Abstract

We investigate a class of stacked metasurfaces where the interaction between layers is dominated by their respective far-field response. Using a semi-analytic scattering matrix approach, we exploit the Fabry-Perot-type response for different layer distances to show the spectral tunability of the resonant effect. This method presents a faster and more intuitive route to modeling Fabry-Perot-type effects than rigorous numerical simulations. The results are illustrated for a chiral metasurface stack that exhibits asymmetric transmission. Here, the effect of asymmetric transmission is highly sensitive to the layer distance, which is used as a free parameter in our model. To prove our theoretical findings we fabricate two variants of the stack with different layer distances and show that far-field interaction between layers is sufficient to generate the effect while being accessible by semi-analytic modeling. The analyticity of the approach is promising for designing sophisticated layered media containing stacks of arbitrary metasurfaces.

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

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  1. A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
    [Crossref]
  2. C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Physical Review A - Atomic, Molecular, and Optical Physics 82, 1–9 (2010).
    [Crossref]
  3. C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials,” IEEE Antennas and Propagation Magazine 54, 10–35 (2012).
    [Crossref]
  4. I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
    [Crossref] [PubMed]
  5. M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
    [Crossref]
  6. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nature Materials 13, 139–150 (2014).
    [Crossref] [PubMed]
  7. P. Genevet, F. Capasso, F. Aieta, M. Khorasaninejad, and R. Devlin, “Recent advances in planar optics: from plasmonic to dielectric metasurfaces,” Optica 4, 139 (2017).
    [Crossref]
  8. C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
    [Crossref]
  9. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13, 024006 (2011).
    [Crossref]
  10. S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
    [Crossref]
  11. K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
    [Crossref] [PubMed]
  12. Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Applied Physics Letters 110, 221108 (2017).
    [Crossref]
  13. E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Applied Physics Letters 93, 1–4 (2008).
    [Crossref]
  14. M. V. Gorkunov, A. A. Ezhov, V. V. Artemov, O. Y. Rogov, and S. G. Yudin, “Extreme optical activity and circular dichroism of chiral metal hole arrays,” Applied Physics Letters 104, 8–12 (2014).
    [Crossref]
  15. M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
    [Crossref] [PubMed]
  16. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
    [Crossref]
  17. C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
    [Crossref] [PubMed]
  18. J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal N-helices,” Optics Express 20, 26012 (2012).
    [Crossref] [PubMed]
  19. M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
    [Crossref]
  20. K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
    [Crossref] [PubMed]
  21. E. Plum and N. I. Zheludev, “Chiral mirrors,” Applied Physics Letters 106, 221901 (2015).
    [Crossref]
  22. L. Cong, N. Xu, W. Zhang, and R. Singh, “Polarization control in terahertz metasurfaces with the lowest order rotational symmetry,” Advanced Optical Materials 3, 1176–1183 (2015).
    [Crossref]
  23. G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
    [Crossref]
  24. C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: Asymmetric transmission of light,” Physical Review Letters 113, 1–5 (2014).
    [Crossref]
  25. J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
    [Crossref]
  26. Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
    [Crossref]
  27. J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
    [Crossref] [PubMed]
  28. M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
    [Crossref] [PubMed]
  29. K. Hannam, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Broadband chiral metamaterials with large optical activity,” Physical Review B - Condensed Matter and Materials Physics 89, 1–6 (2014).
    [Crossref]
  30. Z. Wu and Y. Zheng, “Moiré Chiral Metamaterials,” Advanced Optical Materials 1700034, 1700034 (2017).
    [Crossref]
  31. Y. Zhao, M. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nature Communications 3, 870 (2012).
    [Crossref] [PubMed]
  32. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
    [Crossref]
  33. C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Physical Review B 75, 195111 (2007).
    [Crossref]
  34. C. Menzel, J. Sperrhake, and T. Pertsch, “Efficient treatment of stacked metasurfaces for optimizing and enhancing the range of accessible optical functionalities,” Physical Review A 93, 063832 (2016).
    [Crossref]
  35. A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
    [Crossref]
  36. T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
    [Crossref]
  37. Y. Zhao, N. Engheta, and A. Alù, “Homogenization of plasmonic metasurfaces modeled as transmission-line loads,” Metamaterials 5, 90–96 (2011).
    [Crossref]
  38. J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
    [Crossref]
  39. N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Optics Express 18, 7569 (2010).
    [Crossref] [PubMed]
  40. R. M. Redheffer, “On a certain linear fractional transformation,” Journal of Mathematics and Physics 39, 269–286 (1960).
    [Crossref]
  41. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” Journal of the Optical Society of America A 13, 1024–1035 (1996).
    [Crossref]
  42. R. J. Potton, “Reciprocity in optics,” Reports on Progress in Physics 67, 717–754 (2004).
    [Crossref]
  43. N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Physical Review B 72, 045138 (2005).
    [Crossref]
  44. E. Pshenay-Severin, M. Falkner, C. Helgert, and T. Pertsch, “Ultra broadband phase measurements on nanostructured metasurfaces,” Applied Physics Letters 104, 8–13 (2014).
    [Crossref]
  45. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” Journal of the Optical Society of America A 14, 2758 (1997).
    [Crossref]
  46. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” Journal of the Optical Society of America A 11, 2494 (1994).
    [Crossref]
  47. D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
    [Crossref]
  48. T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
    [Crossref]

