Abstract

Holographic optics are an essential tool for the control of light, generating highly complex and tailored light field distributions that can represent physical objects or abstract information. Conceptually, a hologram is a region of space in which an arbitrary phase shift and amplitude variation are added to an incident reference wave at every spatial location, such that the reference wave will produce a desired field distribution as it scatters from the medium. Practical holograms are composed of materials, however, which have limited properties that constrain the possible field distributions. Here, we show it is possible to produce a hologram with continuous phase distribution and a non-uniform amplitude variation at every point by leveraging resonant metamaterial elements and constraining the hologram’s pixels to match the elements’ resonant behavior. We demonstrate the viability of the resonant metamaterial approach with a single layer, co-polarized holographic metasurface that produces an image at millimeter wavelengths (92.5 GHz) despite the elements’ limited phase range and coupled amplitude dependency.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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2016 (2)

2015 (6)

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107(20), 204104 (2015).
[Crossref]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

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

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54(31), 9343–9353 (2015).
[Crossref] [PubMed]

P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015).
[Crossref] [PubMed]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

2014 (4)

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, D. R. Smith, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am. A 31(10), 2109–2119 (2014).
[Crossref] [PubMed]

M. Johnson, P. Bowen, N. Kundtz, and A. Bily, “Discrete-dipole approximation model for control and optimization of a holographic metamaterial antenna,” Appl. Opt. 53(25), 5791–5799 (2014).
[Crossref] [PubMed]

2013 (7)

G. Lipworth, A. Mrozack, J. Hunt, D. L. Marks, T. Driscoll, D. Brady, and D. R. Smith, “Metamaterial apertures for coherent computational imaging on the physical layer,” J. Opt. Soc. Am. A 30(8), 1603–1612 (2013).
[Crossref] [PubMed]

S. Zhong and S. He, “Ultrathin and lightweight microwave absorbers made of mu-near-zero metamaterials,” Sci. Rep. 3, 2083 (2013).
[Crossref] [PubMed]

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

A. Pors and S. I. Bozhevolnyi, “Plasmonic metasurfaces for efficient phase control in reflection,” Opt. Express 21(22), 27438–27451 (2013).
[Crossref] [PubMed]

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110(19), 197401 (2013).
[Crossref] [PubMed]

2012 (3)

X. Li, S. Xiao, B. Cai, Q. He, T. J. Cui, and L. Zhou, “Flat metasurfaces to focus electromagnetic waves in reflection geometry,” Opt. Lett. 37(23), 4940–4942 (2012).
[Crossref] [PubMed]

S. Larouche, Y.-J. Tsai, T. Tyler, N. M. Jokerst, and D. R. Smith, “Infrared metamaterial phase holograms,” Nat. Mater. 11(5), 450–454 (2012).
[Crossref] [PubMed]

P. T. Bowern, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14(3), 033038 (2012).
[Crossref]

2011 (2)

B. J. Arritt, D. R. Smith, and T. Khraishi, “Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters,” J. Appl. Phys. 109(7), 1–5 (2011).
[Crossref]

Y.-J. Tsai, S. Larouche, T. Tyler, G. Lipworth, N. M. Jokerst, and D. R. Smith, “Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths,” Opt. Express 19(24), 24411–24423 (2011).
[Crossref] [PubMed]

2010 (2)

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

W. Freese, T. Kämpfe, E.-B. Kley, and A. Tünnermann, “Design of binary subwavelength multiphase level computer generated holograms,” Opt. Lett. 35(5), 676–678 (2010).
[Crossref] [PubMed]

2008 (2)

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

2005 (2)

1999 (1)

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

1973 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

1966 (1)

1962 (1)

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

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] [PubMed]

Arritt, B. J.

B. J. Arritt, D. R. Smith, and T. Khraishi, “Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters,” J. Appl. Phys. 109(7), 1–5 (2011).
[Crossref]

Bai, B.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Bily, A.

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

Bowen, P.

Bowern, P. T.

P. T. Bowern, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14(3), 033038 (2012).
[Crossref]

Bozhevolnyi, S. I.

