Abstract

Diffractive optical elements are ultra-thin optical components required for constructing very compact optical 3D sensors. However, the required wide-angle diffractive 2D fan-out gratings have been elusive due to design challenges. Here, we introduce a new strategy for optimizing such high-performance and wide-angle diffractive optical elements, offering unprecedented control over the power distribution among the desired diffraction orders with only low requirements with respect to computational power. The microstructure surfaces were designed by an iterative gradient optimization procedure based on an adjoint-state method, capable to account for application-dependent target functions while ensuring compatibility with existing fabrication processes. The results of the experimental characterization confirm the simulated tailored power distributions and optical efficiencies of the fabricated elements.

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

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2020 (3)

2019 (3)

J. Sung, G.-Y. Lee, and B. Lee, “Progresses in the practical metasurface for holography and lens,” Nanophotonics 8(10), 1701–1718 (2019).
[Crossref]

F. Wang, Z. Zhang, R. Wang, X. Zeng, X. Yang, S. Lv, F. Zhang, D. Xue, J. Yan, and X. Zhang, “Distortion measurement of optical system using phase diffractive beam splitter,” Opt. Express 27(21), 29803–29816 (2019).
[Crossref]

H. Hao, Z. Tingting, S. Qiang, and Y. Xiaodong, “Wide angle 2D beam splitter design based on vector diffraction theory,” Opt. Commun. 434, 28–35 (2019).
[Crossref]

2018 (8)

O. Barlev and M. A. Golub, “Multifunctional binary diffractive optical elements for structured light projectors,” Opt. Express 26(16), 21092 (2018).
[Crossref]

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
[Crossref]

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(1), 1700302 (2018).
[Crossref]

J. Wang, Y. Shi, T. Hughes, Z. Zhao, and S. Fan, “Adjoint-based optimization of active nanophotonic devices,” Opt. Express 26(3), 3236 (2018).
[Crossref]

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

A. Michaels and E. Yablonovitch, “Leveraging continuous material averaging for inverse electromagnetic design,” Opt. Express 26(24), 31717 (2018).
[Crossref]

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

A. Junker and K.-H. Brenner, “Achieving a high mode count in the exact electromagnetic simulation of diffractive optical elements,” J. Opt. Soc. Am. A 35(3), 377–385 (2018).
[Crossref]

2017 (2)

W. Iff, T. Kämpfe, Y. Jourlin, and A. V. Tishchenko, “Memory sparing fast scattering formalism for rigorous diffraction modeling,” J. Opt. 19(7), 075602 (2017).
[Crossref]

T. Hughes, G. Veronis, K. P. Wootton, R. Joel England, and S. Fan, “Method for computationally efficient design of dielectric laser accelerator structures,” Opt. Express 25(13), 15414 (2017).
[Crossref]

2016 (1)

A. Kordecki, H. Palus, and A. Bal, “Practical vignetting correction method for digital camera with measurement of surface luminance distribution,” Signal, Image Video Process. 10(8), 1417–1424 (2016).
[Crossref]

2015 (1)

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).
[Crossref]

2014 (1)

2013 (2)

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693 (2013).
[Crossref]

X. Qian and O. Sigmund, “Topological design of electromechanical actuators with robustness toward over- and under-etching,” Comput. Methods Appl. Mech. Eng. 253, 237–251 (2013).
[Crossref]

2012 (1)

V. Liu and S. Fan, “S 4: A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012).
[Crossref]

2011 (2)

S. Thibault, A. Arfaoui, and P. Desaulniers, “Cross-diffractive optical elements for wide angle geometric camera calibration,” Opt. Lett. 36(24), 4770 (2011).
[Crossref]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43(6), 767–784 (2011).
[Crossref]

2008 (1)

2007 (1)

2006 (1)

2004 (2)

T. Vallius, P. Vahimaa, and M. Honkanen, “Electromagnetic approach to the thin element approximation,” J. Mod. Opt. 51(14), 2079–2092 (2004).
[Crossref]

D. Feng, Y. Yan, G. Jin, and S. Fan, “Design and fabrication of continuous-profile diffractive micro-optical elements as a beam splitter,” Appl. Opt. 43(29), 5476–5480 (2004).
[Crossref]

2002 (2)

M. Skeren, I. Richter, and P. Fiala, “Iterative Fourier transform algorithm: comparison of various approaches,” J. Mod. Opt. 49(11), 1851–1870 (2002).
[Crossref]

S. Bühling and F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49(11), 1871–1892 (2002).
[Crossref]

2001 (1)

2000 (2)

1998 (1)

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

1997 (1)

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
[Crossref]

1996 (1)

1995 (2)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12(5), 1068–1076 (1995).
[Crossref]

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. on Sci. Comput. 16(5), 1190–1208 (1995).
[Crossref]

1994 (2)

1988 (1)

Arbabi, A.

