Abstract

We demonstrate optimization of optical metasurfaces over 105–106 degrees of freedom in two and three dimensions, 100–1000+ wavelengths (λ) in diameter, with 100+ parameters per λ2. In particular, we show how topology optimization, with one degree of freedom per high-resolution “pixel,” can be extended to large areas with the help of a locally periodic approximation that was previously only used for a few parameters per λ2. In this way, we can computationally discover completely unexpected metasurface designs for challenging multi-frequency, multi-angle problems, including designs for fully coupled multi-layer structures with arbitrary per-layer patterns. Unlike typical metasurface designs based on subwavelength unit cells, our approach can discover both sub- and supra-wavelength patterns and can obtain both the near and far fields.

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

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References

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2018 (10)

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12, 659 (2018).
[Crossref]

V.-C. Su, C. H. Chu, G. Sun, and D. P. Tsai, “Advances in optical metasurfaces: fabrication and applications,” Opt. Express 26, 13148–13182 (2018).
[Crossref]

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-optimized multilayered metaoptics,” Phys. Rev. Appl. 9, 044030 (2018).
[Crossref]

R. Pestourie, C. Pérez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, “Inverse design of large-area metasurfaces,” Opt. Express 26, 33732–33747 (2018).
[Crossref]

C. Pérez-Arancibia, R. Pestourie, and S. G. Johnson, “Sideways adiabaticity: beyond ray optics for slowly varying metasurfaces,” Opt. Express 26, 30202–30230 (2018).
[Crossref] [PubMed]

F. Capasso, “The future and promise of flat optics: a personal perspective,” Nanophotonics 7, 953–957 (2018).
[Crossref]

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, 220 (2018).
[Crossref] [PubMed]

S. Wang, P. C. Wu, V.-C. Su, Y.-C. Lai, M.-K. Chen, H. Y. Kuo, B. H. Chen, Y. H. Chen, T.-T. Huang, J.-H. Wang, R.-M. Lin, C.-H. Kuan, T. Li, Z. Wang, S. Zhu, and D. P. Tsai, “A broadband achromatic metalens in the visible,” Nat. Nanotechnol. 13, 227 (2018).
[Crossref] [PubMed]

S. Shrestha, A. C. Overvig, M. Lu, A. Stein, and N. Yu, “Broadband achromatic dielectric metalenses,” Light. Sci. Appl. 7, 85 (2018).
[Crossref]

N. Zhao, S. Verweij, W. Shin, and S. Fan, “Accelerating convergence of an iterative solution of finite difference frequency domain problems via schur complement domain decomposition,” Opt. Express 26, 16925–16939 (2018).
[Crossref] [PubMed]

2017 (10)

P. S. Venkataram, J. Hermann, A. Tkatchenko, and A. W. Rodriguez, “Unifying microscopic and continuum treatments of van der Waals and Casimir interactions,” Phys. Rev. Lett. 118, 266802 (2017).
[Crossref] [PubMed]

Y. E. Lee, O. D. Miller, M. H. Reid, S. G. Johnson, and N. X. Fang, “Computational inverse design of non-intuitive illumination patterns to maximize optical force or torque,” Opt. Express 25, 6757–6766 (2017).
[Crossref] [PubMed]

S. M. Kamali, E. Arbabi, A. Arbabi, Y. Horie, M. Faraji-Dana, and A. Faraon, “Angle-multiplexed metasurfaces: Encoding independent wavefronts in a single metasurface under different illumination angles,” Phys. Rev. X 7, 041056 (2017).