Brady, D.

Brady, D. J.

Brown, B. R.

Brunton, S. L.

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Extremum-seeking control of the beam pattern of a reconfigurable holographic metamaterial antenna,” J. Opt. Soc. Am. A 33(1), 59–68 (2016).
[Crossref] [PubMed]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

Cai, B.

Capasso, 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] [PubMed]

P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Cheah, K.-W.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Chen, S.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Chen, X.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Chu, D. C.

Colburn, J. S.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

Cui, T. J.

Cummer, S. A.

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

Da Huang, J. S.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Dolling, G.

Driscoll, T.

Eleftheriades, V.

Enkrich, C.

Ensworth, J.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Epstein, A.

Fainman, Y.

Fienup, J. R.

Fong, B. H.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

Freese, W.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

Genevet, P.

P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015).
[Crossref] [PubMed]

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

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Gollub, J.

Gollub, J. N.

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107(20), 204104 (2015).
[Crossref]

Goodman, J. W.

Gowda, V. R.

Grbic, A.

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110(19), 197401 (2013).
[Crossref] [PubMed]

Hand, T. H.

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

He, Q.

He, S.

S. Zhong and S. He, “Ultrathin and lightweight microwave absorbers made of mu-near-zero metamaterials,” Sci. Rep. 3, 2083 (2013).
[Crossref] [PubMed]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Holden, J.

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Huang, L.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Hunt, J.

Imani, M. F.

Jin, G.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Johnson, M.

Johnson, M. C.

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Extremum-seeking control of the beam pattern of a reconfigurable holographic metamaterial antenna,” J. Opt. Soc. Am. A 33(1), 59–68 (2016).
[Crossref] [PubMed]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

Jokerst, N. M.

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Kämpfe, T.

Kats, M. A.

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

Khraishi, T.

B. J. Arritt, D. R. Smith, and T. Khraishi, “Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters,” J. Appl. Phys. 109(7), 1–5 (2011).
[Crossref]

Kildishev, A. V.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Kim, H.-C.

Kley, E.-B.

Kundtz, N.

Kundtz, N. B.

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Extremum-seeking control of the beam pattern of a reconfigurable holographic metamaterial antenna,” J. Opt. Soc. Am. A 33(1), 59–68 (2016).
[Crossref] [PubMed]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

P. T. Bowern, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14(3), 033038 (2012).
[Crossref]

Kutz, N. J.

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Extremum-seeking control of the beam pattern of a reconfigurable holographic metamaterial antenna,” J. Opt. Soc. Am. A 33(1), 59–68 (2016).
[Crossref] [PubMed]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Larouche, S.

Lee, J. S.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Leith, E. N.

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

Levy, U.

Li, J.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Li, X.

Linden, S.

Lipworth, G.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54(31), 9343–9353 (2015).
[Crossref] [PubMed]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, D. R. Smith, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am. A 31(10), 2109–2119 (2014).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

G. Lipworth, A. Mrozack, J. Hunt, D. L. Marks, T. Driscoll, D. Brady, and D. R. Smith, “Metamaterial apertures for coherent computational imaging on the physical layer,” J. Opt. Soc. Am. A 30(8), 1603–1612 (2013).
[Crossref] [PubMed]

Y.-J. Tsai, S. Larouche, T. Tyler, G. Lipworth, N. M. Jokerst, and D. R. Smith, “Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths,” Opt. Express 19(24), 24411–24423 (2011).
[Crossref] [PubMed]

Lohmann, A. W.

Marks, D. L.

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Mrozack, A.

Mühlenbernd, H.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Ni, X.

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Nomura, T.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Odabasi, H.

Ottusch, J. J.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Pendry, J. B.

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Pfeiffer, C.

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110(19), 197401 (2013).
[Crossref] [PubMed]

Pors, A.

Qiu, C.-W.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Reynolds, M. S.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, D. R. Smith, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am. A 31(10), 2109–2119 (2014).
[Crossref] [PubMed]

Robbins, D. J.

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Rose, A.

Sajuyigbe, S.