Arbabi, E.

Arfaoui, A.

Ayliffe, M. H.

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(6), 374–377 (2015).
[Crossref]

Bacher, A.

Bal, A.

A. Kordecki, H. Palus, and A. Bal, “Practical vignetting correction method for digital camera with measurement of surface luminance distribution,” Signal, Image Video Process. 10(8), 1417–1424 (2016).
[Crossref]

Barlev, O.

Bauer, M.

Bernier, E.

Bhargava, S.

Brenner, K.-H.

Brosseau, D. F.

Bryngdahl, O.

Bühling, S.

S. Bühling and F. Wyrowski, “Improved transmission design algorithms by utilizing variable-strength projections,” J. Mod. Opt. 49(11), 1871–1892 (2002).
[Crossref]

Byrd, R. H.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
[Crossref]

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. on Sci. Comput. 16(5), 1190–1208 (1995).
[Crossref]

Cai, W.

Capasso, F.

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

Chen, Z.

Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
[Crossref]

Deán-Ben, X. L.

Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
[Crossref]

Desaulniers, P.

Doshay, S.

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
[Crossref]

Fan, J. A.

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
[Crossref]

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(1), 1700302 (2018).
[Crossref]

Fan, S.

Faraon, A.

Feng, D.

Fiala, P.

M. Skeren, I. Richter, and P. Fiala, “Iterative Fourier transform algorithm: comparison of various approaches,” J. Mod. Opt. 49(11), 1851–1870 (2002).
[Crossref]

Fiddy, M. A.

Fiebelkorn, R.

R. Vandenhouten, A. Hermerschmidt, and R. Fiebelkorn, “Design and quality metrics of point patterns for coded structured light illumination with diffractive optical elements in optical 3D sensors,” in Digital Optical Technologies 2017, vol. 10335B. C. Kress and P. Schelkens, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 264–276.

Gaylord, T. K.

Gérard, P.

Golub, M. A.

Goodman, J. W.

J. W. Goodman, “Introduction to fourier optics,” Introduction to Fourier optics, 3rd ed., by JW Goodman. (Roberts & Co. Publishers, Englewood, CO, 2005) 1 (2005).

Gottschalk, S.

Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
[Crossref]

Grann, E. B.

Griessbach, D.

Groever, B.

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

Guilhem, M.

P. Twardowski, B. Serio, V. Raulot, and M. Guilhem, “Three-dimensional shape measurement based on light patterns projection using diffractive optical elements,” in Micro-Optics 2010, vol. 7716H. Thienpont, P. V. Daele, J. Mohr, and H. Zappe, eds., International Society for Optics and Photonics (SPIE, 2010), pp. 704–711.

Hao, H.

H. Hao, Z. Tingting, S. Qiang, and Y. Xiaodong, “Wide angle 2D beam splitter design based on vector diffraction theory,” Opt. Commun. 434, 28–35 (2019).
[Crossref]

Häringer, D.

Heggarty, K.

Hermerschmidt, A.

M. Bauer, D. Griessbach, A. Hermerschmidt, S. Krüger, M. Scheele, and A. Schischmanow, “Geometrical camera calibration with diffractive optical elements,” Opt. Express 16(25), 20241–20248 (2008).
[Crossref]

A. Hermerschmidt, S. Krüger, and G. Wernicke, “Binary diffractive beam splitters with arbitrary diffraction angles,” Opt. Lett. 32(5), 448–450 (2007).
[Crossref]

R. Vandenhouten, A. Hermerschmidt, and R. Fiebelkorn, “Design and quality metrics of point patterns for coded structured light illumination with diffractive optical elements in optical 3D sensors,” in Digital Optical Technologies 2017, vol. 10335B. C. Kress and P. Schelkens, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 264–276.

Honkanen, M.

