED and MD Resonances. Simultaneous ED and MD modes have demonstrated several interesting phenomena including directional scattering8−10,30 and negative index media.7 However, spectral tuning of both ED and MD resonances to a single target wavelength is difficult, if not cumbersome. DL-assisted engineering would substantially reduce time and effort taken to find such conditions. We could successfully find conditions with simultaneous ED and MD modes using our network (Figure 5a) with near-zero backward scattering (Figure 5b). Nanoparticles with a large scattering cross section without backward scattering would be useful for many photonic devices that require long photon trapping time near acceptors such as solar cells. Under dipole approximation, zero backward scattering occurs when ED and MD are in-phase (the first Kerker condition), and zero forward scattering occurs when ED and MD are out-of-phase (the second Kerker

Figure 3. Spectrally tuned ED resonances. Spectra of user-drawn inputs (solid lines) and predicted responses (open circles) obtained from the provided design parameters; ED (red) and MD (black). Insets and bottoms are the radiation patterns and electric field distributions at resonant wavelengths of (a) 400, (b) 700, and (c) 900 nm.

from spectrally tuned sharp ED resonance, especially due to the limited plasmonic materials and host media. We further confirmed that the found mode is an ED mode from field and radiation patterns (Figure 3). 2.3. Finding Spectrally Isolated MD Resonances. In general, subwavelength nanoparticles exhibit dominantly an ED mode at plane-wave incidence. Recently, strong higherorder multipole modes, such as MD and electric quadrupole, have been observed in subwavelength structured plasmonic nanoparticles3 as well as subwavelength high-index dielectric 24266


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to only ED and MD modes. By additionally introducing higher-order modes to the input spectra, many designs with interesting phenomena may also be designed. However, it is not always possible to find a structure with an arbitrary spectrum possibly because of the limited geometry of the three-layered spherical core−shell nanoparticles and the size limit. If the target application is fixed, it should be better to directly use optical properties, such as directionality, for directional scattering and effective refractive index for negative index media, instead of general extinction spectra. Nevertheless, this work shows great potential of DL-based inverse design by introducing materials as parameters and thereby significantly extending the degree of freedom of the design possibility. Also, as data throughput increases, it is expected that many new designs will be available that were difficult to design empirically or intuitively. Figure 5. Spectrally overlapped ED and MD resonances. (a) Input spectra (solid lines) and predicted responses (open circles); ED (red) and MD (black). (b) Scattering radiation pattern and (c) electric (|E|) and magnetic (|H|) field distributions at the wavelength of 600 nm. (d) Effective parameters of permittivity (top), permeability (middle), and refractive index (bottom) of the provided core−shell nanoparticle with filling fraction f = 0.7405

3. CONCLUSIONS In conclusion, we have provided the first simultaneous inverse design of materials and structural parameters of core−shell nanoparticles based on independently given ED and MD spectra. Materials are indexed by numbering and designed by DL, along with the structural parameters of the thicknesses. This combined classification (material) and regression (thickness) problem is solved by our devised loss function. To show the capability of our DL network, we specifically demonstrate a DL-assisted inverse design for three dipole resonance engineering problems with on-demand applications using spherical core−shell nanoparticles: (1) spectral tuning of ED resonances for sensing applications; (2) finding spectrally isolated MD resonances; and (3) finding spectrally overlapped ED and MD resonances for directional scattering or negative index media. We present individual engineering of ED and MD spectra to discuss their relevance to physical phenomena. We believe that our concept of DL-assisted simultaneous inverse design of materials and structures will lead to the development of the field of nanophotonics by allowing efficient inverse design of nanophotonic structures.

condition).30 This principle has made high-efficiency metasurface31,32 and perfect absorber33 possible. Another interesting phenomenon that would benefit from DL-assisted engineering is design of negative index media. To obtain negative index media, ED and MD resonances need to be simultaneously tuned to a target wavelength. To overcome this difficulty, Pendry proposed chiral media as a possible route to negative index media without simultaneously engineering ED and MD resonances.34 Our designed core−shell nanoparticle exhibits a negative refractive index n = − 0.118 + 1.684i as well as simultaneously negative effective permittivity and permeability at the target wavelength of 600 nm (Figure 5c). Here, effective permittivity and permeability are obtained using Clausius− Mossotti relation (eq 3) μ − μ0 ϵeff − ϵ0 α α = f E 3 , eff = f M3 ϵeff + 2ϵ0 4πR μeff + 2μ0 4πR (3)

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ASSOCIATED CONTENT

* Supporting Information S

where αE and αM are the electric and magnetic polarizabilities. 2.5. Discussion. With a single NVIDIA GTX 1080Ti, it takes a total of 400 min for a whole deep-learning process, which includes 300 min for data construction and 100 min for training the network. However, this is a one-time cost; once the network is trained, the trained NN can provide structural designs within 1 s without additional training processes. On the other hand, using a conventional inverse design method of genetic algorithm optimization, it takes ∼187 min to find one optimal design for a given input. This method requires additional optimization processes for other design tasks. This is the main advantage of our method over conventional optimization methods, which significantly reduces the computational time and cost for multiple design tasks. So far, multiplexed input spectra with different polarizations have been demonstrated in DL;15,17 however, multiplexed input spectra with multipole-decomposed spectra have not been reported. We believe that this approach could be used to directly engineer many physically intriguing phenomena because simple extinction spectra cannot give deeper physical insights, which the multipole-decomposed spectra may provide. In this present study, we limited the input spectra

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsami.9b05857. Details on the deep-learning procedure, effect of the spectrum network, design parameters used in the main text, additional examples, inverse design using arbitrary input spectra, response to gradual change of input spectra, and extinction from spherical core−shell nanoparticles (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Sunae So: 0000-0001-8606-2234 Jungho Mun: 0000-0002-7290-6253 Junsuk Rho: 0000-0002-2179-2890 Author Contributions ¶

S.S. and J.M. contributed equally to this work.

Notes

The authors declare no competing financial interest. 24267


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ACKNOWLEDGMENTS This work is financially supported by the national Research Foundation grants (NRF-2019R1A2C3003129, CAMM2019M3A6B3030637, NRF-2018M3D1A1058998, and NRF2015R1A5A1037668) funded by the Ministry of Science and ICT, Korea. S.S. acknowledges global Ph.D fellowship (NRF2017H1A2A1043322) from the NRF-MSIT, Korea.

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Simultaneous Inverse Design of Materials and Structures via Deep

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