Abstract

Large-area metasurfaces composed of discrete wavelength-scale scatterers present an extremely large number of degrees of freedom to engineer an optical element. While these degrees of freedom provide tremendous design flexibility, they also present a central challenge in metasurface design: how to optimally leverage these degrees of freedom towards a desired optical function. Inverse design is an attractive solution for this challenge. Here, we report an inverse design method exploiting T-matrix scattering of ellipsoidal scatterers. Multi-functional, polarization multiplexed metasurfaces were designed using this approach. We also optimized the efficiency of an existing high numerical aperture (0.83) metalens using the proposed method, and report an increase in efficiency from 26% to 32%.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (4)

B. Yang, T. Liu, H. Guo, S. Xiao, and L. Zhou, “High-performance meta-devices based on multilayer meta-atoms: interplay between the number of layers and phase coverage,” Sci. Bull. 64(12), 823–835 (2019).
[Crossref]

E. Bayati, A. Zhan, S. Colburn, M. V. Zhelyeznyakov, and A. Majumdar, “Role of refractive index in metalens performance,” Appl. Opt. 58(6), 1460–1466 (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]

A. Zhan, R. Gibson, J. Whitehead, E. Smith, J. R. Hendrickson, and A. Majumdar, “Controlling three-dimensional optical fields via inverse mie scattering,” Sci. Adv. 5(10), eaax4769 (2019).
[Crossref]

2018 (5)

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]

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

H. Guo, J. Lin, M. Qiu, J. Tian, Q. Wang, Y. Li, S. Sun, Q. He, S. Xiao, and L. Zhou, “Flat optical transparent window: mechanism and realization based on metasurfaces,” J. Phys. D: Appl. Phys. 51(7), 074001 (2018).
[Crossref]

S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (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(3), 220–226 (2018).
[Crossref]

2017 (9)

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref]

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11(7), 415–420 (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(4), 041056 (2017).
[Crossref]

L. Hsu, M. Dupré, A. Ndao, J. Yellowhair, and B. Kanté, “Local phase method for designing and optimizing metasurface devices,” Opt. Express 25(21), 24974–24982 (2017).
[Crossref]

V. Egorov, M. Eitan, and J. Scheuer, “Genetically optimized all-dielectric metasurfaces,” Opt. Express 25(3), 2583–2593 (2017).
[Crossref]

S. C. Ligon, R. Liska, J. Stampfl, M. Gurr, and R. Mülhaupt, “Polymers for 3d printing and customized additive manufacturing,” Chem. Rev. 117(15), 10212–10290 (2017).
[Crossref]

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, “Celes: Cuda-accelerated simulation of electromagnetic scattering by large ensembles of spheres,” J. Quant. Spectrosc. Radiat. Transfer 199, 103–110 (2017).
[Crossref]

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref]

2016 (2)

S. Xiao, H. Mühlenbernd, G. Li, M. Kenney, F. Liu, T. Zentgraf, S. Zhang, and J. Li, “Helicity-preserving omnidirectional plasmonic mirror,” Adv. Opt. Mater. 4(5), 654–658 (2016).
[Crossref]

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).
[Crossref]

2015 (8)

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).
[Crossref]

S. Xiao, F. Zhong, H. Liu, S. Zhu, and J. Li, “Flexible coherent control of plasmonic spin-hall effect,” Nat. Commun. 6(1), 8360 (2015).
[Crossref]

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).
[Crossref]

K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[Crossref]

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 mm2 footprint,” Nat. Photonics 9(6), 378–382 (2015).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

P. Hansen and L. Hesselink, “Accurate adjoint design sensitivities for nano metal optics,” Opt. Express 23(18), 23899–23923 (2015).
[Crossref]

2013 (2)

2012 (2)

S. Sun, Q. He, S. Xiao, Q. Xu, X. Li, and L. Zhou, “Gradient-index meta-surfaces as a bridge linking propagating waves and surface waves,” Nat. Mater. 11(5), 426–431 (2012).
[Crossref]

V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012).
[Crossref]

2011 (2)

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 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(6054), 333–337 (2011).
[Crossref]

1996 (1)

1995 (1)

1988 (1)

J. B. Schneider and I. C. Peden, “Differential cross section of a dielectric ellipsoid by the T-matrix extended boundary condition method,” IEEE Trans. Antennas Propag. 36(9), 1317–1321 (1988).
[Crossref]

1971 (1)

P. C. Waterman, “Symmetry, unitarity, and geometry in electromagnetic scattering,” Phys. Rev. D 3(4), 825–839 (1971).
[Crossref]

Aieta, F.