T. Vallius, P. Vahimaa, and M. Honkanen, “Electromagnetic approach to the thin element approximation,” J. Mod. Opt. 51(14), 2079–2092 (2004).
[Crossref]

Hughes, T.

Hugonin, J.-P.

J.-P. Hugonin and P. Lalanne, Reticolo software for grating analysis, (Institut d’Optique, Orsay, France, 2005).

Iff, W.

W. Iff, T. Kämpfe, Y. Jourlin, and A. V. Tishchenko, “Memory sparing fast scattering formalism for rigorous diffraction modeling,” J. Opt. 19(7), 075602 (2017).
[Crossref]

Jakobs, P.-J.

Jiang, J.

Jin, G.

Jin, W.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Joel England, R.

Jourlin, Y.

W. Iff, T. Kämpfe, Y. Jourlin, and A. V. Tishchenko, “Memory sparing fast scattering formalism for rigorous diffraction modeling,” J. Opt. 19(7), 075602 (2017).
[Crossref]

Junker, A.

Jurek, M. P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

Jureller, J. E.

Kämpfe, T.

W. Iff, T. Kämpfe, Y. Jourlin, and A. V. Tishchenko, “Memory sparing fast scattering formalism for rigorous diffraction modeling,” J. Opt. 19(7), 075602 (2017).
[Crossref]

Kim, H. Y.

Kirk, A. G.

Kordecki, A.

A. Kordecki, H. Palus, and A. Bal, “Practical vignetting correction method for digital camera with measurement of surface luminance distribution,” Signal, Image Video Process. 10(8), 1417–1424 (2016).
[Crossref]

Krüger, S.

Kwon, H.

Lacroix, F.

Lagoudakis, K. G.

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).
[Crossref]

Lalanne, P.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45(7), 1357–1374 (1998).
[Crossref]

P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13(4), 779 (1996).
[Crossref]

J.-P. Hugonin and P. Lalanne, Reticolo software for grating analysis, (Institut d’Optique, Orsay, France, 2005).

Lalau-Keraly, C. M.

Lazarov, B. S.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43(6), 767–784 (2011).
[Crossref]

Lee, B.

J. Sung, G.-Y. Lee, and B. Lee, “Progresses in the practical metasurface for holography and lens,” Nanophotonics 8(10), 1701–1718 (2019).
[Crossref]

Lee, G.-Y.

J. Sung, G.-Y. Lee, and B. Lee, “Progresses in the practical metasurface for holography and lens,” Nanophotonics 8(10), 1701–1718 (2019).
[Crossref]

Lin, Z.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

Liu, V.

V. Liu and S. Fan, “S 4: A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012).
[Crossref]

Liu, Z.

Loncar, M.

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

Lu, J.

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).
[Crossref]

Lu, P.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
[Crossref]

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. on Sci. Comput. 16(5), 1190–1208 (1995).
[Crossref]

Lv, S.

Mansouree, M.

Mc Larney, B.

Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
[Crossref]

McClung, A.

Meyrueis, P.

Michaels, A.

Miller, O. D.

Moharam, M. G.

Molesky, S.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Morris, G. M.

Nguyen, G.-N.

Nocedal, J.

C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
[Crossref]

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. on Sci. Comput. 16(5), 1190–1208 (1995).
[Crossref]

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Nordin, G.

Palus, H.

A. Kordecki, H. Palus, and A. Bal, “Practical vignetting correction method for digital camera with measurement of surface luminance distribution,” Signal, Image Video Process. 10(8), 1417–1424 (2016).
[Crossref]

Petykiewicz, J.

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).
[Crossref]

Pfeiffer, P.

Phan, T.

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
[Crossref]

Piggott, A. Y.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[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. Photonics 9(6), 374–377 (2015).
[Crossref]

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Pommet, D. A.

Qian, X.

X. Qian and O. Sigmund, “Topological design of electromechanical actuators with robustness toward over- and under-etching,” Comput. Methods Appl. Mech. Eng. 253, 237–251 (2013).
[Crossref]

Qiang, S.