2017 (6)

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Applied Physics Letters 110, 221108 (2017).
[Crossref]

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
[Crossref] [PubMed]

Z. Wu and Y. Zheng, “Moiré Chiral Metamaterials,” Advanced Optical Materials 1700034, 1700034 (2017).
[Crossref]

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

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

2016 (1)

C. Menzel, J. Sperrhake, and T. Pertsch, “Efficient treatment of stacked metasurfaces for optimizing and enhancing the range of accessible optical functionalities,” Physical Review A 93, 063832 (2016).
[Crossref]

2015 (3)

E. Plum and N. I. Zheludev, “Chiral mirrors,” Applied Physics Letters 106, 221901 (2015).
[Crossref]

L. Cong, N. Xu, W. Zhang, and R. Singh, “Polarization control in terahertz metasurfaces with the lowest order rotational symmetry,” Advanced Optical Materials 3, 1176–1183 (2015).
[Crossref]

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

2014 (8)

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

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: Asymmetric transmission of light,” Physical Review Letters 113, 1–5 (2014).
[Crossref]

J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
[Crossref]

Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
[Crossref]

M. V. Gorkunov, A. A. Ezhov, V. V. Artemov, O. Y. Rogov, and S. G. Yudin, “Extreme optical activity and circular dichroism of chiral metal hole arrays,” Applied Physics Letters 104, 8–12 (2014).
[Crossref]

K. Hannam, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Broadband chiral metamaterials with large optical activity,” Physical Review B - Condensed Matter and Materials Physics 89, 1–6 (2014).
[Crossref]

E. Pshenay-Severin, M. Falkner, C. Helgert, and T. Pertsch, “Ultra broadband phase measurements on nanostructured metasurfaces,” Applied Physics Letters 104, 8–13 (2014).
[Crossref]

2013 (1)

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

2012 (7)

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

Y. Zhao, M. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nature Communications 3, 870 (2012).
[Crossref] [PubMed]

J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal N-helices,” Optics Express 20, 26012 (2012).
[Crossref] [PubMed]

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
[Crossref]

K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
[Crossref] [PubMed]

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
[Crossref] [PubMed]

2011 (4)

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13, 024006 (2011).
[Crossref]

C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
[Crossref]

Y. Zhao, N. Engheta, and A. Alù, “Homogenization of plasmonic metasurfaces modeled as transmission-line loads,” Metamaterials 5, 90–96 (2011).
[Crossref]

2010 (5)

J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
[Crossref]

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Optics Express 18, 7569 (2010).
[Crossref] [PubMed]

A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
[Crossref]

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
[Crossref]

