Abstract

Metasurfaces are an emerging technology that may supplant many of the conventional optics found in imaging devices, displays, and precision scientific instruments. Here, we develop a method for designing optical systems composed of multiple unique metasurfaces aligned in sequence and separated by distances much larger than the design wavelengths. Our approach is based on computational inverse design, also known as the adjoint-gradient method. This technique enables thousands or millions of independent design variables (e.g., the shapes of individual meta-atoms) to be optimized in parallel, with little or no intervention required by the user. The assumptions underlying our method are as follows: we use the local periodic approximation to determine the phase-response of a given meta-atom, we use the scalar wave approximation to propagate light fields between metasurface layers, and we do not consider multiple reflections between metasurface layers (analogous to a sequential-optics ray-tracer). To demonstrate the broad applicability of our method, we use it to design an achromatic doublet metasurface lens, a spectrally-multiplexed holographic element, and an ultra-compact optical neural network for classifying handwritten digits.

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

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2019 (5)

R. J. Lin, V. Su, S. Wang, M. K. Chen, T. L. Chung, Y. H. Chen, H. Y. Kuo, J. Chen, J. Chen, Y. Huang, J. Wang, C. H. Chum P, C. Wu, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “Achromatic metalens array for full-colour light-field imaging,” Nat. Nanotechnol. 14(3), 227–231 (2019).
[Crossref]

T. Phan, D. Sell, E. W. Wang, S. Doshay, K. Edee, J. Yang, and J. A. Fan, “High-efficiency, large-area topology-optimized metasurfaces,” Light: Sci. Appl. 8(1), 48 (2019).
[Crossref]

W. T. Chen, A. Y. Zhu, J. Sisler, Z. Bharwani, and F. Capasso, “A broadband achromatic polarization-insensitive metalens consisting of anisotropic nanostructures,” Nat. Commun. 10(1), 355 (2019).
[Crossref]

E. Hershko, L. E. Weiss, T. Michaeli, and Y. Shechtman, “Multicolor localization microscopy and point-spread-function engineering by deep learning,” Opt. Express 27(5), 6158–6183 (2019).
[Crossref]

Z. Lin, V. Liu, R. Pestourie, and S. G. Johnson, “Topology optimization of freeform large-area metasurfaces,” Opt. Express 27(11), 15765–15775 (2019).
[Crossref]

2018 (20)

L. Su, R. Trivedi, N. V. Sapra, A. Y. Piggott, D. Vercruysse, and J. Vučković, “Fully-automated optimization of grating couplers,” Opt. Express 26(4), 4023–4034 (2018).
[Crossref]

A. Zhan, T. K. Fryett, S. Colburn, and A. Majumdar, “Inverse design of optical elements based on arrays of dielectric spheres,” Appl. Opt. 57(6), 1437–1446 (2018).
[Crossref]

T. W. Hughes, M. Minkov, Y. Shi, and S. Fan, “Training of photonic neural networks through in situ backpropagation and gradient measurement,” Optica 5(7), 864–871 (2018).
[Crossref]

M. Miscuglio, A. Mehrabian, Z. Hu, S. I. Azzam, J. George, A. V. Kildishev, M. Pelton, and V. J. Sorger, “All-optical nonlinear activation function for photonic neural networks,” Opt. Mater. Express 8(12), 3851–3863 (2018).
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W. T. Chen, A. Y. Zhu, V. Sanjeev, M. Khorasaninejad, Z. Shi, E. Lee, and F. Capasso, “A broadband achromatic metalens for focusing and imaging in the visible,” Nat. Nanotechnol. 13(3), 220–226 (2018).
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W. T. Chen, A. Y. Zhu, J. Sisler, Y. Huang, K. M. A. Yousef, E. Lee, C. Qiu, and F. Capasso, “Broadband achromatic metasurface-refractive optics,” Nano Lett. 18(12), 7801–7808 (2018).
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2017 (15)