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Schmalenberg, P.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Seetharam, K.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Shalaev, V. M.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Sievenpiper, D. F.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

Sleasman, T.

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107(20), 204104 (2015).
[Crossref]

Smith, D. R.

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107(20), 204104 (2015).
[Crossref]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, A. Rose, O. Yurduseven, V. R. Gowda, M. F. Imani, H. Odabasi, P. Trofatter, J. Gollub, and D. R. Smith, “Comprehensive simulation platform for a metamaterial imaging system,” Appl. Opt. 54(31), 9343–9353 (2015).
[Crossref] [PubMed]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, D. R. Smith, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am. A 31(10), 2109–2119 (2014).
[Crossref] [PubMed]

J. Hunt, J. Gollub, T. Driscoll, G. Lipworth, A. Mrozack, M. S. Reynolds, D. J. Brady, D. R. Smith, and D. R. Smith, “Metamaterial microwave holographic imaging system,” J. Opt. Soc. Am. A 31(10), 2109–2119 (2014).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

G. Lipworth, A. Mrozack, J. Hunt, D. L. Marks, T. Driscoll, D. Brady, and D. R. Smith, “Metamaterial apertures for coherent computational imaging on the physical layer,” J. Opt. Soc. Am. A 30(8), 1603–1612 (2013).
[Crossref] [PubMed]

P. T. Bowern, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14(3), 033038 (2012).
[Crossref]

S. Larouche, Y.-J. Tsai, T. Tyler, N. M. Jokerst, and D. R. Smith, “Infrared metamaterial phase holograms,” Nat. Mater. 11(5), 450–454 (2012).
[Crossref] [PubMed]

B. J. Arritt, D. R. Smith, and T. Khraishi, “Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters,” J. Appl. Phys. 109(7), 1–5 (2011).
[Crossref]

Y.-J. Tsai, S. Larouche, T. Tyler, G. Lipworth, N. M. Jokerst, and D. R. Smith, “Design and fabrication of a metamaterial gradient index diffraction grating at infrared wavelengths,” Opt. Express 19(24), 24411–24423 (2011).
[Crossref] [PubMed]

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Soukoulis, C. M.

Stewart, W. J.

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Tan, Q.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Tang, W.

W. Tang and H. Zhao, “A meander line resonator to realize negative index materials,” in Proceedings of IEEE Conference on Antennas and Propagation Society International Symposium (IEEE, 2008), pp. 1–4.

Trofatter, P.

Tsai, C.-H.

Tsai, Y.-J.

Tünnermann, A.

Tyler, T.

Upatnieks, J.

Urzhumov, Y.

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

Visher, J. L.

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

Wegener, M.

Xiao, S.

Yu, N.

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Yurduseven, O.

Zentgraf, T.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Zhang, H.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Zhang, S.

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

Zhao, H.

W. Tang and H. Zhao, “A meander line resonator to realize negative index materials,” in Proceedings of IEEE Conference on Antennas and Propagation Society International Symposium (IEEE, 2008), pp. 1–4.

Zhong, S.

S. Zhong and S. He, “Ultrathin and lightweight microwave absorbers made of mu-near-zero metamaterials,” Sci. Rep. 3, 2083 (2013).
[Crossref] [PubMed]

Zhou, J. F.

Zhou, L.

Appl. Opt. (4)

Appl. Phys. Lett. (2)

T. Sleasman, M. F. Imani, J. N. Gollub, and D. R. Smith, “Dynamic metamaterial aperture for microwave imaging,” Appl. Phys. Lett. 107(20), 204104 (2015).
[Crossref]

T. H. Hand, J. Gollub, S. Sajuyigbe, D. R. Smith, and S. A. Cummer, “Characterization of complementary electric field coupled resonant surfaces,” Appl. Phys. Lett. 93(21), 1–3 (2008).
[Crossref]

IBM J. Res. Develop. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The kinoform: a new wavefront reconstruction device,” IBM J. Res. Develop. 13(2), 150–155 (1969).
[Crossref]

IEEE Trans. Antenn. Propag. (2)