C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Physical Review A - Atomic, Molecular, and Optical Physics 82, 1–9 (2010).
[Crossref]

2009 (2)

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
[Crossref]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[Crossref] [PubMed]

2008 (1)

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Applied Physics Letters 93, 1–4 (2008).
[Crossref]

2007 (4)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
[Crossref]

C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials 1, 62–80 (2007).
[Crossref]

C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Physical Review B 75, 195111 (2007).
[Crossref]

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials 1, 2–11 (2007).
[Crossref]

2005 (1)

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Physical Review B 72, 045138 (2005).
[Crossref]

2004 (1)

R. J. Potton, “Reciprocity in optics,” Reports on Progress in Physics 67, 717–754 (2004).
[Crossref]

1997 (1)

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” Journal of the Optical Society of America A 14, 2758 (1997).
[Crossref]

1996 (1)

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” Journal of the Optical Society of America A 13, 1024–1035 (1996).
[Crossref]

1994 (1)

E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” Journal of the Optical Society of America A 11, 2494 (1994).
[Crossref]

1960 (1)

R. M. Redheffer, “On a certain linear fractional transformation,” Journal of Mathematics and Physics 39, 269–286 (1960).
[Crossref]

Adamo, G.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Applied Physics Letters 110, 221108 (2017).
[Crossref]

Aieta, F.

Alaee, R.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

Alù, A.

Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
[Crossref]

Y. Zhao, M. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nature Communications 3, 870 (2012).
[Crossref] [PubMed]

Y. Zhao, N. Engheta, and A. Alù, “Homogenization of plasmonic metasurfaces modeled as transmission-line loads,” Metamaterials 5, 90–96 (2011).
[Crossref]

Andryieuski, A.

A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
[Crossref]

Artemov, V. V.

M. V. Gorkunov, A. A. Ezhov, V. V. Artemov, O. Y. Rogov, and S. G. Yudin, “Extreme optical activity and circular dichroism of chiral metal hole arrays,” Applied Physics Letters 104, 8–12 (2014).
[Crossref]

Bai, B.

K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
[Crossref] [PubMed]

Belkin, M.

Y. Zhao, M. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nature Communications 3, 870 (2012).
[Crossref] [PubMed]

Booth, J.

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

Brener, I.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Capasso, F.

Chipouline, A.

J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
[Crossref]

Cong, L.

L. Cong, N. Xu, W. Zhang, and R. Singh, “Polarization control in terahertz metasurfaces with the lowest order rotational symmetry,” Advanced Optical Materials 3, 1176–1183 (2015).
[Crossref]

Cui, T. J.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
[Crossref]

Decker, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
[Crossref]

Deubel, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
[Crossref]

Devlin, R.

Dietrich, K.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
[Crossref] [PubMed]

Dominguez, J.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Dregely, D.

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
[Crossref]

Duan, X.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
[Crossref] [PubMed]

Economou, E. N.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

Engheta, N.

Y. Zhao, N. Engheta, and A. Alù, “Homogenization of plasmonic metasurfaces modeled as transmission-line loads,” Metamaterials 5, 90–96 (2011).
[Crossref]

Ezhov, A. A.

M. V. Gorkunov, A. A. Ezhov, V. V. Artemov, O. Y. Rogov, and S. G. Yudin, “Extreme optical activity and circular dichroism of chiral metal hole arrays,” Applied Physics Letters 104, 8–12 (2014).
[Crossref]

Falkner, M.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

E. Pshenay-Severin, M. Falkner, C. Helgert, and T. Pertsch, “Ultra broadband phase measurements on nanostructured metasurfaces,” Applied Physics Letters 104, 8–13 (2014).
[Crossref]

C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

Fang, S.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Fedotov, V. A.

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13, 024006 (2011).
[Crossref]

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Applied Physics Letters 93, 1–4 (2008).
[Crossref]

Filter, R.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

Fofang, N. T.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Gansel, J. K.

J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal N-helices,” Optics Express 20, 26012 (2012).
[Crossref] [PubMed]

Genevet, P.