D. Lin, M. Melli, E. Poliakov, P. St. Hilaire, S. Dhuey, C. Peroz, S. Cabrini, M. Brongersma, and M. Klug, “Optical metasurfaces for high angle steering at visible wavelengths,” Sci. Rep. 7(1), 2286 (2017).
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D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
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J. P. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
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A. Y. Piggot, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
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Y. Shen, N. C. Harris, S. Skirlo, M. Prabhu, T. Baehr-Jones, M. Hochberg, X. Sun, S. Zhao, H. Larochelle, D. Englund, and M. Soljačić, “Deep learning with coherent nanophotonic circuits,” Nat. Photonics 11(7), 441–446 (2017).
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B. Groever, W. T. Chen, and F. Capasso, “Meta-lens doublet in the visible region,” Nano Lett. 17(8), 4902–4907 (2017).
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S. Liu, A. Vaskin, S. Campione, O. Wolf, M. B. Sinclair, J. Reno, G. A. Keeler, I. Staude, and I. Brener, “Huygens’ metasurfaces enabled by magnetic dipole resonance tuning in split dielectric nanoresonators,” Nano Lett. 17(7), 4297–4303 (2017).
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O. Avayu, E. Almeida, Y. Prior, and T. Ellenbogen, “Composite functional metasurfaces for multispectral achromatic optics,” Nat. Commun. 8(1), 14992 (2017).
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S. Wang, P. C. Wu, V. Su, Y. Lai, C. H. Chu, J. Chen, S. Lu, J. Chen, B. Xu, C. Kuan, T. Li, S. Zhu, and D. P. Tsai, “Broadband achromatic optical metasurface devices,” Nat. Commun. 8(1), 187 (2017).
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M. Khorasaninejad, Z. Shi, A. Y. Zhu, W. T. Chen, V. Sanjeev, A. Zaidi, and F. Capasso, “Achromatic metalens over 60 nm bandwidth in the visible and metalens with reverse chromatic dispersion,” Nano Lett. 17(3), 1819–1824 (2017).
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P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4(1), 139–152 (2017).
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V. Egorov, M. Eitan, and J. Scheuer, “Genetically optimized all-dielectric metasurfaces,” Opt. Express 25(3), 2583–2593 (2017).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces,” Optica 4(6), 625–632 (2017).
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J. Yang and J. A. Fan, “Topology-optimized metasurfaces: impact of initial geometric layout,” Opt. Lett. 42(16), 3161–3164 (2017).
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2016 (13)

B. Wang, F. Dong, Q. Li, D. Yang, C. Sun, J. Chen, Z. Song, L. Xu, W. Chu, Y. Xiao, Q. Gong, and Y. Li, “Visible-frequency dielectric metasurfaces for multiwavelength achromatic and highly dispersive holograms,” Nano Lett. 16(8), 5235–5240 (2016).
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A. Arbabi, E. Arbabi, S. M. Kamali, Y. Horie, S. Han, and A. Faraon, “Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations,” Nat. Commun. 7(1), 13682 (2016).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength metasurfaces through spatial multiplexing,” Sci. Rep. 6(1), 32803 (2016).
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C. Smith, M. Huisman, M. Siemons, D. Grünwald, and S. Stallinga, “Simultaneous measurement of emission color and 3D position of single molecules,” Opt. Express 24(5), 4996–5013 (2016).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “High efficiency double-wavelength dielectric metasurface lenses with dichroic birefringent meta-atoms,” Opt. Express 24(16), 18468–18477 (2016).
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A. Jesacher, S. Bernet, and M. Ritsche-Marte, “Colored point spread function engineering for parallel confocal microscopy,” Opt. Express 24(24), 27395–27402 (2016).
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P. Wang, N. Mohammad, and R. Menon, “Chromatic-aberration-corrected diffractive lenses for ultra-broadband focusing,” Sci. Rep. 6(1), 21545 (2016).
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Y. Shechtman, L. E. Weiss, A. S. Backer, M. Y. Lee, and W. E. Moerner, “Multicolor localization microscopy by point-spread-function engineering,” Nat. Photonics 10(9), 590–594 (2016).
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50. R. C. Devlin, M. Khorasaninejad, W. T. Chen, J. Oh, and F. Capasso, “Broadband high-efficiency dielectric metasurfaces for the visible spectrum,” Proc. Natl. Acad. Sci. U. S. A. 113(38), 10473–10478 (2016).
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M. Khorasaninejad, A. Y. Zhu, C. Roques-Carmes, W. T. Chen, J. Oh, I. Mishra, R. C. Devlin, and F. Capasso, “Polarization-insensitive metalenses at visible wavelengths,” Nano Lett. 16(11), 7229–7234 (2016).
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A. Zhan, S. Colburn, R. Trivedi, T. K. Fryett, C. M. Dodson, and A. Majumdar, “Low-contrast dielectric metasurface optics,” ACS Photonics 3(2), 209–214 (2016).
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S. M. Kamali, A. Arbabi, E. Arbabi, Y. Horie, and A. Faraon, “Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces,” Nat. Commun. 7(1), 11618 (2016).
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M. Khorasaninejad, W. T. Chen, R. C. Devlin, J. Oh, A. Y. Zhu, and F. Capasso, “Metalenses at visible wavelengths: diffraction-limited focusing and subwavelength resolution imaging,” Science 352(6290), 1190–1194 (2016).
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2015 (5)

