Abstract

The design of intrinsically flat two-dimensional optical components, i.e., metasurfaces, generally requires an extensive parameter search to target the appropriate scattering properties of their constituting building blocks. Such design methodologies neglect important near-field interaction effects, playing an essential role in limiting the device performance. Optimization of transmission, phase-addressing and broadband performances of metasurfaces require new numerical tools. Additionally, uncertainties and systematic fabrication errors should be analysed. These estimations, of critical importance in the case of large production of metaoptics components, are useful to further project their deployment in industrial applications. Here, we report on a computational methodology to optimize metasurface designs. We complement this computational methodology by quantifying the impact of fabrication uncertainties on the experimentally characterized components. This analysis provides general perspectives on the overall metaoptics performances, giving an idea of the expected average behavior of a large number of devices.

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

Full Article  |  PDF Article

Corrections

22 February 2019: A typographical correction was made to the title.


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References

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S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4, eaar2114 (2018).
[Crossref] [PubMed]

F. Ding, Y. Chen, and S. I. Bozhevolnyi, “Metasurface-based polarimeters,” Appl. Sci. 8, 594 (2018).
[Crossref]

H. Pahlevaninezhad, M. Khorasaninejad, Y.-W. Huang, Z. Shi, L. P. Hariri, D. C. Adams, V. Ding, A. Zhu, C.-W. Qiu, F. Capasso, and M. J. Suter, “Nano-optic endoscope for high-resolution optical coherence tomography in vivo,” Nat. Photonics 12, 540 (2018).
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[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).
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[Crossref]

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[Crossref]

R. Sawant, P. Bhumkar, A. Zhu, P. Ni, F. Capasso, and P. Genevet, “Mitigating chromatic dispersion with hybrid optical metasurfaces,” Adv. Mater. 31, 1805555 (2018).

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[Crossref] [PubMed]

2017 (9)

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[Crossref]

V. Egorov, M. Eitan, and J. Scheuer, “Genetically optimized all-dielectric metasurfaces,” Opt. Express 25, 2583–2593 (2017).
[Crossref]

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]

J. R. Ong, H. S. Chu, V. H. Chen, A. Y. Zhu, and P. Genevet, “Freestanding dielectric nanohole array metasurface for mid-infrared wavelength applications,” Opt. Lett. 42, 2639–2642 (2017).
[Crossref] [PubMed]

S. Lanteri, C. Scheid, and J. Viquerat, “Analysis of a generalized dispersive model coupled to a DGTD method with application to nanophotonics,” SIAM J. Sci. Comp. 39, A831–A859 (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]

K. D. Donda and R. S. Hegde, “Rapid design of wide-area heterogeneous electromagnetic metasurfaces beyond the unit-cell approximation,” Prog. In Electromagn. Res. M 60, 1–10 (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).

H. Zhao, B. Quan, X. Wang, C. Gu, J. Li, and Y. Zhang, “Demonstration of orbital angular momentum multiplexing and demultiplexing based on a metasurface in the terahertz band,” ACS Photonics 5, 1726–1732 (2017).
[Crossref]

2016 (6)

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]

W. T. Chen, P. Török, M. R. Foreman, C. Y. Liao, W.-Y. Tsai, P. R. Wu, and D. P. Tsai, “Integrated plasmonic metasurfaces for spectropolarimetry,” Nanotechnology 27, 224002 (2016).
[Crossref] [PubMed]

M. Mehmood, S. Mei, S. Hussain, K. Huang, S. Siew, L. Zhang, T. Zhang, X. Ling, H. Liu, J. Teng, A. Danner, S. Zhang, and C. Qiu, “Visible-frequency metasurface for structuring and spatially multiplexing optical vortices,” Adv. Mater. 28, 2533–2539 (2016).
[Crossref] [PubMed]

H.-T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Reports on Prog. Phys. 79, 076401 (2016).
[Crossref]

J. B. Mueller, K. Leosson, and F. Capasso, “Ultracompact metasurface in-line polarimeter,” Optica 3, 42–47 (2016).
[Crossref]

