Abstract

Deep learning has risen to the forefront of many fields in recent years, overcoming challenges previously considered intractable with conventional means. Materials discovery and optimization is one such field, but significant challenges remain, including the requirement of large labeled datasets and one-to-many mapping that arises in solving the inverse problem. Here we demonstrate modeling of complex all-dielectric metasurface systems with deep neural networks, using both the metasurface geometry and knowledge of the underlying physics as inputs. Our deep learning network is highly accurate, achieving an average mean square error of only 1.16 × 10−3 and is over five orders of magnitude faster than conventional electromagnetic simulation software. We further develop a novel method to solve the inverse modeling problem, termed fast forward dictionary search (FFDS), which offers tremendous controls to the designer and only requires an accurate forward neural network model. These techniques significantly increase the viability of more complex all-dielectric metasurface designs and provide opportunities for the future of tailored light matter interactions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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F. Lussier, D. Missirlis, J. P. Spatz, and J.-F. Masson, “Machine learning driven surface-enhanced raman scattering optophysiology reveals multiplexed metabolite gradients near cells,” ACS Nano 13, 1403–1411 (2019).
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2018 (6)

D. Liu, Y. Tan, E. Khoram, and Z. Yu, “Training Deep Neural Networks for the Inverse Design of Nanophotonic Structures,” ACS Photonics 5, 1365–1369 (2018).
[Crossref]

I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, and H. Suchowski, “Plasmonic nanostructure design and characterization via Deep Learning,” Light. Sci. Appl. 760 (2018).
[Crossref]

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[Crossref]

W. Ma, F. Cheng, and Y. Liu, “Deep-Learning-Enabled On-Demand Design of Chiral Metamaterials,” ACS Nano 12, 6326–6334 (2018).
[Crossref] [PubMed]

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5, 2295–2304 (2018).
[Crossref]

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 44206 (2018).
[Crossref] [PubMed]

2017 (6)

X. Liu, K. Fan, I. V. Shadrivov, and W. J. Padilla, “Experimental realization of a terahertz all-dielectric metasurface absorber,” Opt. Express 25, 191–201 (2017).
[Crossref] [PubMed]

K. Fan, J. Y. Suen, X. Liu, and W. J. Padilla, “All-dielectric metasurface absorbers for uncooled terahertz imaging,” Optica 4, 601–604 (2017).
[Crossref]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

H. W. Lin, M. Tegmark, and D. Rolnick, “Why does deep and cheap learning work so well?” J. Stat. Phys. 168, 1223–1247 (2017).
[Crossref]

2016 (1)

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

2015 (2)

J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano 9, 4308–4315 (2015).
[Crossref] [PubMed]

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[Crossref] [PubMed]

2014 (1)

M. Muja and D. G. Lowe, “Scalable nearest neighbor algorithms for high dimensional data,” IEEE Trans. Pattern Analysis Mach. Intell. 36, 2227–2240 (2014).
[Crossref]

2011 (1)

X. Liu, T. Tyler, T. Starr, A. F. Starr, N. M. Jokerst, and W. J. Padilla, “Taming the blackbody with infrared metamaterials as selective thermal emitters,” Phys. Rev. Lett. 107, 045901 (2011).
[Crossref] [PubMed]

2008 (1)

H. Kabir, Y. Wang, M. Yu, and Q. Zhang, “Neural Network Modeling and Applications to Microwave Design,” IEEE Trans. Microw. Theory Tech. 56, 867 (2008).
[Crossref]

1989 (2)

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks 2, 359–366 (1989).
[Crossref]

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Contr. Sign. Syst. 2, 303–314 (1989).
[Crossref]

Abbott, D.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Abdollahrameazni, S.

Y. Kiarashinejad, S. Abdollahrameazni, and A. Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures,” arXiv preprint arXiv:1902.03865 (2019).

Abdollahramezani, S.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simulations0, 1900088.

Adibi, A.

Y. Kiarashinejad, S. Abdollahrameazni, and A. Adibi, “Deep learning approach based on dimensionality reduction for designing electromagnetic nanostructures,” arXiv preprint arXiv:1902.03865 (2019).