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786 (2017).
[Crossref]

C. V. Poulton, A. Yaacobi, D. B. Cole, M. J. Byrd, M. Raval, D. Vermeulen, and M. R. Watts, “Coherent solid-state LIDAR with silicon photonic optical phased arrays,” Opt. Lett. 42, 4091–4094 (2017).
[Crossref] [PubMed]

A. Pick, B. Zhen, O. D. Miller, C. W. Hsu, F. Hernandez, A. W. Rodriguez, M. Soljačić, and S. G. Johnson, “General theory of spontaneous emission near exceptional points,” Opt. Express 25, 12325–12348 (2017).
[Crossref] [PubMed]

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17, 3752–3757 (2017).
[Crossref] [PubMed]

M. Khorasaninejad, W. T. Chen, A. Y. Zhu, J. Oh, R. C. Devlin, C. Roques-Carmes, I. Mishra, and F. Capasso, “Visible wavelength planar metalenses based on titanium dioxide,” IEEE J. Sel. Top. Quantum Electron. 23, 43–58 (2017).
[Crossref]

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, 1819–1824 (2017).
[Crossref] [PubMed]

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, 625–632 (2017).
[Crossref]

2016 (6)

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, 1190–1194 (2016).
[Crossref] [PubMed]

Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, “Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals,” Phys. Rev. Lett. 117, 107402 (2016).
[Crossref] [PubMed]

N. Rivera, I. Kaminer, B. Zhen, J. D. Joannopoulos, and M. Soljačić, “Shrinking light to allow forbidden transitions on the atomic scale,” Science 353, 263–269 (2016).
[Crossref] [PubMed]

Z. Lin, X. Liang, M. Lončar, S. G. Johnson, and A. W. Rodriguez, “Cavity-enhanced second-harmonic generation via nonlinear-overlap optimization,” Optica 3, 233–238 (2016).
[Crossref]

S. Tao, J. Cheng, and H. Mosallaei, “An integral equation based domain decomposition method for solving large-size substrate-supported aperiodic plasmonic array platforms,” MRS Commun. 6, 105–115 (2016).
[Crossref]

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, 7229–7234 (2016).
[Crossref] [PubMed]

2015 (4)

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, 374–377 (2015).
[Crossref]

M. Zhou, B. S. Lazarov, F. Wang, and O. Sigmund, “Minimum length scale in topology optimization by geometric constraints,” Comput. Methods Appl. Mech. Eng. 293, 266–282 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref] [PubMed]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref] [PubMed]

2014 (1)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13, 139 (2014).
[Crossref] [PubMed]

2013 (2)

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013).
[Crossref] [PubMed]

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 197401 (2013).
[Crossref]

2012 (4)

G. Kim, J. A. Domínguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20, 2814–2823 (2012).
[Crossref] [PubMed]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref] [PubMed]

V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233–2244 (2012).
[Crossref]

2011 (3)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5, 308–321 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).

A. W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Frequency-selective near-field radiative heat transfer between photonic crystal slabs: a computational approach for arbitrary geometries and materials,” Phys. Rev. Lett. 107, 114302 (2011).
[Crossref] [PubMed]

2010 (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[Crossref]

2002 (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[Crossref]

1994 (1)

T. F. Chan and T. P. Mathew, “Domain decomposition algorithms,” Acta Numer. 3, 61–143 (1994).
[Crossref]

1987 (1)

K. Svanberg, “The method of moving asymptotes — a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359–373 (1987).
[Crossref]

1981 (1)

Aieta, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347, 1342–1345 (2015).
[Crossref] [PubMed]

M. Khorasaninejad, F. Aieta, P. Kanhaiya, M. A. Kats, P. Genevet, D. Rousso, and F. Capasso, “Achromatic metasurface lens at telecommunication wavelengths,” Nano Lett. 15, 5358–5362 (2015).
[Crossref] [PubMed]

F. Aieta, P. Genevet, M. A. Kats, N. Yu, R. Blanchard, Z. Gaburro, and F. Capasso, “Aberration-free ultrathin flat lenses and axicons at telecom wavelengths based on plasmonic metasurfaces,” Nano Lett. 12, 4932–4936 (2012).
[Crossref] [PubMed]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science 334, 333–337 (2011).

Arbabi, A.