Giessen, H.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
[Crossref] [PubMed]

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
[Crossref]

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Optics Express 18, 7569 (2010).
[Crossref] [PubMed]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
[Crossref]

Gippius, N. A.

N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Optics Express 18, 7569 (2010).
[Crossref] [PubMed]

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
[Crossref]

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Physical Review B 72, 045138 (2005).
[Crossref]

Gonzales, E.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Gordon, J. A.

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

Gorkunov, M. V.

M. V. Gorkunov, A. A. Ezhov, V. V. Artemov, O. Y. Rogov, and S. G. Yudin, “Extreme optical activity and circular dichroism of chiral metal hole arrays,” Applied Physics Letters 104, 8–12 (2014).
[Crossref]

Granet, G.

T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
[Crossref]

Grbic, A.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: Asymmetric transmission of light,” Physical Review Letters 113, 1–5 (2014).
[Crossref]

Guan, C.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Guan, C. Y.

J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
[Crossref]

Guo, L. J.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: Asymmetric transmission of light,” Physical Review Letters 113, 1–5 (2014).
[Crossref]

Hannam, K.

K. Hannam, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Broadband chiral metamaterials with large optical activity,” Physical Review B - Condensed Matter and Materials Physics 89, 1–6 (2014).
[Crossref]

Helgert, C.

E. Pshenay-Severin, M. Falkner, C. Helgert, and T. Pertsch, “Ultra broadband phase measurements on nanostructured metasurfaces,” Applied Physics Letters 104, 8–13 (2014).
[Crossref]

K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
[Crossref] [PubMed]

C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
[Crossref]

Hentschel, M.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
[Crossref] [PubMed]

M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
[Crossref]

Holloway, C. L.

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

Ishihara, T.

N. A. Gippius, S. G. Tikhodeev, and T. Ishihara, “Optical properties of photonic crystal slabs with an asymmetrical unit cell,” Physical Review B 72, 045138 (2005).
[Crossref]

Kafesaki, M.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

Kaschke, J.

J. Kaschke, J. K. Gansel, and M. Wegener, “On metamaterial circular polarizers based on metal N-helices,” Optics Express 20, 26012 (2012).
[Crossref] [PubMed]

Katsarakis, N.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

Kenanakis, G.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

Khorasaninejad, M.

Kim, J.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Kim, S.-J.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Kivshar, Y.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Kivshar, Y. S.

M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
[Crossref]

K. Hannam, D. A. Powell, I. V. Shadrivov, and Y. S. Kivshar, “Broadband chiral metamaterials with large optical activity,” Physical Review B - Condensed Matter and Materials Physics 89, 1–6 (2014).
[Crossref]

Kley, E. B.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
[Crossref]

Kley, E.-B.

K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
[Crossref] [PubMed]

C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

Konishi, K.

K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
[Crossref] [PubMed]

Konstantinidis, G.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
[Crossref]

Kriegler, C. E.

Kuester, E. F.

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

Kuwata-Gonokami, M.

K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
[Crossref] [PubMed]

Lalanne, P.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
[Crossref]

Lavrinenko, A.

A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
[Crossref]

Lederer, F.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
[Crossref]

C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
[Crossref]

A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
[Crossref]

J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
[Crossref]

C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Physical Review A - Atomic, Molecular, and Optical Physics 82, 1–9 (2010).
[Crossref]

Lee, B.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Lee, K.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Lee, Y.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Lehr, D.

D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
[Crossref]

K. Dietrich, D. Lehr, C. Helgert, A. Tünnermann, and E.-B. Kley, “Circular dichroism from chiral nanomaterial fabricated by on-edge lithography,” Advanced Materials 24, OP321–OP325 (2012).
[Crossref] [PubMed]

Li, L.

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” Journal of the Optical Society of America A 14, 2758 (1997).
[Crossref]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” Journal of the Optical Society of America A 13, 1024–1035 (1996).
[Crossref]

Li, X.

Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
[Crossref]

Li, Y.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Linden, S.

M. Decker, M. Ruther, C. E. Kriegler, J. Zhou, C. M. Soukoulis, S. Linden, and M. Wegener, “Strong optical activity from twisted-cross photonic metamaterials,” Opt. Lett. 34, 2501–2503 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
[Crossref]

Liu, N.

M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
[Crossref] [PubMed]

Liu, S.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Luan, K.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Luk, T. S.

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Lv, W.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

Ma, H. F.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
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Malureanu, R.

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C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
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A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
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C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced Jones calculus for the classification of periodic metamaterials,” Physical Review A - Atomic, Molecular, and Optical Physics 82, 1–9 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
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C. Menzel, J. Sperrhake, and T. Pertsch, “Efficient treatment of stacked metasurfaces for optimizing and enhancing the range of accessible optical functionalities,” Physical Review A 93, 063832 (2016).
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M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
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C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
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C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
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E. Plum and N. I. Zheludev, “Chiral mirrors,” Applied Physics Letters 106, 221901 (2015).
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C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
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T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
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C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
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C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
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A. Andryieuski, C. Menzel, C. Rockstuhl, R. Malureanu, F. Lederer, and A. Lavrinenko, “Homogenization of resonant chiral metamaterials,” Physical Review B 82, 235107 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
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Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
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J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
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T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Physical Review B 84, 115142 (2011).
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C. Menzel, J. Sperrhake, and T. Pertsch, “Efficient treatment of stacked metasurfaces for optimizing and enhancing the range of accessible optical functionalities,” Physical Review A 93, 063832 (2016).
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M. Decker, I. Staude, M. Falkner, J. Dominguez, D. N. Neshev, I. Brener, T. Pertsch, and Y. S. Kivshar, “High-efficiency dielectric Huygens’ surfaces,” Advanced Optical Materials 3, 813–820 (2015).
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I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
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Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
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J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
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C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Physical Review B 75, 195111 (2007).
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D. Lehr, R. Alaee, R. Filter, K. Dietrich, T. Siefke, C. Rockstuhl, F. Lederer, E. B. Kley, and A. Tünnermann, “Plasmonic nanoring fabrication tuned to pitch: efficient, deterministic, and large scale realization of ultra-small gaps for next generation plasmonic devices,” Applied Physics Letters 105, 143110 (2014).
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C. Helgert, E. Pshenay-Severin, M. Falkner, C. Menzel, C. Rockstuhl, E.-B. Kley, A. Tünnermann, F. Lederer, and T. Pertsch, “Chiral metamaterial composed of three-dimensional plasmonic nanostructures,” Nano Letters 11, 4400–4404 (2011).
[Crossref] [PubMed]

C. Menzel, C. Helgert, C. Rockstuhl, E. B. Kley, A. Tünnermann, T. Pertsch, and F. Lederer, “Asymmetric transmission of linearly polarized light at optical metamaterials,” Physical Review Letters 104, 1–4 (2010).
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J. Petschulat, A. Chipouline, A. Tünnermann, T. Pertsch, C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Simple and versatile analytical approach for planar metamaterials,” Physical Review B 82, 075102 (2010).
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K. Konishi, B. Bai, Y. Toya, J. Turunen, Y. P. Svirko, and M. Kuwata-Gonokami, “Surface-plasmon enhanced optical activity in two-dimensional metal chiral networks,” Optics Letters 37, 4446 (2012).
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M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
[Crossref]

Wang, Y.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Applied Physics Letters 110, 221108 (2017).
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Wang, Z.