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3, 628–633 (2016).
[Crossref]

2015 (5)

A. Pors, M. G. Nielsen, and S. I. Bozhevolnyi, “Plasmonic metagratings for simultaneous determination of stokes parameters,” Optica 2, 716–723 (2015).
[Crossref]

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, 7069 (2015).
[Crossref] [PubMed]

D. Wen, F. Yue, G. Li, G. Zheng, K. Chan, S. Chen, M. Chen, K. F. Li, P. W. H. Wong, K. W. Cheah, E. Y. B. Pun, S. Zhang, and X. Chen, “Helicity multiplexed broadband metasurface holograms,” Nat. Commun. 6, 8241 (2015).
[Crossref] [PubMed]

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. B. Giles, “Multilevel monte carlo methods,” Acta Numer. 24, 259–328 (2015).
[Crossref]

2014 (1)

S. Collin, “Nanostructure arrays in free-space: optical properties and applications,” Reports on Prog. Phys. 77, 126402 (2014).
[Crossref]

2013 (2)

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

A. C. Austin and C. D. Sarris, “Efficient analysis of geometrical uncertainty in the fdtd method using polynomial chaos with application to microwave circuits,” IEEE Transactions on Microw. Theory Tech. 61, 4293–4301 (2013).
[Crossref]

2012 (1)

C. J. Chang-Hasnain and W. Yang, “High-contrast gratings for integrated optoelectronics,” Adv. Opt. Photonics 4, 379–440 (2012).
[Crossref]

2010 (1)

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4, 466 (2010).
[Crossref]

2008 (1)

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. on Numer. Analysis 46, 2309–2345 (2008).
[Crossref]

2007 (1)

R. Gaignaire, S. Clénet, B. Sudret, and O. Moreau, “3-d spectral stochastic finite element method in electromagnetism,” IEEE Transactions on Magn. 43, 1209–1212 (2007).
[Crossref]

2006 (2)

C. Chauviere, J. S. Hesthaven, and L. Lurati, “Computational modeling of uncertainty in time-domain electromagnetics,” SIAM J. on Sci. Comput. 28, 751–775 (2006).
[Crossref]

P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a coupled bloch-mode insight,” J. Light. Technol. 24, 2442–2449 (2006).
[Crossref]

2005 (2)

H. G. Matthies and A. Keese, “Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations,” Comput. Methods Appl. Mech. Eng. 194, 1295–1331 (2005).
[Crossref]

U. Levy, H.-C. Kim, C.-H. Tsai, and Y. Fainman, “Near-infrared demonstration of computer-generated holograms implemented by using subwavelength gratings with space-variant orientation,” Opt. Lett. 30, 2089–2091 (2005).
[Crossref] [PubMed]

2004 (2)

I. Babuska, R. Tempone, and G. E. Zouraris, “Galerkin finite element approximations of stochastic elliptic partial differential equations,” SIAM J. on Numer. Analysis 42, 800–825 (2004).
[Crossref]

C. F. Mateus, M. C. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62/spl mu/m) using a subwavelength grating,” IEEE Photonics Technol. Lett. 16, 1676–1678 (2004).
[Crossref]

1981 (1)

M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” JOSA 71, 811–818 (1981).
[Crossref]

1969 (1)

V. A. Epanechnikov, “Non-parametric estimation of a multivariate probability density,” Theory Probab. & Its Appl. 14, 153–158 (1969).
[Crossref]

Adams, D. C.

H. Pahlevaninezhad, M. Khorasaninejad, Y.-W. Huang, Z. Shi, L. P. Hariri, D. C. Adams, V. Ding, A. Zhu, C.-W. Qiu, F. Capasso, and M. J. Suter, “Nano-optic endoscope for high-resolution optical coherence tomography in vivo,” Nat. Photonics 12, 540 (2018).
[Crossref]

Aieta, F.

P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139–152 (2017).
[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]

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).