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[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(6054), 333–337 (2011).
[Crossref]

Arbabi, A.

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11(7), 415–420 (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(4), 041056 (2017).
[Crossref]

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).
[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(1), 7069 (2015).
[Crossref]

Arbabi, E.

A. Arbabi, E. Arbabi, Y. Horie, S. M. Kamali, and A. Faraon, “Planar metasurface retroreflector,” Nat. Photonics 11(7), 415–420 (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(4), 041056 (2017).
[Crossref]

Babinec, T. M.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

Bagheri, M.

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).
[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(1), 7069 (2015).
[Crossref]

Ball, A. J.

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).
[Crossref]

Bayati, E.

Bhargava, S.

Capasso, F.

R. Pestourie, C. Pérez-Arancibia, Z. Lin, W. Shin, F. Capasso, and S. G. Johnson, “Inverse design of large-area metasurfaces,” Opt. Express 26(26), 33732–33747 (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(3), 220–226 (2018).
[Crossref]

F. Aieta, M. A. Kats, P. Genevet, and F. Capasso, “Multiwavelength achromatic metasurfaces by dispersive phase compensation,” Science 347(6228), 1342–1345 (2015).
[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(6054), 333–337 (2011).
[Crossref]

Chen, W. T.

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).
[Crossref]

Cheng, Q.

K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
[Crossref]

Colburn, S.

E. Bayati, A. Zhan, S. Colburn, M. V. Zhelyeznyakov, and A. Majumdar, “Role of refractive index in metalens performance,” Appl. Opt. 58(6), 1460–1466 (2019).
[Crossref]

S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (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]

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref]

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).
[Crossref]

S. Colburn and A. Majumdar, “Simultaneous varifocal and broadband achromatic computational imaging using quartic metasurfaces,” (2019).

Cui, T. J.

K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
[Crossref]

Davis, L. D.

M. I. Mishchenko, L. D. Davis, and A. A. Lacis, Scattering, Absoprtion, and Emission of Light by Small Particles (NASA Goddard Institute for Space Studies, 2002).

Dodson, C. M.

A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
[Crossref]

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).
[Crossref]

Doicu, A.

A. Doicu, T. Wriedt, and J. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources (Springer, 2006).

Donelli, M.

Dong, D. S.

K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
[Crossref]

Dupré, M.

Egel, A.

D. Theobald, A. Egel, G. Gomard, and U. Lemmer, “Plane-wave coupling formalism for T-matrix simulations of light scattering by nonspherical particles,” Phys. Rev. A 96(3), 033822 (2017).
[Crossref]

A. Egel, L. Pattelli, G. Mazzamuto, D. S. Wiersma, and U. Lemmer, “Celes: Cuda-accelerated simulation of electromagnetic scattering by large ensembles of spheres,” J. Quant. Spectrosc. Radiat. Transfer 199, 103–110 (2017).
[Crossref]

Egorov, V.

Eitan, M.

Eremin, J. A.

A. Doicu, T. Wriedt, and J. A. Eremin, Light Scattering by Systems of Particles: Null-Field Method with Discrete Sources (Springer, 2006).

Fan, J. A.

J. A. Fan, “High performance metasurfaces based on inverse design (conference presentation),” (2017).

Fan, S.

V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012).
[Crossref]

Faraji-Dana, M.