Z. Wang, B. H. Teh, Y. Wang, G. Adamo, J. Teng, and H. Sun, “Enhancing circular dichroism by super chiral hot spots from a chiral metasurface with apexes,” Applied Physics Letters 110, 221108 (2017).
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J. H. Shi, H. F. Ma, C. Y. Guan, Z. P. Wang, and T. J. Cui, “Broadband chirality and asymmetric transmission in ultrathin 90°-twisted Babinet-inverted metasurfaces,” Physical Review B - Condensed Matter and Materials Physics 89, 1–7 (2014).
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[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Advanced Materials 19, 207–210 (2007).
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N. A. Gippius, T. Weiss, S. G. Tikhodeev, and H. Giessen, “Resonant mode coupling of optical resonances in stacked nanostructures,” Optics Express 18, 7569 (2010).
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T. Weiss, N. A. Gippius, S. G. Tikhodeev, G. Granet, and H. Giessen, “Efficient calculation of the optical properties of stacked metamaterials with a Fourier modal method,” Journal of Optics A: Pure and Applied Optics 11, 114019 (2009).
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Z. Wu and Y. Zheng, “Moiré Chiral Metamaterials,” Advanced Optical Materials 1700034, 1700034 (2017).
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L. Cong, N. Xu, W. Zhang, and R. Singh, “Polarization control in terahertz metasurfaces with the lowest order rotational symmetry,” Advanced Optical Materials 3, 1176–1183 (2015).
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[Crossref]

Yun, H.

J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
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J.-G. Yun, S.-J. Kim, H. Yun, K. Lee, J. Sung, J. Kim, Y. Lee, and B. Lee, “Broadband ultrathin circular polarizer at visible and near-infrared wavelengths using a non-resonant characteristic in helically stacked nano-gratings,” Optics Express 25, 14260 (2017).
[Crossref] [PubMed]

Zhang, C.

C. Pfeiffer, C. Zhang, V. Ray, L. J. Guo, and A. Grbic, “High performance bianisotropic metasurfaces: Asymmetric transmission of light,” Physical Review Letters 113, 1–5 (2014).
[Crossref]

Zhang, W.

L. Cong, N. Xu, W. Zhang, and R. Singh, “Polarization control in terahertz metasurfaces with the lowest order rotational symmetry,” Advanced Optical Materials 3, 1176–1183 (2015).
[Crossref]

Zhao, R.

G. Kenanakis, R. Zhao, A. Stavrinidis, G. Konstantinidis, N. Katsarakis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Flexible chiral metamaterials in the terahertz regime: a comparative study of various designs,” Optical Materials Express 2, 1702 (2012).
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Y. Zhao, J. Shi, L. Sun, X. Li, and A. Alù, “Alignment-free three-dimensional optical metamaterials,” Advanced Materials 26, 1439–1445 (2014).
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Zheludev, N. I.

E. Plum and N. I. Zheludev, “Chiral mirrors,” Applied Physics Letters 106, 221901 (2015).
[Crossref]

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Asymmetric transmission: a generic property of two-dimensional periodic patterns,” J. Opt. 13, 024006 (2011).
[Crossref]

E. Plum, V. A. Fedotov, and N. I. Zheludev, “Optical activity in extrinsically chiral metamaterial,” Applied Physics Letters 93, 1–4 (2008).
[Crossref]

Zheng, Y.

Z. Wu and Y. Zheng, “Moiré Chiral Metamaterials,” Advanced Optical Materials 1700034, 1700034 (2017).
[Crossref]

Zhou, J.

Zhu, Z.

S. Fang, K. Luan, H. F. Ma, W. Lv, Y. Li, Z. Zhu, C. Guan, J. Shi, and T. J. Cui, “Asymmetric transmission of linearly polarized waves in terahertz chiral metamaterials,” Journal of Applied Physics 121, 033103 (2017).
[Crossref]

ACS Nano (1)

I. Staude, A. E. Miroshnichenko, M. Decker, N. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisks,” ACS Nano 7, 7824–7832 (2013).
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Advanced Materials (3)

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M. Schäferling, D. Dregely, M. Hentschel, and H. Giessen, “Tailoring enhanced optical chirality: design principles for chiral plasmonic nanostructures,” Physical Review X 2, 031010 (2012).
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M. Hentschel, M. Schäferling, X. Duan, H. Giessen, and N. Liu, “Chiral plasmonics,” Science Advances 3, e1602735 (2017).
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Figures (5)