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3, 628–633 (2016).
[Crossref]

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, 7069 (2015).
[Crossref] [PubMed]

E. Arbabi, S. M. Kamali, A. Arbabi, and A. Faraon, “Full stokes imaging polarimetry using dielectric metasurfaces,” arXiv preprint arXiv:1803.03384 (2018).

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]

E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Multiwavelength polarization-insensitive lenses based on dielectric metasurfaces with meta-molecules,” Optica 3, 628–633 (2016).
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Figures (12)

Fig. 1
Fig. 1 (a) Illustrative schematic of the angular deflection property of a phase gradient metasurface. Metasurfaces based on arrays of unit cells distributed in a periodic manner with Γ > λ, allowing discrete diffracted modes only. In the most general case, and assuming an incident light at normal incidence, in transmission, the light will be deflected into the different orders (permitted by the periodicity of the microscopic structure). A single diffracted beam is achievable by controlling the light diffracted by each unit cell. In (b), the device works essentially as conventional echelette blazed grating. Replacing the periodic echelette with a subwavelength array of nanoridges (c), we create a metasurface. A strong influence of the nanoridge response as a function of their width is obtained for TM polarized incident light. The width of a periodic unit cell is denoted by Γ, the centers of the ridges by x = [x1, . . ., xi, . . ., xN], where i = 1, . . ., N is the index of the nanoridge and N the total number of nanoridges. In the same manner, we define the widths of all ridges by δx = [δx1, .., δxi, .., δxN]. (d) Typical broadband response of the transmission efficiency for an optimized metasurface obtained using two different electromagnetic simulation solvers, the Discontinuous Galerkin Time-Domain solver (DGTD) solver and the Rigorous Coupled Wave Analysis (RCWA) solver in orange dashed.
Fig. 2
Fig. 2 Diffraction efficiency spectra for θ = 27°, h = 1000 nm, d = 1322 nm, �� = 90 nm, and varying N. The design wavelength is λdesign = 600 nm. The upper and lower plots correspond to a TM and TE polarization, respectively.
Fig. 3
Fig. 3 Diffraction efficiency spectra for θ = 27°, �� = 90 nm, different N and varying grating height h. The design wavelength and height are λdesign = 600 nm and h = 1000 nm, respectively.
Fig. 4
Fig. 4 Top: Diffraction efficiency spectra for h = 1000 nm, �� = 50nm (left) and �� = 90 nm (right), and varying design angles θ and Γ(θ), respectively. The design wavelength is λdesign = 600 nm. Bottom: Angle dependence of η−1. The number of nanoridges N varies depending on the angle. The design wavelength is again λdesign = 600 nm.
Fig. 5
Fig. 5 Experimental setup and k-space microscopy images of metagrating deflection properties. (a) the broadband light transmitting through the metasurface is collected with a high microscope objective (N A = 0.9) of an inverted microscope. A modified 4f lensing system relays the k-space information to the entrance slit of an imaging spectrometer. (b) shows a typical k-space image of the light transmitted through the metagrating when the spectrometer is set in imaging mode, i.e. by looking at the back focal plane image reflected toward the CCD camera by the diffraction grating oriented to the specular reflection angle. The deflected fields at m = −1 diffraction order appears as an elongated spot around 45°. The red circle indicates the NA of the microscope objective. (c) The spectral response of the components is characterized by closing the spectrometer entrance slit and spectrally dispersing the vertical k-space information. The image displays both spectral information (along the x dimension) and transverse ky momentum information. The m = −1 has higher intensity indicating strong blazing behaviour as expected from the design. Note also the typical k-space angular dispersion properties of the metagrating as function of the wavelength.
Fig. 6
Fig. 6 Example of a fabricated metasurface. (a, top) The schematics indicate the fabrication work flow, starting from GaN growth on Sapphire substrate, followed by eBeam lithography on HSQ resist which is cured and used as a hard mask for RIE etching. The excess of resist is then washed away using BOE etch. (b) A typical scanning electron micrograph of a fabricated structure. (c) Zoomed SEM images of the different nanoridge gratings with increasing number of elements per period from 2 to 6. In the large scale figure, the assemblies of nanoridges highlighted in yellow represent unit cells of the metasurfaces. In the inset, a zoomed of unit cells revealing the sub-unit nanoridges chosen to discretize the 2π phase ramp. In this example, the unit cells are composed of 2 to 6 nanoridges indicated by the colors ranging from red to violet. (d) Fourier plane optical transmission curves. The red curves represent the metasurface transmission as a function of the deflection angles for various nanoridges per periods. The data have been measured at 600 nm. (e) Summary of the deflection efficiencies as function of the number of elements per period.
Fig. 7
Fig. 7 Measured diffraction efficiency spectra for TM and TE incident polarizations. The structure is composed of 5 nanoridges per unit cell as presented in Fig. 6 for θ = 27°, h = 1000 nm, �� = 90 nm, and varying polarization. The design wavelength is λdesign = 600 nm. (a) and (b) are raw experimental k space images obtained with a microscope objective (0.9 NA). (c) are normalized efficiencies for S and P polarizations.
Fig. 8
Fig. 8 Input and output probability density functions.
Fig. 9
Fig. 9 Broadband UQ results for uniform input distributions and N = 5 nanoridges per period.
Fig. 10
Fig. 10 Sketches of considered numerical models with N = 5 nanoridges per period. Black solid lines illustrate periodic boundary conditions. Black dots indicate uncertain parameters. Top: unit cell consisting of n = 1 grating as considered in Section 3.1 and 3.2. Middle, bottom: increased number of gratings per unit cell (top: n = 2; bottom: n = 4) in order to systematically reduce the influence of the periodic boundary conditions (Section 3.3).
Fig. 11
Fig. 11 Effect of periodic boundary conditions on UQ results. Mean value (left) and standard deviation (right). The periodic part consists of an increasing number of gratings n. The mean value decreases for an increasing number of gratings before saturating after roughly n = 15. The standard deviation and hence, the uncertainty due to the manufacturing are decreasing continuously.
Fig. 12
Fig. 12 Uncertainty quantification in quasi-periodic structures.