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(4), 041056 (2017).
[Crossref]

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A. Zhan, R. Gibson, J. Whitehead, E. Smith, J. R. Hendrickson, and A. Majumdar, “Controlling three-dimensional optical fields via inverse mie scattering,” Sci. Adv. 5(10), eaax4769 (2019).
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S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (2018).
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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).
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A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
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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. Xiao, H. Mühlenbernd, G. Li, M. Kenney, F. Liu, T. Zentgraf, S. Zhang, and J. Li, “Helicity-preserving omnidirectional plasmonic mirror,” Adv. Opt. Mater. 4(5), 654–658 (2016).
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Zhao, J.

K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
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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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S. Xiao, F. Zhong, H. Liu, S. Zhu, and J. Li, “Flexible coherent control of plasmonic spin-hall effect,” Nat. Commun. 6(1), 8360 (2015).
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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. Xiao, H. Mühlenbernd, G. Li, M. Kenney, F. Liu, T. Zentgraf, S. Zhang, and J. Li, “Helicity-preserving omnidirectional plasmonic mirror,” Adv. Opt. Mater. 4(5), 654–658 (2016).
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S. Xiao, F. Zhong, H. Liu, S. Zhu, and J. Li, “Flexible coherent control of plasmonic spin-hall effect,” Nat. Commun. 6(1), 8360 (2015).
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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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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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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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S. Colburn, A. Zhan, and A. Majumdar, “Metasurface optics for full-color computational imaging,” Sci. Adv. 4(2), eaar2114 (2018).
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A. Zhan, R. Gibson, J. Whitehead, E. Smith, J. R. Hendrickson, and A. Majumdar, “Controlling three-dimensional optical fields via inverse mie scattering,” Sci. Adv. 5(10), eaax4769 (2019).
[Crossref]

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B. Yang, T. Liu, H. Guo, S. Xiao, and L. Zhou, “High-performance meta-devices based on multilayer meta-atoms: interplay between the number of layers and phase coverage,” Sci. Bull. 64(12), 823–835 (2019).
[Crossref]

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K. Wang, J. Zhao, Q. Cheng, D. S. Dong, and T. J. Cui, “Broadband and broad-angle low-scattering metasurface based on hybrid optimization algorithm,” Sci. Rep. 4(1), 5935 (2015).
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A. Zhan, S. Colburn, C. M. Dodson, and A. Majumdar, “Metasurface freeform nanophotonics,” Sci. Rep. 7(1), 1673 (2017).
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[Crossref]

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S. Colburn and A. Majumdar, “Simultaneous varifocal and broadband achromatic computational imaging using quartic metasurfaces,” (2019).

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Figures (6)

