Abstract

We present a computational framework for efficient optimization-based “inverse design” of large-area “metasurfaces” (subwavelength-patterned surfaces) for applications such as multi-wavelength/multi-angle optimizations, and demultiplexers. To optimize surfaces that can be thousands of wavelengths in diameter, with thousands (or millions) of parameters, the key is a fast approximate solver for the scattered field. We employ a “locally periodic” approximation in which the scattering problem is approximated by a composition of periodic scattering problems from each unit cell of the surface, and validate it against brute-force Maxwell solutions. This is an extension of ideas in previous metasurface designs, but with greatly increased flexibility, e.g. to automatically balance tradeoffs between multiple frequencies or to optimize a photonic device given only partial information about the desired field. Our approach even extends beyond the metasurface regime to non-subwavelength structures where additional diffracted orders must be included (but the period is not large enough to apply scalar diffraction theory).

© 2018 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)

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

K. H. Matlack, M. Serra-Garcia, A. Palermo, S. D. Huber, and C. Daraio, “Designing perturbative metamaterials from discrete models,” Nat. Mater. 17, 323 (2018).
[Crossref] [PubMed]

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] [PubMed]

B. Groever, C. Roques-Carmes, S. J. Byrnes, and F. Capasso, “Substrate aberration and correction for meta-lens imaging: an analytical approach,” Appl. Opt. 57, 2973–2980 (2018).
[Crossref] [PubMed]

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

M. Kim and G. V. Eleftheriades, “Design and demonstration of impedance-matched dual-band chiral metasurfaces,” Sci. Rep. 8, 3449 (2018).
[Crossref] [PubMed]

S. Tcvetkova, D.-H. Kwon, A. Díaz-Rubio, and S. Tretyakov, “Near-perfect conversion of a propagating plane wave into a surface wave using metasurfaces,” Phys. Rev. B 97, 115447 (2018).
[Crossref]

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]

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. 4, eaar4206 (2018).
[Crossref] [PubMed]

J. Yang, D. Sell, and J. A. Fan, “Freeform metagratings based on complex light scattering dynamics for extreme, high efficiency beam steering,” Ann. Phys. 530, 1700302 (2018).
[Crossref]

2017 (9)

O. P. Bruno, E. Garza, and C. Pérez-Arancibia, “Windowed Green function method for nonuniform open-waveguide problems,” IEEE Trans. Antennas Propag. 65, 4684–4692 (2017).
[Crossref]

O. P. Bruno and C. Pérez-Arancibia, “Windowed Green function method for the helmholtz equation in the presence of multiply layered media,” Proc. R. Soc. A 473, 20170161 (2017).
[Crossref] [PubMed]

J. Cheng, S. Inampudi, and H. Mosallaei, “Optimization-based dielectric metasurfaces for angle-selective multifunctional beam deflection,” Sci. Rep. 7, 12228 (2017).
[Crossref] [PubMed]

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 (2017).
[Crossref]

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photon. 11, 415–420 (2017).
[Crossref]

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]

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

D. Sell, J. Yang, S. Doshay, and J. A. Fan, “Periodic dielectric metasurfaces with high-efficiency, multiwavelength functionalities,” Adv. Opt. Mater. 5, 1700645 (2017).
[Crossref]

2016 (2)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotech. 11, 23–36 (2016).
[Crossref]

O. P. Bruno, M. Lyon, C. Pérez-Arancibia, and C. Turc, “Windowed Green function method for layered-media scattering,” SIAM J. Appl. Math. 76, 1871–1898 (2016).
[Crossref]

2015 (5)

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]

S. Tretyakov, “Metasurfaces for general transformations of electromagnetic fields,” Phil. Trans. R. Soc. A 373, 20140362 (2015).
[Crossref] [PubMed]

K. Achouri, M. A. Salem, and C. Caloz, “General metasurface synthesis based on susceptibility tensors,” IEEE Trans. Antennas Propag. 63, 2977–2991 (2015).
[Crossref]