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simulations0, 1900088.

Anguelov, D.

W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C. Y. Fu, and A. C. Berg, “SSD: Single shot multibox detector,” in European Conference on Computer Vision, (2016), pp. 21–37.

Arieli, U.

I. Malkiel, M. Mrejen, A. Nagler, U. Arieli, L. Wolf, and H. Suchowski, “Plasmonic nanostructure design and characterization via Deep Learning,” Light. Sci. Appl. 760 (2018).
[Crossref]

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436 (2015).
[Crossref] [PubMed]

Berg, A. C.

W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C. Y. Fu, and A. C. Berg, “SSD: Single shot multibox detector,” in European Conference on Computer Vision, (2016), pp. 21–37.

Bhaskaran, M.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Brener, I.

J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano 9, 4308–4315 (2015).
[Crossref] [PubMed]

Cano-Renteria, F.

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 44206 (2018).
[Crossref] [PubMed]

Carrasco, E.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Chang, C. C.

C. C. Chang, L. Huang, J. Nogan, and H. T. Chen, “Invited Article: Narrowband terahertz bandpass filters employing stacked bilayer metasurface antireflection structures,” APL Photonics 3051602 (2018).
[Crossref]

Chang, M.-W.

J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” arXiv preprint arXiv:1810.04805 (2018).

Chen, C.

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5, 2295–2304 (2018).
[Crossref]

Chen, H. T.

C. C. Chang, L. Huang, J. Nogan, and H. T. Chen, “Invited Article: Narrowband terahertz bandpass filters employing stacked bilayer metasurface antireflection structures,” APL Photonics 3051602 (2018).
[Crossref]

Chen, K.

T. Mikolov, I. Sutskever, K. Chen, G. Corrado, and J. Dean, “Distributed Representations of Words and Phrases and their Compositionality,” in Advances in Neural Information Processing Systems, (NIPS, 2013), pp. 1–9.

Cheng, F.

W. Ma, F. Cheng, and Y. Liu, “Deep-Learning-Enabled On-Demand Design of Chiral Metamaterials,” ACS Nano 12, 6326–6334 (2018).
[Crossref] [PubMed]

Clune, J.

J. Yosinski, J. Clune, A. Nguyen, T. Fuchs, and H. Lipson, “Understanding neural networks through deep visualization,” arXiv preprint arXiv:1506.06579 (2015).

Corrado, G.

T. Mikolov, I. Sutskever, K. Chen, G. Corrado, and J. Dean, “Distributed Representations of Words and Phrases and their Compositionality,” in Advances in Neural Information Processing Systems, (NIPS, 2013), pp. 1–9.

Cybenko, G.

G. Cybenko, “Approximation by superpositions of a sigmoidal function,” Math. Contr. Sign. Syst. 2, 303–314 (1989).
[Crossref]

Dean, J.

T. Mikolov, I. Sutskever, K. Chen, G. Corrado, and J. Dean, “Distributed Representations of Words and Phrases and their Compositionality,” in Advances in Neural Information Processing Systems, (NIPS, 2013), pp. 1–9.

Decker, M.

J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano 9, 4308–4315 (2015).
[Crossref] [PubMed]

DeLacy, B. G.

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 44206 (2018).
[Crossref] [PubMed]

Devlin, J.

J. Devlin, M.-W. Chang, K. Lee, and K. Toutanova, “Bert: Pre-training of deep bidirectional transformers for language understanding,” arXiv preprint arXiv:1810.04805 (2018).

Duan, Z.

S. Sun, Z. Zhou, C. Zhang, Y. Gao, Z. Duan, S. Xiao, and Q. Song, “All-dielectric full-color printing with tio2 metasurfaces,” ACS Nano 11, 4445–4452 (2017). PMID: .
[Crossref] [PubMed]

Dumoulin, V.

V. Dumoulin and F. Visin, “A guide to convolution arithmetic for deep learning,” arXiv preprint arXiv:1603.07285 (2016).

Dyck, O.