Fig. 1.
Fig. 1. A. Mie scattering schematic. Light is incident onto the set of ellipsoidal scatterers. Each scatterer has an associated T-matrix. The incident field onto each scatterer is described by the incident field $E_{in}$ and the scattered fields from all other scatterers. The inter-particle coupling is represented by the matrix $W_{n''n'}^{i''i'}$ which describes the coupling between spheres $i'$ and $i''$ . B. Design parameters for ellipsoidal scatterers. The semi-major axes are taken to always be aligned with the particle frame: semi-major axis $a$ is aligned with the $x_{part}$ axis, $b$ with the $y_{part}$ axis, and $c$ with the $z_{part}$ axis. The rotation $\phi$ is about the z-axis, with the counterclockwise direction defined as a positive rotation.
Fig. 2.
Fig. 2. Verification of the analytical T-matrix derivatives. A shows the error between the analytical T-matrix derivative and the numerical derivative with respect to semi-major axis $a$ , B with respect to $b$ , C with respect to $c$ . Figure 2D shows the T-matrix derivative with respect to the azimuthal rotation of the ellipsoid $\phi$ . As the step size of the numerical approximation to the derivative gets smaller, the mean error between numerical and analytical derivative gets closer to 0, which implies that the analytical derivatives are valid.
Fig. 3.
Fig. 3. A Final distribution of scatterers with periodicity $450$ nm for the inverse designed lens. semi-major axes $a$ and $b$ are allowed to range between $40$ and $150$ nm. Semi-major axis $c$ is allowed to range between $40$ and $300$ nm. B the field cross-section in the x-z plane at $y=0 \mu m$ , C the cross-section in the x-y plane at $z=10\mu m$ . D shows the Gaussian fit to the field at the focal spot $z=10 \mu m$ along $x=0$ . In order to calculate the lens efficiency, the full-width at half-maximum (FWHM) was calculated for the fitted Gaussians. The integral of the field intensity around the disk $d=3 \times FWHM$ about the center of the focal spot was calculated, and then divided by the total incident field intensity. The units of all plots are arbitrary light intensity units. The efficiency of the inverse designed lens was calculated to be $3.38\%$ .
Fig. 4.
Fig. 4. A Scatterer distribution of the polarization multiplexed lens. Lattice periodicity is 650 nm, radii were limited to range from 40 nm to 292.5 nm for the $a$ and $b$ axes, and $0$ to $357.5 nm$ for the c axis. For the initial condition, all of the semi-major axis radii were set to $250 nm$ , and the rotations were set to $0$ radians. In the final parameter distribution, the scatterers look very similar, and indeed, the minimum semi-major axis radius in the design is $\sim 205 nm$ and the maximum is $\sim 289 nm$ . B-D Are field distributions correspond to x-polarized light, and E-G correspond to y-polarized light. B,E are scattered field slices in the x-z plane at $y=0\mu m$ . C,F are x-y profiles at each focal spot. C is a slice at $z=20 \mu m$ , and E is a slice at $z=30 \mu m$ . D,G are Gaussian fits at each focal spot.
Fig. 5.
Fig. 5. Figs. A-D correspond to the forward designed lens, and Figs. E-H to the optimized lens. A,E are the scatterer distributions. E,F are the x-z slices of the resulting field profile at $y=0\mu m$ . C,G correspond to the x-y field slice at $z=10\mu m$ . D,H are the Gaussians fitted to the field profiles at their focal spot with $y=0\mu m$ . The forward design lens efficiency was determined to be $25.59\%$ , and the optimized efficiency was calculated to be $32.00\%$
Fig. 6.
Fig. 6. Transmission of individual scatterers with periodic boundary conditions as a function of the radius of the ellipsoids (semi-major axes $a = b$ ). A is the plot of the complex transmission of the ellipsoids used from section 4.1 and 5. Ellipsoid height is fixed to be $600 nm$ . The lattice constant is $450 nm$ . B transmission response for ellipsoids outlined in section 4.2. Ellipsoidal height is fixed at $715nm$ with a lattice constant of $650nm$ .

Equations (62)