Fig. 1
Fig. 1 Rendered sketch of a stack of gold nano-wire metasurfaces embedded in glass.
Fig. 2
Fig. 2 (a) SEM picture of a FIB cut revealing the metasurface layers of the stack. The nano-particles in this image were colored golden during postprocessing to enhance visibility. (b) SEM image of the lower layer showing axis-aligned nano-wires. (c) SEM image of the upper layer showing nano-wires rotated counterclockwise by 60 .
Fig. 3
Fig. 3 Experimental and simulation results of the stack in Jones matrix form: a) T xx, b) T xy, c) T yx, d) T yy. Blue curves show transmittance and orange ones phase. Measured data is represented by solid and dashed lines showing results for the 345 nm stack and 40 nm stack, respectively. Results obtained via SASA are marked with crosses and those from rigorous FMM calculations with circles.
Fig. 4
Fig. 4 Sketch of the stack construction with S-matrices using eq. (9). Dashed lines mark the interfaces of the stack to the surrounding medium. Arrows point in the direction of (forward) propagation along the z-axis. While spacer (spac.), and cladding (clad.) have defined thicknesses, the two metasurface layers (MS   l, MS   u) are assumed to be infinitely thin.
Fig. 5
Fig. 5 (a)-(b) Linear asymmetric transmission (a) and ellipticity (b) derived from measurement (solid lines) and SASA results (dashed lines). Red indicates x-polarized and blue y-polarized incident light. (c)-(e) Asymmetric transmission and ellipticity scanned over the spacer thickness from 40 nm to 1000 nm. (c) Asymmetric transmission for x-polarization (y-polarization as its negative counterpart was omitted); (d) and (e) display ellipticity for x- and y-polarization. The red line in each surface plot marks a spacer thickness of d sp = 345 nm, corresponding to solid lines in (a) and (b).