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. Photon. 9, 374–377 (2015).
[Crossref]

2014 (2)

A. Epstein and G. V. Eleftheriades, “Floquet-Bloch analysis of refracting Huygens metasurfaces,” Phys. Rev. B 90, 235127 (2014).
[Crossref]

A. Epstein and G. V. Eleftheriades, “Passive lossless Huygens metasurfaces for conversion of arbitrary source field to directive radiation,” IEEE Trans. Antennas Propag. 62, 5680–5695 (2014).
[Crossref]

2013 (4)

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

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]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1232009 (2013).
[Crossref] [PubMed]

N. Yu, P. Genevet, F. Aieta, M. A. Kats, R. Blanchard, G. Aoust, J.-P. Tetienne, Z. Gaburro, and F. Capasso, “Flat optics: Controlling wavefronts with optical antenna metasurfaces,” IEEE J. Sel. Top. Quantum Electron. 19, 4700423 (2013).
[Crossref]

2012 (5)

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]

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54, 10–35 (2012).
[Crossref]

M. Costabel and F. Le Louër, “Shape derivatives of boundary integral operators in electromagnetic scattering. part i: Shape differentiability of pseudo-homogeneous boundary integral operators,” IEOT 72, 509–535 (2012).

M. Costabel and F. Le Louër, “Shape derivatives of boundary integral operators in electromagnetic scattering. part ii: Application to scattering by a homogeneous dielectric obstacle,” IEOT 73, 17–48 (2012).

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]

2011 (1)

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).
[Crossref] [PubMed]

2010 (1)

2009 (2)

C. L. Holloway, A. Dienstfrey, E. F. Kuester, J. F. O’Hara, A. K. Azad, and A. J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials,” Metamaterials 3, 100–112 (2009).
[Crossref]

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

2003 (1)

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antennas Propag. 51, 2641–2651 (2003).
[Crossref]

2002 (2)

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[Crossref]

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

2001 (1)

N. J. Champagne, J. G. Berryman, and H. M. Buettner, “FDFD: A 3D finite-difference frequency-domain code for electromagnetic induction tomography,” J. Comput. Phys. 170, 830–848 (2001).
[Crossref]

1999 (1)

1998 (1)

T. Gerstner and M. Griebel, “Numerical integration using sparse grids,” Numer. Algorithms 18, 209 (1998).
[Crossref]

1995 (1)

R. Mittra and U. Pekel, “A new look at the perfectly matched layer (PML) concept for the reflectionless absorption of electromagnetic waves,” IEEE Microw. Wirel. Compon. Lett. 5, 84–86 (1995).

Achouri, K.

K. Achouri, M. A. Salem, and C. Caloz, “General metasurface synthesis based on susceptibility tensors,” IEEE Trans. Antennas Propag. 63, 2977–2991 (2015).
[Crossref]

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]

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[Crossref]

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ACS Photonics (1)

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]

Adv. Opt. Mater. (1)

D. Sell, J. Yang, S. Doshay, and J. A. Fan, “Periodic dielectric metasurfaces with high-efficiency, multiwavelength functionalities,” Adv. Opt. Mater. 5, 1700645 (2017).
[Crossref]

Ann. Phys. (1)

J. Yang, D. Sell, and J. A. Fan, “Freeform metagratings based on complex light scattering dynamics for extreme, high efficiency beam steering,” Ann. Phys. 530, 1700302 (2018).
[Crossref]

Appl. Opt. (2)

Eng. Optim. (1)

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Figures (8)