H. Hao, Z. Tingting, S. Qiang, and Y. Xiaodong, “Wide angle 2D beam splitter design based on vector diffraction theory,” Opt. Commun. 434, 28–35 (2019).
[Crossref]

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P. Twardowski, B. Serio, V. Raulot, and M. Guilhem, “Three-dimensional shape measurement based on light patterns projection using diffractive optical elements,” in Micro-Optics 2010, vol. 7716H. Thienpont, P. V. Daele, J. Mohr, and H. Zappe, eds., International Society for Optics and Photonics (SPIE, 2010), pp. 704–711.

Razansky, D.

Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
[Crossref]

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Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
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Rodriguez, A. W.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
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Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

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Scherer, N. F.

Schischmanow, A.

Sell, D.

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(1), 1700302 (2018).
[Crossref]

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
[Crossref]

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P. Twardowski, B. Serio, V. Raulot, and M. Guilhem, “Three-dimensional shape measurement based on light patterns projection using diffractive optical elements,” in Micro-Optics 2010, vol. 7716H. Thienpont, P. V. Daele, J. Mohr, and H. Zappe, eds., International Society for Optics and Photonics (SPIE, 2010), pp. 704–711.

Shi, Y.

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M. Skeren, I. Richter, and P. Fiala, “Iterative Fourier transform algorithm: comparison of various approaches,” J. Mod. Opt. 49(11), 1851–1870 (2002).
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G. Zhang, Q. Song, L. Wei, and X. Yin, “Inverse optimization for designing wide angle diffractive optical element,” in Eleventh International Conference on Information Optics and Photonics (CIOP 2019), H. Wang, ed. (SPIE, 2019), December 2019, p. 117.

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J. Sung, G.-Y. Lee, and B. Lee, “Progresses in the practical metasurface for holography and lens,” Nanophotonics 8(10), 1701–1718 (2019).
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Thibault, S.

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H. Hao, Z. Tingting, S. Qiang, and Y. Xiaodong, “Wide angle 2D beam splitter design based on vector diffraction theory,” Opt. Commun. 434, 28–35 (2019).
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P. Twardowski, B. Serio, V. Raulot, and M. Guilhem, “Three-dimensional shape measurement based on light patterns projection using diffractive optical elements,” in Micro-Optics 2010, vol. 7716H. Thienpont, P. V. Daele, J. Mohr, and H. Zappe, eds., International Society for Optics and Photonics (SPIE, 2010), pp. 704–711.

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R. Vandenhouten, A. Hermerschmidt, and R. Fiebelkorn, “Design and quality metrics of point patterns for coded structured light illumination with diffractive optical elements in optical 3D sensors,” in Digital Optical Technologies 2017, vol. 10335B. C. Kress and P. Schelkens, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 264–276.

Veronis, G.

Vuckovic, J.

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[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. Photonics 9(6), 374–377 (2015).
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D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
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F. Wang, Z. Zhang, R. Wang, X. Zeng, X. Yang, S. Lv, F. Zhang, D. Xue, J. Yan, and X. Zhang, “Distortion measurement of optical system using phase diffractive beam splitter,” Opt. Express 27(21), 29803–29816 (2019).
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F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidiscip. Optim. 43(6), 767–784 (2011).
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G. Zhang, Q. Song, L. Wei, and X. Yin, “Inverse optimization for designing wide angle diffractive optical element,” in Eleventh International Conference on Information Optics and Photonics (CIOP 2019), H. Wang, ed. (SPIE, 2019), December 2019, p. 117.

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Yablonovitch, E.

Yan, J.

Yan, Y.

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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(1), 1700302 (2018).
[Crossref]

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
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Yin, X.

G. Zhang, Q. Song, L. Wei, and X. Yin, “Inverse optimization for designing wide angle diffractive optical element,” in Eleventh International Conference on Information Optics and Photonics (CIOP 2019), H. Wang, ed. (SPIE, 2019), December 2019, p. 117.

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Zhang, F.

Zhang, G.

G. Zhang, Q. Song, L. Wei, and X. Yin, “Inverse optimization for designing wide angle diffractive optical element,” in Eleventh International Conference on Information Optics and Photonics (CIOP 2019), H. Wang, ed. (SPIE, 2019), December 2019, p. 117.