A. Arbabi, Y. Horie, A. J. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat. Commun. 6(1), 7069 (2015).
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A. Arbabi, Y. Horie, M. Bagheri, and A. Faraon, “Dielectric metasurfaces for complete control of phase and polarization with subwavelength spatial resolution and high transmission,” Nat. Nanotechnol. 10(11), 937–943 (2015).
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M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Adv. Opt. Mater. 3(6), 813–820 (2015).
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K. E. Chong, I. Staude, A. James, J. Dominguez, S. Liu, S. Campione, G. S. Subramania, T. S. Luk, M. Decker, D. N. Neshev, I. Brener, and Y. S. Kivshar, “Polarization-independent silicon metadevices for efficient optical wavefront control,” Nano Lett. 15(8), 5369–5374 (2015).
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A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
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2014 (5)

V. Ganapati, O. D. Miller, and E. Yablonovitch, “Light trapping textures designed by electromagnetic optimization for subwavelength thick solar cells,” IEEE J. Photovolt. 4(1), 175–182 (2014).
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A. C. R. Niederberger, D. A. Fattal, N. R. Gauger, S. Fan, and R. G. Beausoleil, “Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints,” Opt. Express 22(11), 12971–12981 (2014).
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D. Lin, P. Fan, E. Hasman, and M. L. Brongersma, “Dielectric gradient metasurface optical elements,” Science 345(6194), 298–302 (2014).
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2013 (4)

J. Lu and J. Vučković, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
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C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
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C. Pfeiffer and A. Grbic, “Cascaded metasurfaces for complete phase and polarization control,” Appl. Phys. Lett. 102(23), 231116 (2013).
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D. Fattal, Z. Peng, T. Tran, S. Vo, M. Fiorentino, J. Brug, and R. G. Beausoleil, “A multi-directional backlight for a wide-angle glasses-free three-dimensional display,” Nature 495(7441), 348–351 (2013).
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2011 (2)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 308–321 (2011).
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E. Schonbrun, K. Seo, and K. Crozier, “Reconfigurable imaging systems using elliptical nanowires,” Nano Lett. 11(10), 4299–4303 (2011).
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2010 (1)

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010).
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2003 (1)

E. Hasman, V. Kleiner, G. Biener, and A. Niv, “Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics,” Appl. Phys. Lett. 82(3), 328–330 (2003).
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2002 (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
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2000 (1)

M. B. Giles and N. A. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbul. Combust. 65(3/4), 393–415 (2000).
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1999 (2)

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, and D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” J. Aircr. 36(1), 61–74 (1999).
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1998 (1)

Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proc. IEEE 86(11), 2278–2324 (1998).
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1993 (1)

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L. J. Rudin, S. Osher, and E. Fatemi, “Nonlinear total-variation-based noise removal algorithms,” Phys. D 60(1-4), 259–268 (1992).
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1991 (1)