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, 625–632 (2017).
[Crossref]

S. M. Kamali, E. Arbabi, A. Arbabi, Y. Horie, M. Faraji-Dana, and A. Faraon, “Angle-multiplexed metasurfaces: Encoding independent wavefronts in a single metasurface under different illumination angles,” Phys. Rev. X 7, 041056 (2017).

Arbabi, E.

S. M. Kamali, E. Arbabi, A. Arbabi, Y. Horie, M. Faraji-Dana, and A. Faraon, “Angle-multiplexed metasurfaces: Encoding independent wavefronts in a single metasurface under different illumination angles,” Phys. Rev. X 7, 041056 (2017).

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, 625–632 (2017).
[Crossref]

Babinec, T. M.

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, 374–377 (2015).
[Crossref]

Bermel, P.

A. W. Rodriguez, O. Ilic, P. Bermel, I. Celanovic, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Frequency-selective near-field radiative heat transfer between photonic crystal slabs: a computational approach for arbitrary geometries and materials,” Phys. Rev. Lett. 107, 114302 (2011).
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A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
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Figures (5)

Fig. 1
Fig. 1 An arbitrary aperiodic multi-layered meta-structure (top) is approximated by solving a set of periodic scattering problems (bottom), one for each unit cell (shaded areas), to obtain the approximate near fields over the entire metasurface.
Fig. 2
Fig. 2 Multi-layered 2D metalens concentrator (NA=0.51) which can combine 11 incident angles to a single focus. The lens consists of five layers of TiO2, with thicknesses of 0.10λ, 0.14λ, 0.16λ, 0.20λ and 0.24λ respectively, situated above and within the silica substrate. Three different portions of the lens have been magnified for an easy viewing of the device geometry; note the scale bars. Full-wave simulations of the entire structure reveal diffraction-limited focusing for the 11 incident angles with average transmission efficiency of ≈ 40%.
Fig. 3
Fig. 3 Multi-layered 2D metalens (NA=0.45) chromatically corrected at the wavelengths λ1 = 650 nm, λ2 = 540 nm and λ3 = 470 nm. The lens consists of two layers of TiO2, with thicknesses of 0.5λ1 and 0.2λ1 respectively, situated above and within the silica substrate. Three different portions of the lens have been magnified for an easy viewing of the device geometry; note the scale bars. The far field profiles are obtained by full-wave simulations of the entire structure (the upper panel) and by locally periodic approximation (the lower panel), exhibiting at- or near-diffraction limited focusing.
Fig. 4
Fig. 4 Monochromatic 3D cylindrical metalens (NA=0.71) with aperiodic cells along the x axis. The lens consists of a single TiO2 layer above the silica substrate. Three different portions have been magnified for easy viewing; note the scale bars. The shaded area shows an example of a λ × λ cell. A full-wave simulation of the entire structure shows the diffraction-limited focusing.
Fig. 5
Fig. 5 Monochromatic 3D metalens (NA=0.37). A few portions of the lens have been magnified for easy viewing; note the scale bars. The lens consists of a single TiO2 layer above the silica substrate. The far field profile is obtained by locally periodic approximation, showing diffraction-limited focusing.

Equations (9)

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max { ¯ ( r ) } f ( E ; ¯ ) = | G 0 ( r 0 , r s ) E ( r s ) d r s | 2 0 ¯ 1 .
× μ 1 × E ω 2 E = i ω J
¯ f ( r ) = 2 ω 2 ( st bg ) Re { g * E ˜ ( r ) E ( r ) }
g = G 0 ( r 0 , r ) E ( r ) d r ,
× μ 1 × E ˜ ω 2 E ˜ = G 0 ( r 0 , r ) δ ( r r s ) .
max ¯ min i { f ω i , J i } ,
max ¯ , t t
t f ω i , J i 0 .
r ( ) = | × μ 1 × E LPA ω 2 E LPA + i ω J | 2