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Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
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C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
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C. Zhu, R. H. Byrd, P. Lu, and J. Nocedal, “Algorithm 778: L-BFGS-B: Fortran subroutines for large-scale bound-constrained optimization,” ACM Trans. Math. Softw. 23(4), 550–560 (1997).
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ACS Photonics (1)

D. Sell, J. Yang, E. W. Wang, T. Phan, S. Doshay, and J. A. Fan, “Ultra-High-Efficiency Anomalous Refraction with Dielectric Metasurfaces,” ACS Photonics 5(6), 2402–2407 (2018).
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X. Qian and O. Sigmund, “Topological design of electromechanical actuators with robustness toward over- and under-etching,” Comput. Methods Appl. Mech. Eng. 253, 237–251 (2013).
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V. Liu and S. Fan, “S 4: A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183(10), 2233–2244 (2012).
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M. Skeren, I. Richter, and P. Fiala, “Iterative Fourier transform algorithm: comparison of various approaches,” J. Mod. Opt. 49(11), 1851–1870 (2002).
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Z. Chen, B. Mc Larney, J. Rebling, X. L. Deán-Ben, Q. Zhou, S. Gottschalk, and D. Razansky, “High-Speed Large-Field Multifocal Illumination Fluorescence Microscopy,” Laser Photonics Rev. 14(1), 1900070 (2020).
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Nanophotonics (1)

J. Sung, G.-Y. Lee, and B. Lee, “Progresses in the practical metasurface for holography and lens,” Nanophotonics 8(10), 1701–1718 (2019).
[Crossref]

Nat. Photonics (2)

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).
[Crossref]

S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Opt. Commun. (1)

H. Hao, Z. Tingting, S. Qiang, and Y. Xiaodong, “Wide angle 2D beam splitter design based on vector diffraction theory,” Opt. Commun. 434, 28–35 (2019).
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F. Wang, Z. Zhang, R. Wang, X. Zeng, X. Yang, S. Lv, F. Zhang, D. Xue, J. Yan, and X. Zhang, “Distortion measurement of optical system using phase diffractive beam splitter,” Opt. Express 27(21), 29803–29816 (2019).
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Opt. Lett. (3)

Optica (1)

Phys. Rev. Appl. (1)

Z. Lin, B. Groever, F. Capasso, A. W. Rodriguez, and M. Lončar, “Topology-Optimized Multilayered Metaoptics,” Phys. Rev. Appl. 9(4), 044030 (2018).
[Crossref]

SIAM J. on Sci. Comput. (1)

R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound Constrained Optimization,” SIAM J. on Sci. Comput. 16(5), 1190–1208 (1995).
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R. Vandenhouten, A. Hermerschmidt, and R. Fiebelkorn, “Design and quality metrics of point patterns for coded structured light illumination with diffractive optical elements in optical 3D sensors,” in Digital Optical Technologies 2017, vol. 10335B. C. Kress and P. Schelkens, eds., International Society for Optics and Photonics (SPIE, 2017), pp. 264–276.

P. Twardowski, B. Serio, V. Raulot, and M. Guilhem, “Three-dimensional shape measurement based on light patterns projection using diffractive optical elements,” in Micro-Optics 2010, vol. 7716H. Thienpont, P. V. Daele, J. Mohr, and H. Zappe, eds., International Society for Optics and Photonics (SPIE, 2010), pp. 704–711.

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Figures (7)