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S. M. Kamali, E. Arbabi, A. Arbabi, and A. Faraon, “A review of dielectric optical metasurfaces for wavefront control,” Nanophotonics 7(6), 1041–1068 (2018).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces,” Optica 4(6), 625–632 (2017).
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E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “High efficiency double-wavelength dielectric metasurface lenses with dichroic birefringent meta-atoms,” Opt. Express 24(16), 18468–18477 (2016).
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S. M. Kamali, A. Arbabi, E. Arbabi, Y. Horie, and A. Faraon, “Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces,” Nat. Commun. 7(1), 11618 (2016).
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S. M. Kamali, A. Arbabi, E. Arbabi, Y. Horie, and A. Faraon, “Decoupling optical function and geometrical form using conformal flexible dielectric metasurfaces,” Nat. Commun. 7(1), 11618 (2016).
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A. Arbabi, E. Arbabi, S. M. Kamali, Y. Horie, S. Han, and A. Faraon, “Miniature optical planar camera based on a wide-angle metasurface doublet corrected for monochromatic aberrations,” Nat. Commun. 7(1), 13682 (2016).
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A. Arbabi, Y. Horie, A. J. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat. Commun. 6(1), 7069 (2015).
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N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
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A. Arbabi, Y. Horie, A. J. Ball, M. Bagheri, and A. Faraon, “Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays,” Nat. Commun. 6(1), 7069 (2015).
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J. P. Balthasar Mueller, N. A. Rubin, R. C. Devlin, B. Groever, and F. Capasso, “Metasurface polarization optics: independent phase control of arbitrary orthogonal states of polarization,” Phys. Rev. Lett. 118(11), 113901 (2017).
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N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50(12), 2751–2758 (2002).
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S. Vo, D. Fattal, W. V. Sorin, Z. Peng, T. Tran, M. Fiorentino, and R. G. Beausoleil, “Sub-wavelength grating lenses with a twist,” IEEE Photonics Technol. Lett. 26(13), 1375–1378 (2014).
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Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proc. IEEE 86(11), 2278–2324 (1998).
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Bernet, S.

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Figures (8)

Fig. 1.
Fig. 1. Overview of metasurface optimization. (a) Electron micrograph of a representative metasurface lens, reprinted from [53]. This metasurface is composed of multiple TiO2 nanopillars (meta-atoms) arranged in a rectangular lattice. By adjusting the widths of individual meta-atoms, the output phase is controlled. (b) General schematic of the types of designs considered in this paper: A series of glass substrates containing metasurfaces on either side are cascaded to form an optical system. Individual meta-atom geometries must be optimized to achieve a desired output intensity at the image plane. (c) In this paper, we consider square, TiO2 meta-atoms of constant height, and tunable width w. Our simulations assume a lattice period of 400 nm. (d) An FDTD parameter sweep plots output phase as a function of meta-atom width. Results were smoothed using a moving average filter to remove resonances at specific widths/wavelengths. This dataset is used for all of the design examples included later in the paper. The individual plots are color-coded according to wavelength, with the bluest wavelength corresponding to λ = 480 nm, the reddest wavelength corresponding to λ = 640 nm, and 20 nm increments between wavelengths. (e) Using finite differences, the derivative of phase with respect to meta-atom width is estimated.
Fig. 2.
Fig. 2. Inverse design of an achromatic doublet. (a) Axial schematic of design. (b) Optimized meta-atom widths of front and back metasurface. Widths of individual meta-atoms are plotted as a function of aperture position. (c) Unwrapped phase across metasurface apertures at λ = 640 nm. (d) Cross-section of focal plane intensity at all design wavelengths. Individual plots are color-coded according to wavelength, with the bluest wavelength corresponding to λ = 480 nm, the reddest wavelength corresponding to λ = 640 nm, and 20 nm increments between wavelengths. (e) Transmission and focusing efficiency calculations at all design wavelengths.
Fig. 3.
Fig. 3. XZ-slices of the point spread function for doublet and singlet metalenses. (a) XZ-slices of focal-region intensity for an optimized metasurface doublet. (b) XZ-slices of focal-region intensity for an optimized metasurface singlet. Intensity color scale for Figs. 3(a) and 3(b) are identical.
Fig. 4.
Fig. 4. Inverse design of a spectrally-multiplexed holographic element. (a) Axial schematic of design. (b) Optimized design of front and back metasurfaces. Widths of individual meta-atoms are plotted. (c) Top: Simulated output intensity distributions for optimized design. Bottom: Target output intensity distributions.
Fig. 5.
Fig. 5. An optical neural network for handwritten digit classification. (a) Schematic of design. Handwritten digits are projected on the front of the network. Light propagates through the optical system and is incident on an array of ten detectors at the image plane. Digits are classified based on which detector receives the most intensity. (b) Simulated test inputs and outputs of optimized design. A different detector receives the majority of output intensity, depending upon which digit is projected at the input. (c) Confusion matrix for MNIST testing dataset. (d) Optimized metasurface designs. Widths of individual meta-atoms are plotted.
Fig. 6.
Fig. 6. Modelling multiple reflections between metasurface layers: Achromatic metalens doublet revisited. (a) Simulation schematic: The input light beam is first reflected off the back metasurface, then the front metasurface then transmitted through the back surface (blue arrows). The resulting electric field is added to the contribution from purely transmitted light (red lines). (b) Difference in image-plane intensity between simulation considering reflected light, and purely transmitted light only. Compare results with those plotted in Fig. 3(d).
Fig. 7.
Fig. 7. Modelling errors associated with the local periodic approximation. (a) Simulation schematic: A single meta-atom of 200 nm width is surrounded on either side by meta-atoms of different widths. (b) Phase changes associated with neighboring meta-atoms of different widths. All phase values are reported relative to the case in which the all meta-atoms have an identical width of 200 nm. (c) FDTD simulation results. The real components of x-polarized and y-polarized input light are shown. In descending order, each row of panels shows results for {85 nm, 200 nm, 300 nm, and 375 nm} neighboring meta-atom widths. The meta-atoms and substrate boundaries are outlined in white, and the plane at which the output phase is measured is indicated with a black dotted line.
Fig. 8.
Fig. 8. Misalignment within an optical neural network reduces classification accuracy. (a) Simulation schematic: The third substrate is axially misaligned by 20 µm. (b) A representative input image, and output intensity distributions associated with the perfectly aligned, and misaligned optical systems respectively. (c) Confusion matrix associated with running test images through the misaligned optical neural network.