B. H. Fong, J. S. Colburn, J. J. Ottusch, J. L. Visher, and D. F. Sievenpiper, “Scalar and tensor holographic artificial impedance surfaces,” IEEE Trans. Antenn. Propag. 58(10), 3212–3221 (2010).
[Crossref]

M. C. Johnson, S. L. Brunton, N. B. Kundtz, and N. J. Kutz, “Sidelobe canceling for reconfigurable holographic metamaterial antenna,” IEEE Trans. Antenn. Propag. 63(1), 1881–1886 (2015).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

J. B. Pendry, J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

J. Appl. Phys. (1)

B. J. Arritt, D. R. Smith, and T. Khraishi, “Equivalent circuit analysis of metamaterial strain-dependent effective medium parameters,” J. Appl. Phys. 109(7), 1–5 (2011).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Nat. Commun. (2)

L. Huang, X. Chen, H. Mühlenbernd, H. Zhang, S. Chen, B. Bai, Q. Tan, G. Jin, K.-W. Cheah, C.-W. Qiu, J. Li, T. Zentgraf, and S. Zhang, “Three dimensional Optical Holography using a plasmonic metasurface,” Nat. Commun. 4, 2808 (2013).
[Crossref]

X. Ni, A. V. Kildishev, and V. M. Shalaev, “Metasurface holograms for visible light,” Nat. Commun. 4, 2807 (2013).
[Crossref]

Nat. Mater. (2)

S. Larouche, Y.-J. Tsai, T. Tyler, N. M. Jokerst, and D. R. Smith, “Infrared metamaterial phase holograms,” Nat. Mater. 11(5), 450–454 (2012).
[Crossref] [PubMed]

N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014).
[Crossref] [PubMed]

Nature (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref] [PubMed]

New J. Phys. (1)

P. T. Bowern, T. Driscoll, N. B. Kundtz, and D. R. Smith, “Using a discrete dipole approximation to predict complete scattering of complicated metamaterials,” New J. Phys. 14(3), 033038 (2012).
[Crossref]

Opt. Express (2)

Opt. Lett. (4)

Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35(2), 237–246 (1972).

Phys. Rev. Lett. (2)

C. Pfeiffer and A. Grbic, “Metamaterial Huygens’ surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110(19), 197401 (2013).
[Crossref] [PubMed]

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Rep. Prog. Phys. (1)

P. Genevet and F. Capasso, “Holographic optical metasurfaces: a review of current progress,” Rep. Prog. Phys. 78(2), 024401 (2015).
[Crossref] [PubMed]

Sci. Rep. (3)

S. Zhong and S. He, “Ultrathin and lightweight microwave absorbers made of mu-near-zero metamaterials,” Sci. Rep. 3, 2083 (2013).
[Crossref] [PubMed]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Da Huang, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Magnetic metamaterial superlens for increased range wireless power transfer,” Sci. Rep. 4, 8–13 (2014).
[Crossref]

G. Lipworth, J. Ensworth, K. Seetharam, J. S. Lee, P. Schmalenberg, T. Nomura, M. S. Reynolds, D. R. Smith, and Y. Urzhumov, “Quasi-static magnetic field shielding using longitudinal mu-near-zero metamaterials,” Sci. Rep. 5, 12764 (2015).
[Crossref] [PubMed]

Science (2)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339(6125), 1232009 (2013).
[Crossref] [PubMed]

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

Other (5)

J. D. Jackson, Classical electrodynamics (Vol. 3) (Wiley, 1998).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Co., 2005).

W. Tang and H. Zhao, “A meander line resonator to realize negative index materials,” in Proceedings of IEEE Conference on Antennas and Propagation Society International Symposium (IEEE, 2008), pp. 1–4.