Fig. 1.
Fig. 1. (a) Scheme of two-dimension diffractive beam splitter generating $7\times 7$ spot array. (b) Uniformity of $7\times 7$ diffractive beam splitter designed by TEA-based IFTA as a function of maximum diffraction angle when the grating period $\Lambda$ decreases. The Uniformity error and NRMS calculated using Eq. (9) and (10). The insets show the layout of the single unit cell. Black represents dielectric material and white represents air. The maximum angle is at $(3,3)^{\textrm {th}}$ diffracted beam and the grating’s normal vector.
Fig. 2.
Fig. 2. (a) The surface profile of a grating structure in a single period. (b) Schematic of the forward and adjoint simulations in RCWA.
Fig. 3.
Fig. 3. Filtering and projection of an example design density. (a) the initial design density $\boldsymbol{\rho }$ before processing. (b) the density after applying the spatial filter, $\boldsymbol{\tilde {\rho }}$ . (c) the density after applying projection, $\boldsymbol{\bar {\tilde {\rho }}}$ . (d) the final relative permittivity $\boldsymbol{\epsilon }$ .
Fig. 4.
Fig. 4. Theoretical analysis of large-angle diffractive beam splitters. Plots of the figure of merit over the course of the optimization process of $7\times 7$ (a) and $7\times 5$ (b) diffractive beam splitter. In the bottom figures, the calculated efficiency of diffractive beam splitter before and after optimization.
Fig. 5.
Fig. 5. The target diffraction pattern for diffractive beam splitter with designated intensity distribution. The target efficiency depends on the groups, where group A, B, C, D, E, B’, C’, D’ and E’ have 1.0, 1.5, 2.0, 2.5, 1.0, 1.5, 2.0, 2.5 and 1.0 of intensity ratio, respectively. (see Table 1)
Fig. 6.
Fig. 6. Theoretical analysis of diffractive beam splitters generating the tailored spot array. (a) a plot of the figure of merit over the course of the optimization process for the beam splitter with tailored intensity distribution. The bottoms figures show refractive indices distributions of the device at different stages of the optimization process. (b) the calculated efficiency of diffractive beam splitter after optimization.
Fig. 7.
Fig. 7. Experimental characterization of the diffractive beam splitter with designated intensity distribution. (a) Scanning electron microscopy image of the diffractive beam splitter. Top insets: magnified tilted-view image of a grating unit cell. (b) The measured (orange line) and simulated (blue bar) efficiency of a beam splitter with tailored power distribution. The dash lines indicate the groups and their target efficiency

Tables (3)

Tables Icon

Table 1. The target efficiency depends on groups in beam splitter with the tailored intensity distribution

Tables Icon

Table 2. Comparison with the theoretical and experimental properties of the 7 × 7 and 7 × 5 beam splitters. The calculated efficiency takes into account the loss from Fresnel reflection in the air-SiO2 substrate interface.

Tables Icon

Table 3. Comparison with the simulated and experimental properties of the beam splitters with tailored power distribution. The simulated efficiency take into account the loss from Fresnel reflection in the air-SiO2 substrate interface.

Equations (17)

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F = m = M M ( η m η o b j ) 2
η = | t | 2 = 1 4 | Λ | 2 | Λ [ E ( r ) × H i ( r ) E i ( r ) × H ( r ) ] n z d r | 2
Δ η = 1 | Λ | { t Λ [ Δ E ( r ) × H i ( r ) E i ( r ) × Δ H ( r ) ] n z d r }
η ϵ ( r ) = 1 | Λ | { t Λ [ G ^ E P ( r , r ) E ( r ) × H i ( r ) E i ( r ) × G ^ H P ( r , r ) E ( r ) ] n z d r }
η ϵ ( r ) = 1 | Λ | ( t Λ { G ^ E P ( r , r ) [ H i ( r ) × n z ] G ^ E M ( r , r ) [ E i ( r ) × n z ] } d r E ( r ) ) = 1 | Λ | [ t E a d j ( r ) E ( r ) ]
ρ ~ i = j N i W i j ρ j j N i W i j
W i j = R | r i r j |
ρ ~ ¯ i = tanh ( β γ ) + tanh ( β [ ρ ~ i γ ] ) tanh ( β γ ) + tanh ( β [ 1 γ ] )
U E = η ~ m a x η ~ m i n η ~ m a x + η ~ m i n
σ = 1 M N ( η ~ m , n η ~ o b j ) 2
M A P D = 1 M N | η m , n S η m , n E η m , n S |
× E = i k 0 μ ( r ) H × H = i k 0 ϵ ( r ) E
× ( E + Δ E ) = i k 0 [ μ ( r ) + Δ μ ( r ) ] ( H + Δ H ) × ( H + Δ H ) = i k 0 [ ϵ ( r ) + Δ ϵ ( r ) ] ( E + Δ E ) .
× Δ E = i k 0 [ μ ( r ) Δ H + Δ μ ( r ) H ] × Δ H = i k 0 [ ϵ ( r ) Δ E + Δ ϵ ( r ) E ]
P ( r ) = Δ ϵ ( r ) E ( r ) M ( r ) = Δ μ ( r ) H ( r )
Δ E ( r ) = G ^ E P ( r , r ) P ( r ) + G ^ E M ( r , r ) M ( r ) Δ H ( r ) = G ^ H P ( r , r ) P ( r ) + G ^ H M ( r , r ) M ( r )
G ^ E P ( r , r ) = G ^ E P ( r , r ) G ^ H P ( r , r ) = G ^ E M ( r , r ) .

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