Fig. 1
Fig. 1 Schematic of our design method: exact Maxwell scattering solutions for a set of periodic unit cells (top) are composed into an approximate solution for an arbitrary aperiodic composition (right), and this approximation is then used for large-scale optimization to determine the metasurface parameters to maximize a given objective (e.g. the focal intensity, bottom). The performance of the final design can then be fed back into adjusting the design of the unit cell (left).
Fig. 2
Fig. 2 Left: an arbitrary aperiodic metasurface (top) is approximated by solving a set of periodic scattering problems (bottom), one for each unit cell, to obtain the scattered field just above the surface (horizontal line segments). Right: 0th diffracted-order amplitude (top) and phase (bottom) of periodic subproblems as a function of the pillar width. This is precomputed for several widths (markers) and interpolated as needed.
Fig. 3
Fig. 3 Bottom: geometry of a metasurface designed for a 5-degree incident plane wave of wavelength 532 nm and focal length 14.7 µm (numerical aperture of 0.3) using the wavefront method. This design produces a field with the needed phase (middle). Top: |Ez|2 intensity plot shows focusing to the target focal spot.
Fig. 4
Fig. 4 Left: |Ez|2 intensity plot of the scattered field of a metalens using our locally periodic approximate solver showing good agreement with a brute force calculation (middle). Right: the field sections computed by the two solvers show perfect agreement close to the focal lines (top). This agreement starts to deteriorate the closer the section is to the metasurface (bottom).
Fig. 5
Fig. 5 Bottom: the focal line of the scattered field for the three target wavelengths (blue, green, and red) show a clear focusing on the target focal axis. Middle: sensitivity plot for the focal length with respect to the wavelength show chromatic aberration, and each wavelength objective creates a “spurious focus” (local maximum along along the focal axis) on the focal axis at other wavelengths. The red spots represent the foci for each wavelength, and we clearly see chromatic aberration. Top: intensity at the target spot vs. wavelength.
Fig. 6
Fig. 6 Bottom: focal lines for the three target wavelengths (blue, green and red) focus on points sixty microns apart. Top: the field produced by our design focuses on the desired foci, the high-intensity regions for blue (left) and red (right) are tilted because their foci are off-axis.
Fig. 7
Fig. 7 Left: the focal lines for 0-degrees, 3-degrees, 6-degrees, and 9-degrees angles of incidence show four foci with the maximum intensity at the target focal spot (at x = 0), the other three peaks correspond to the other three target angles. Right: corresponding produced field around the foci, the focal spot becomes more tilted as the angle of incidence increases.
Fig. 8
Fig. 8 Bottom: the geometry (left) from intensity optimization shows big variations in the width of the pillar, and produce good focusing when simulated with a brute force simulation (right), or our locally periodic solver (middle) which includes the diffractive orders ±1. Top: the geometry (left) from wavefront optimization shows poor focusing both using our locally periodic solver (middle) or a brute force calculation (right). All the intensity plots have the same color scale.

Equations (7)

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[ J K ] = δ ( y y 0 ) [ n ^ × H n ^ × E ]
E z ( x ) = surface G ( x , x ) E z ( x ) d x
min s 0 , ϕ 0 , p | s ( p ( x ) ) s 0 a ( x ) e i ϕ ( x ) + i ϕ 0 | 2 d x ,
max parameters [ min λ wavelengths objective ( parameters , λ ) ] .
max t , parameters t subject to t objective ( parameters , λ ) for λ wavelengths .
f p = 2 ( ( s ( p ( x ) ) s 0 a ( x ) e i ϕ ( x ) + i ϕ 0 ) * s ( p ( x ) ) d x ) f s 0 = 2 ( ( s ( p ( x ) ) s 0 a ( x ) e i ϕ ( x ) + i ϕ 0 ) * a ( x ) e i ϕ ( x ) + i ϕ 0 d x ) f ϕ 0 = 2 ( ( s ( p ( x ) ) s 0 a ( x ) e i ϕ ( x ) + i ϕ 0 ) * i s 0 a ( x ) e i ϕ ( x ) + i ϕ 0 d x ) ,
f p = 2 ( ( y = y 0 G ( x , ( x , 0 ) ) s ( p ( x ) ) d x ) * y = y 0 G ( x , ( x , 0 ) ) s ( p ( x ) ) d x ) .

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