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E i n i = n a n i ψ n ( 1 ) ( r r i )
E s c a t i = n b n i ψ n ( 3 ) ( r r i )
E i n i ( r ) = E i n ( r ) + i i E s c a t i ( r )
b n i = T n n i i a n i
M n n i i b n i = T n n i i a i n , n i
M n n i i = δ i i δ n n T n n i i W n n i i
f P j = 2 R e { ( λ n i ) T ( T n n i i P j a i n , n i + T n n i i P j W n n i i b n i ) }
T ( S ( P ) ) = T ( S ( a , b , c , ϕ ) ) = T ( a , b , c , ϕ )
P T N = T ( P + Δ P ) T ( P ) Δ P + O ( Δ P 2 )
e r r o r = m e a n ( i j | P T i , j N P T i , j A | )
f ( b ( P ) , P ) = ( I T I A ( x , y , z = F ) ) 2
η = Ω E ( x , y , z = F ) E ( x , y , z = F ) d x d y x , y E ( x , y , z = 0 μ m ) E ( x , y , z = 0 ) d x d y
Ω := x 2 + y 2 < ( 3 × F W H M ) 2
f = f x + f y
f x = ( I m a x T ( 0 , 0 , 20 μ m ) I A ( 0 , 0 , 20 μ m ) + I m i n T ( 0 , 0 , 30 μ m ) I A ( 0 , 0 , 30 μ m ) ) 2
f y = ( I m a x T ( 0 , 0 , 30 μ m ) I A ( 0 , 0 , 30 μ m ) + I m i n T ( 0 , 0 , 20 μ m ) I A ( 0 , 0 , 20 μ m ) ) 2
min P { a , b , c , ϕ } max ( f x , f y )
ϕ ( x , y ) = 2 π λ ( ( x 2 + y 2 ) + f 2 f )
T = R g Q ( Q ) 1
Q = [ P ¯ R ¯ S ¯ U ¯ ] ,
P ¯ l m l m = i k k s J l m l m ( 21 ) i k 2 J l m l m ( 12 ) ,
R ¯ l m l m = i k k s J l m l m ( 11 ) i k 2 J l m l m ( 22 ) ,
S ¯ l m l m = i k k s J l m l m ( 22 ) i k 2 J l m l m ( 11 ) ,
U ¯ l m l m = i k k s J l m l m ( 12 ) i k 2 J l m l m ( 21 ) ,
J l m l m ( p q ) = ( 1 ) m S d S n ^ ( r ) Ψ p , l , m ( 1 ) ( k s r , θ , ϕ ) × Ψ q , l , m ( 3 ) ( k r , θ , ϕ ) ,
Ψ 1 l m ( ν ) ( r ) = e i m ϕ 2 l ( l + 1 ) b l ( k r ) [ i m π l m ( θ ) θ ^ τ l m ( θ ) ϕ ^ ] ,
Ψ 2 l m ( ν ) ( r ) = e i m ϕ 2 l ( l + 1 ) { l ( l + 1 ) b l ( k r ) k r P l | m | ( c o s θ ) r ^ + 1 k r ( k r b l ( k r ) ) ( k r ) [ τ l m ( θ ) θ ^ + i m π l m ( θ ) ϕ ^ ] } ,
π l m ( θ ) = P l | m | ( c o s θ ) s i n θ , τ l m ( θ ) = P l | m | ( c o s θ ) θ .
d S n ^ ( r ) = r 2 s i n ( θ ) σ ( r ) d θ d ϕ ,
σ ( r ) = r ^ θ ^ 1 r r θ ϕ ^ 1 r s i n θ r θ .
r ( θ , ϕ ) = [ s i n 2 θ ( c o s 2 ϕ a 2 + s i n 2 ϕ b 2 ) + c o s 2 θ c 2 ] 1 / 2
T p = ( R g Q p T Q p ) Q 1 ,
J l m l m ( 11 ) p = i α l m l m ( m π l m τ l m + m π l m τ l m ) [ r ( k b l p j l + k s b l j l p ) + 2 b l j l ] r s i n θ d θ d ϕ ,
J l m l m ( 12 ) p = α l m l m { [ R l m l m ( 12 ) r + ( Θ l m l m ( 12 ) E θ + Φ l m l m ( 12 ) E ϕ ) ρ l , l r ] r p + ( Θ l m l m ( 12 ) E θ p + Φ l m l m ( 12 ) E ϕ p ) ρ l , l } d θ d ϕ ,
J l m l m ( 21 ) p = α l m l m { [ R l m l m ( 21 ) r + ( Θ l m l m ( 21 ) E θ + Φ l m l m ( 21 ) E ϕ ) ρ l , l r ] r p + ( Θ l m l m ( 21 ) E θ p + Φ l m l m ( 21 ) E ϕ p ) ρ l , l } d θ d ϕ ,
J l m l m ( 22 ) p = α l m l m { [ R l m l m ( 22 ) r + Θ l m l m ( 22 ) r E θ + Φ l m l m ( 22 ) r E ϕ ] r p + Θ l m l m ( 22 ) E θ p + Φ l m l m ( 22 ) E ϕ p } d θ d ϕ ,