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

S = ( T ^ f R ^ b R ^ f T ^ b ) .
S sp = diag ( P , P , P , P ) ,
S S sp = ( P T xx f P T xy f P 2 R xx b P 2 R xy b P T yx f P T yy f P 2 R yx b P 2 R yy b R xx f R xy f P T xx b P T xy b R yx f R yy f P T yx b P T yy b ) .
S w = ( T x 0 R x 0 0 T y 0 R y R x 0 T x 0 0 R y 0 T y ) ,
S ˜ w = ( T ˜ xx T ˜ xy R ˜ xx R ˜ xy T ˜ xy T ˜ yy R ˜ xy R ˜ yy R ˜ xx R ˜ xy T ˜ xx T ˜ xy R ˜ xy R ˜ yy T ˜ xy T ˜ yy ) .
Δ x = | T xy f | 2 | T yx f | 2 .
S ¯ = S w * S Sp * S ˜ w = ( T ¯ xx T ¯ xy R ¯ xx b R ¯ xy b T ¯ yx T ¯ yy R ¯ xy b R ¯ yy b R ¯ xx f R ¯ xy f T ¯ xx T ¯ yx R ¯ xy f R ¯ yy f T ¯ xy T ¯ yy )
S i = ( 2 n 1 n 1 + n 2 0 n 1 n 2 n 1 + n 2 0 0 2 n 1 n 1 + n 2 0 n 1 n 2 n 1 + n 2 n 1 n 2 n 1 + n 2 0 2 n 2 n 1 + n 2 0 0 n 1 n 2 n 1 + n 2 0 2 n 2 n 1 + n 2 ) ,
S st = S is S l S sp S u S c S ic ,
ϵ x = 2 | T xx f | | T yx f | | T xx f | 2 + | T yx f | 2 sin  δ x .
S ( d sp ) = ( S is S l ) S sp ( d sp ) ( ( S u S c ) S ic ) ,
T ¯ x x = P T x ( P 2 R y ( R ˜ x y T ˜ x y + R ˜ y y T ˜ x x ) - T ˜ x x ) P 4 R x R y ( R ˜ x y 2 - R ˜ x x R ˜ y y ) + P 2 ( R x R ˜ x x + R y R ˜ y y ) - 1 T ¯ x y = P T y ( P 2 R x ( R ˜ x y T ˜ x x + R ˜ x x T ˜ x y ) - T ˜ x y ) P 4 R x R y ( R ˜ x y 2 - R ˜ x x R ˜ y y ) + P 2 ( R x R ˜ x x + R y R ˜ y y ) - 1 T ¯ y x = P T x ( P 2 R y ( R ˜ x y T ˜ y y + R ˜ y y T ˜ x y ) - T ˜ x y ) P 4 R x R y ( R ˜ x y 2 - R ˜ x x R ˜ y y ) + P 2 ( R x R ˜ x x + R y R ˜ y y ) - 1 T ¯ y y = P T y ( P 2 R x ( R ˜ x y T ˜ x y + R ˜ x x T ˜ y y ) - T ˜ y y ) P 4 R x R y ( R ˜ x y 2 - R ˜ x x R ˜ y y ) + P 2 ( R x R ˜ x x + R y R ˜ y y ) - 1 R ¯ x x b = ( P 4 R x R y ( 2 R ˜ x y T ˜ x x T ˜ x y - R ˜ x x T ˜ x y 2 - R ˜ y y T ˜ x x 2 - R ˜ x x ( R ˜ x y 2 - R ˜ x x R ˜ y y ) ) + P 2 ( R x T ˜ x x 2 + R y T ˜ x y 2 - R ˜ x x ( R x R ˜ x x + R y R ˜ y y ) ) + R ˜ x x ) / ( P 4 R x R y ( R ˜ x x R ˜ y y - R ˜ x y 2 ) - P 2 ( R x R ˜ x x + R y R ˜ y y ) + 1 ) R ¯ x y b = ( P 4 R x R y ( R ˜ x y ( R ˜ x x R ˜ y y + T ˜ x y 2 - R ˜ x y 2 + T ˜ x x T ˜ y y ) - T ˜ x y ( R ˜ y y T ˜ x x + R ˜ x x T ˜ y y ) ) + P 2 ( R x T ˜ x x T ˜ x y + R y T ˜ x y T ˜ y y - R ˜ x y ( R x R ˜ x x + R y R ˜ y y ) ) + R ˜ x y ) / ( P 4 R x R y ( R ˜ x x R ˜ y y - R ˜ x y 2 ) - P 2 ( R x R ˜ x x + R y R ˜ y y ) + 1 ) R ¯ y y b = ( P 4 ( - R x ) R y ( R ˜ y y ( - R ˜ x x R ˜ y y + R ˜ x y 2 + T ˜ x y 2 ) + R ˜ x x T ˜ y y 2 - 2 R ˜ x y T ˜ x y T ˜ y y ) + P 2 ( R x T ˜ x y 2 + R y T ˜ y y 2 - R ˜ y y ( R x R ˜ x x + R y R ˜ y y ) ) + R ˜ y y ) / ( P 4 R x R y ( R ˜ x x R ˜ y y - R ˜ x y 2 ) - P 2 ( R x R ˜ x x + R y R ˜ y y ) + 1 ) R ¯ x x f = R x 2 - T x 2 R x + T x 2 ( 1 - P 2 R y R ˜ y y ) R x ( P 4 R x R y ( R ˜ x x R ˜ y y - R ˜ x y 2 ) - P 2 ( R x R ˜ x x + R y R ˜ y y ) + 1 ) R ¯ x y f = - P 2 T x T y R ˜ x y P 4 R x R y ( R ˜ x y 2 - R ˜ x x R ˜ y y ) + P 2 ( R x R ˜ x x + R y R ˜ y y ) - 1 R ¯ y y f = R y 2 - T y 2 R y + T y 2 ( 1 - P 2 R x R ˜ x x ) R y ( P 4 R x R y ( R ˜ x x R ˜ y y - R ˜ x y 2 ) - P 2 ( R x R ˜ x x + R y R ˜ y y ) + 1 ) .

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