M. Ziatdinov, O. Dyck, A. Maksov, X. Li, X. Sang, K. Xiao, R. R. Unocic, R. Vasudevan, S. Jesse, and S. V. Kalinin, “Deep learning of atomically resolved scanning transmission electron microscopy images: chemical identification and tracking local transformations,” ACS Nano 11, 12742–12752 (2017).
[Crossref] [PubMed]

Erhan, D.

W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C. Y. Fu, and A. C. Berg, “SSD: Single shot multibox detector,” in European Conference on Computer Vision, (2016), pp. 21–37.

Fan, K.

Fergus, R.

M. D. Zeiler, D. Krishnan, G. W. Taylor, and R. Fergus, “Deconvolutional networks,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, (IEEE, 2010), pp. 2528–2535.

Fu, C. Y.

W. Liu, D. Anguelov, D. Erhan, C. Szegedy, S. Reed, C. Y. Fu, and A. C. Berg, “SSD: Single shot multibox detector,” in European Conference on Computer Vision, (2016), pp. 21–37.

Fuchs, T.

J. Yosinski, J. Clune, A. Nguyen, T. Fuchs, and H. Lipson, “Understanding neural networks through deep visualization,” arXiv preprint arXiv:1506.06579 (2015).

Fumeaux, C.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Gao, Y.

S. Sun, Z. Zhou, C. Zhang, Y. Gao, Z. Duan, S. Xiao, and Q. Song, “All-dielectric full-color printing with tio2 metasurfaces,” ACS Nano 11, 4445–4452 (2017). PMID: .
[Crossref] [PubMed]

Gomez, A. N.

A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention Is All You Need,” in Advances in Neural Information Processing Systems, (NIPS, 2017), Nips.

Gutruf, P.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Headland, D.

D. Headland, E. Carrasco, S. Nirantar, W. Withayachumnankul, P. Gutruf, J. Schwarz, D. Abbott, M. Bhaskaran, S. Sriram, J. Perruisseau-Carrier, and C. Fumeaux, “Dielectric resonator reflectarray as high-efficiency nonuniform terahertz metasurface,” ACS Photonics 3, 1019–1026 (2016).
[Crossref]

Hemmatyar, O.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simulations0, 1900088.

Hinton, G.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521, 436 (2015).
[Crossref] [PubMed]

Hinton, G. E.

A. Krizhevsky, I. Sutskever, and G. E. Hinton, “ImageNet Classification with Deep Convolutional Neural Networks,” in Advances in Neural Information Processing Systems, (2012), pp. 1097–1105.

Hornik, K.

K. Hornik, M. Stinchcombe, and H. White, “Multilayer feedforward networks are universal approximators,” Neural Networks 2, 359–366 (1989).
[Crossref]

Huang, L.

C. C. Chang, L. Huang, J. Nogan, and H. T. Chen, “Invited Article: Narrowband terahertz bandpass filters employing stacked bilayer metasurface antireflection structures,” APL Photonics 3051602 (2018).
[Crossref]

Jesse, S.

M. Ziatdinov, O. Dyck, A. Maksov, X. Li, X. Sang, K. Xiao, R. R. Unocic, R. Vasudevan, S. Jesse, and S. V. Kalinin, “Deep learning of atomically resolved scanning transmission electron microscopy images: chemical identification and tracking local transformations,” ACS Nano 11, 12742–12752 (2017).
[Crossref] [PubMed]

Jing, L.

Y. Qu, L. Jing, Y. Shen, M. Qiu, and M. Soljačić, “Migrating knowledge between physical scenarios based on artificial neural networks,” ACS Photonics 6, 1168–1174 (2019).
[Crossref]

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 44206 (2018).
[Crossref] [PubMed]

Joannopoulos, J. D.