S. Gregson, J. McCormick, and C. Parini, Principles of Planar Near-Field Antenna Measurements, Vol. 53 (IET, 2007).

C. A. Balanis, Advanced Engineering Electromagnetics (Wiley, 1989).

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Figures (9)

Fig. 1
Fig. 1 Experimental setup. A W-band, Fraunhofer, metamaterial hologram is excited using an off-axis, free- space illumination scheme. A planar near-field scanner measures the radiated near-fields across a plane parallel to the hologram; the far-field pattern is computed by propagating the near-fields to the far-field region. The inset depicts a section of the aperiodic metamaterial array forming the hologram.
Fig. 2
Fig. 2 Complex Lorentzian polarizability. (a) A resonator with Lorentzian polarizability exhibits the normalized magnitude (blue) and phase (red, degrees), plotted as a function of ω 0 /ω . We define the phase at resonance to be zero degrees. (b) The coupled relationship between magnitude and phase. In this example the resonator’s quality factor was arbitrarily chosen as Q = 10.
Fig. 3
Fig. 3 Metamaterial characterization. (a) Illustration of the experimental setup, with the element design shown in the inset; here τ=25 μm and the unit cell size is 1 mm. Measured S 21 magnitude (b) and phase (c) for elements of various lengths. Interpolated (and normalized) magnitude (d) and phase (e) of S 21 vs. L at 92.5 GHz. Circles mark measured data points.
Fig. 4
Fig. 4 Fields at the hologram plane and far-field images. Phase (a) and magnitude (b) of the reference wave illuminating the hologram. Desired field pattern used in the GS algorithm (c). Phase (d), magnitude (e), and far-field pattern (f) of the simulated hologram generated by the GS algorithm. Measured and back-propagated phase (g) and magnitude (h) of the fabricated hologram and its experimental far-field pattern (i). Far-fields were computed as the Fourier transformed hologram fields. Simulations and measurements shown are reported at 92.5 GHz and 94.75 GHz, respectively.
Fig. 5
Fig. 5 Simulated guided-mode hologram. (a) One possible embodiment of a hologram excited by a guided reference wave utilizes a parallel-plate waveguide excited by a source at the center. The guide mode’s magnitude (b) decays away from the excitation point while the phase (c) propagates with cylindrical phase fronts (assuming no reflections from the edges or perturbations due to the metamaterial elements). The magnitude (d) and phase (e) distribution of the hologram. (f) The resulting (normalized and log-scale) far-field pattern, computed as the aperture fields’ Fourier transform.
Fig. 6
Fig. 6 Laser phases and etching. (a) Our laser’s default phases are contour (red), hatch (blue), and heat (green) – shown her along paths computed by the laser’s software to etch a larger, conventional metamaterial element. (b) For our thin W-band elements, we utilized only the contour (red) phase. Initial unit cell simulations assumed only the copper layer was removed by the laser’s beam (c), these simulations were later revised to account for the fact that the laser penetrates beyond the copper layer and etches into the substrate as well (d).
Fig. 7
Fig. 7 Fabricated sample. (a) The hologram covers a 7.6 cm x 7.6 cm square of 1 mm x 1 mm elements. (b) A confined view of approximately 7% of the total hologram area. (c) Zoomed view of 25 elements, which covers less than 0.5% of the total hologram.
Fig. 8
Fig. 8 Experimental setup. The hologram is illuminated by an off-axis transmitting (Tx) horn. Near-fields are measured by a receiver (Rx) probe across a plane parallel to the hologram plane.
Fig. 9
Fig. 9 The effect of a hologram’s phase constraints on the quality of its image. The resulting far-field patterns from theoretical phase holograms with uniform amplitude and varying phase ranges (a-f). The hologram is illuminated by a normally-incident plane wave, and its pixel’s phases are constrained to lie in a limited range, starting at ±30° (a) and increasing in steps of ±30° (b-f) until a full range of ±180° (f) is reached.

Equations (5)

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m ¯ = α ¯ ¯ U,
α ii = F ω 2 / ( ω 0 2 ω 2 +jωγ ) ,
H=| α |exp(j Φ α )| U |exp(j Φ U )=| α || U |exp[ j( Φ α + Φ U ) ]=| H |exp(j Φ H ).
Φ α = Φ min if Φ ˜ α < Φ min and Φ α = Φ max if Φ ˜ α > Φ max ,
U z ^ × ρ ^ H 0 (2) ( k| ρ ¯ | ),

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