Tables (5)

Tables Icon

Table 1 UQ results for η−1 in the single frequency case and for uniform input distributions together with measured η−1 of the fabricated structure.

Tables Icon

Table 2 Grating geometry. Numerical values of the grating’s geometric parameters for a deflection angle θ = 27°, grating period Γ = 1322 nm and different numbers of nanoridges per period N. The optimizations were carried out using the larger minimal feature size �� = 90 nm.

Tables Icon

Table 3 Grating geometry. Numerical values of the grating’s geometric parameters for different deflection angles θ and hence grating periods Γ. The optimizations were carried out using the smaller minimal feature size �� = 50 nm. Here, we only show the setting with the optimal number of nanoridges per period.

Tables Icon

Table 4 Grating geometry. Numerical values of the grating’s geometric parameters for different deflection angles θ and hence grating periods Γ. The optimizations were carried out using the larger minimal feature size �� = 90 nm. Here, we only show the setting with the optimal number of nanoridges per period.

Tables Icon

Table 5 UQ results for an increasing number of nanoridges n per unit cell. Npar = 2nN denotes the number of uncertain parameters and MMC the number of Monte Carlo samples used to estimate the mean value En and the standard deviation σn.

Equations (6)

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min f N ( α , a ) subject to α l 1 1 , a l 1 = 1 .
𝔼 [ η 1 ] 𝔼 MC [ η 1 ] = 1 M MC m = 1 M MC η 1 ( m ) .
𝕍 [ η 1 ] 𝕍 MC [ η 1 ] = 1 M MC m = 1 M MC ( η 1 ( m ) 𝔼 MC [ η 1 ] ) 2 .
RMS = 𝕍 MC [ η 1 ] M MC .
δ x i = 𝔡 + α i ( Γ 02 N 𝔡 ) .
x i = Γ + δ x i 2 + { 0 i = 1 , j = 1 i 1 δ x j + 𝔡 + a j ( Γ N 𝔡 k = 1 N δ x k ) i > 1 .

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