J. Peurifoy, Y. Shen, L. Jing, Y. Yang, F. Cano-Renteria, B. G. DeLacy, J. D. Joannopoulos, M. Tegmark, and M. Soljačić, “Nanophotonic particle simulation and inverse design using artificial neural networks,” Sci. Adv. 44206 (2018).
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ACS Nano (5)

M. Ziatdinov, O. Dyck, A. Maksov, X. Li, X. Sang, K. Xiao, R. R. Unocic, R. Vasudevan, S. Jesse, and S. V. Kalinin, “Deep learning of atomically resolved scanning transmission electron microscopy images: chemical identification and tracking local transformations,” ACS Nano 11, 12742–12752 (2017).
[Crossref] [PubMed]

F. Lussier, D. Missirlis, J. P. Spatz, and J.-F. Masson, “Machine learning driven surface-enhanced raman scattering optophysiology reveals multiplexed metabolite gradients near cells,” ACS Nano 13, 1403–1411 (2019).
[PubMed]

S. Sun, Z. Zhou, C. Zhang, Y. Gao, Z. Duan, S. Xiao, and Q. Song, “All-dielectric full-color printing with tio2 metasurfaces,” ACS Nano 11, 4445–4452 (2017). PMID: .
[Crossref] [PubMed]

J. Sautter, I. Staude, M. Decker, E. Rusak, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Active tuning of all-dielectric metasurfaces,” ACS Nano 9, 4308–4315 (2015).
[Crossref] [PubMed]

W. Ma, F. Cheng, and Y. Liu, “Deep-Learning-Enabled On-Demand Design of Chiral Metamaterials,” ACS Nano 12, 6326–6334 (2018).
[Crossref] [PubMed]

ACS Photonics (4)

C. Chen and S. Li, “Valence electron density-dependent pseudopermittivity for nonlocal effects in optical properties of metallic nanoparticles,” ACS Photonics 5, 2295–2304 (2018).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 (a) Illustration of the all-dielectric metasurface consisting of a square array of cylindrical resonators. (b) Simulated transmittance spectra for a single sub-unit cell structure consisting of a cylindrical square array (insets) with r = 44.5μm, h = 30μm (orange curve), and r = 44.5μm, h = 42.5μm (blue curve). (c) T(ω) of a 2×2 super unit cell (inset) consisting of three r = 44.5μm, h = 30μm cylinders and one r = 44.5μm, h = 42.5μm cylinder. Black arrows denote modes that result that were not present in (a).
Fig. 2
Fig. 2 An illustration of the neural network architecture. Fully connected layers are fed a set of geometric inputs. Data is then smoothed and upsampled in a learnable manner via transpose convolution layers (top row). The final layer (bottom row) is a convolutional layer, which produces a predicted spectra, shown as the blue curve, compared to the ground truth (red curve).
Fig. 3
Fig. 3 (a)–(l) Network predictions of the frequency dependent transmittance (blue curves) and simulated T(ω) spectra (red curves) demonstrating excellent prediction accuracy for a variety of spectral features and input geometric parameters. The average MSE is listed in each sub-panel. The shaded gray area shows the absolute value of the difference in predicted and simulated transmittance, i.e. |TpredTsim|, shown on the right vertical axis. In (m) a histogram of the MSE for all geometries in the evaluation set is shown, where 95% have MSE ≤ 3.4 × 10−3, as indicated by the dashed vertical gray line.
Fig. 4
Fig. 4 Top panels of (a) – (c) show hand-picked T points (open symbols) demonstrating a trough, cusp behavior, and a flat transmission, respectively. The bottom (a) – (c) panels replot desired T points (open symbols) and show candidate spectra produced by FFDS (blue curves) compared to simulated T(ω), shown as the gray curves. Top and bottom panels (d) – (f) depict of other examples of desired T(ω) and the resulting FFDS result candidates and post-lookup simulated spectra.
Fig. 5
Fig. 5 (a) MSE error for geometrically-constrained cross-validation set after training, as described in the text. Color map shows normalized error for a max function projection (maximum MSE= 0.0057) with interpolation between the sampled points (open gray symbols). Shown in (b) and (c) are average MSE values for different values of input geometric parameter as they approach the training boundary at h = 55μm (b), and r = 55μm (c), with ribbons indicating standard deviation scaled by 1/3.

Tables (2)

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Table 1 Hyperspace parameter values for grid definition. All values are in units of microns.

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Table 2 MSE averaged over three runs for various configurations of the network

Equations (1)

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x * = arg min x i L d ( s i , s * )

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