Equations (14)

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Minimize: C = j = 1 J k = 1 K s = 1 S ( I s λ j , f k I s d e s , λ j , f k ) 2 Subject to: w m i n { w 1 , w 2 , , w M } w m a x
w m = w m + α d C d w m m
E o u t , λ j , f k = ( m = 1 M F P m , λ j F Φ w m , λ j ) E i n , λ j , f k
P ξ , ξ m , λ j = exp ( i 2 π z m n m λ j 1 ( λ j ν ξ / n m ) 2 )
d C d w m = d ϕ w m , λ j d w m d C d ϕ w m , λ j
d C d ϕ s w m , λ j = s = 1 S ( d C d E s o u t , λ j , f k ) ( d E s o u t , λ j , f k d ϕ s w m , λ j ) = 4 { s = 1 S ( E s o u t , λ j , f k ( I s λ j , f k I s d e s , λ j , f k ) ) ( d E s o u t , λ j , f k d ϕ s w m , λ j ) } = 4 { ( E o u t , λ j , f k ( I λ j , f k I d e s , λ j , f k ) ) ( d E o u t , λ j , f k d ϕ s w m , λ j ) }
d E o u t , λ j , f k d ϕ s w m , λ j = ( m = m M F P m , λ j F Φ w m , λ j ) Δ s E m 1 , λ j , f k
Δ ξ , ξ s = { i if  ξ = s 0 Otherwise
E m 1 , λ j , f k = { E i n , λ j , f k if  m 1 = 0 ( m = 1 m 1 F P m , λ j F Φ w m , λ j ) E i n , λ j , f k Otherwise
d C d ϕ s w m , λ j = 4 { ( E o u t , λ j , f k ( I λ j , f k I d e s , λ j , f k ) ) ( m = m M F P m , λ j F Φ w m , λ j ) Δ ξ , ξ s E m 1 , λ j , f k }
a λ j , f k = E o u t , λ j , f k ( I λ j , f k I d e s , λ j , f k )
d C d ϕ s w m , λ j = 4 { E m 1 , λ j , f k Δ s ( m = 0 M m Φ w M m , λ j F P M m , λ j F ) a λ j , f k }
d C d ϕ w m , λ j = 4 { i ( E m 1 , λ j , f k ) T ( m = 0 M m Φ w M m , λ j F P M m , λ j F ) a }
O λ i = 1 s = 1 S ( I s λ i I s d e s , λ i ) 2 s = 1 S ( I s d e s , λ i ) 2

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