α l m l m = ( 1 ) m ( 1 + ( 1 ) m m ) ( 1 + ( 1 ) l + l + 1 2 l ( l + 1 ) l ( l + 1 ) e i ( m m ) ϕ
E θ = c o s 2 ϕ a 2 + s i n 2 ϕ b 2 1 c 2
E ϕ = 1 b 2 1 a 2 ,
ρ l , l = r 3 j l b l ,
R l m l m ( 12 ) r = s i n θ k ( m m π l m π l m + τ l m τ l m ) ( j l ( k r b l ) ( k r ) + r ( k s j l r ( k r b l ) ( k r ) + k j l r ( ( k r b l ) ( k r ) ) ) ) ,
Θ l m l m ( 12 ) = s i n θ k l ( l + 1 ) P l | m | τ l m ,
Φ l m l m ( 12 ) = i s i n θ k l ( l + 1 ) m P l | m | π l m .
R l m l m ( 21 ) r = s i n θ k s ( m m π l m π l m + τ l m τ l m ) ( ( k s r j l ) ( k s r ) b l + r ( k s r ( ( k s r j l ) ( k s r ) ) b l + k b l r ( k s r j l ) ( k s r ) ) ) ,
Θ l m l m ( 21 ) = s i n θ k s l ( l + 1 ) P l | m | τ l m ,
Φ l m l m ( 21 ) = i s i n θ k s l ( l + 1 ) m P l | m | π l m .
Θ l m l m ( 22 ) = i r 2 s i n θ k k s ( m l ( l + 1 ) ( k s r j l ) ( k s r ) b l P l | m | π l m + m l ( l + 1 ) j l ( k r b l ) ( k r ) P l | m | π l m )
Φ l m l m ( 22 ) = r 2 s i n θ k k s ( l ( l + 1 ) j l P l | m | ( k r b l ) ( k r ) τ l m l ( l + 1 ) ( k s r j l ) ( k s r ) τ l m b l P l m )
R l m l m ( 22 ) r = i s i n θ k k s ( m π l m τ l m + m π l m τ l m ) ( k r ( ( k r b l ) ( k r ) ) ( k s r j l ) ( k s r ) + k s r ( ( k s r j l ) ( k s r ) ) ( k r b l ) ( k r ) )
Θ l m l m ( 22 ) r = i s i n θ k k s ( m l ( l + 1 ) P l | m | π l m ( 2 r ( k r b l ) ( k r ) j l + r 2 ( k r ( ( k r b l ) ( k r ) ) j l + k s j l r ( k r b l ) ( k r ) ) ) + m l ( l + 1 ) P l | m | τ l m ( 2 r b l ( k s r j l ) ( k s r ) + r 2 ( k b l r ( k s r j l ) ( k s r ) + k s b l r ( ( k s r j l ) ( k s r ) ) ) ) )
Φ l m l m ( 22 ) r = s i n θ k k s ( l ( l + 1 ) P l | m | τ l m ( 2 r ( k r b l ) ( k r ) j l + r 2 ( r ( ( k r b l ) ( k r ) ) j l + k s j l r ( k r b l ) ( k r ) ) ) l ( l + 1 ) P l | m | τ l m ( 2 r b l ( k s r j l ) ( k s r ) + r 2 ( k b l r ( k s r j l ) ( k s r ) r ( ( k s r j l ) ( k s r ) ) ) ) ) .
x l a b = x p a r t c o s ( ϕ r o t ) + y p a r t s i n ( ϕ r o t )
y l a b = x p a r t s i n ( ϕ r o t ) + y p a r t c o s ( ϕ r o t )
z l a b = z p a r t
O p l m p l m l a b ( α , β , γ ) = m 1 = l l m 2 = l l D m m 1 l ( α , β , γ ) O p l m 1 p l m 2 p a r t i c l e D m 2 m l ( γ , β , α ) ,
D m m l ( α , β , γ ) = e i m α d m m l ( β ) e i m γ ,
d m m l ( β ) = l , m | e i β J y | l , m .
O p l m p l m l a b ( α , 0 , 0 ) = m 1 = l l m 2 = l l D m m 1 l ( α , 0 , 0 ) O p l m 1 , p l m 2 p a r t i c l e D m 2 m l ( 0 , 0 , α ) .
D m m l ( α , 0 , 0 ) = e i m α δ m m
D m m l ( 0 , 0 , γ ) = e i m γ δ m m .
O p l m p l m l a b ( α ) = e i ( m m ) α O p l m p l m p a r t i c l e .
T p l m p l m l a b ( α ) α = i ( m m ) e i ( m m ) α T p l m p l m p a r t i c l e .

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