Abstract

We used the high-order finite element method to numerically study the self-action of light in a nonliner metamaterial composed of gold helices. It is shown that such a metamaterial can selectively reflect the waves with certain circular polarization. The impact of the nonlinear optical response of gold on the value of the transmission coefficient and the frequency range, in which this selective reflection occurs, is studied.

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

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References

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  1. N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27
  2. P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).
  3. Y. Huang, Y. Zhou, and S.-T. Wu, “Broadband circular polarizer using stacked chiral polymer films,” Opt. Express 15(10), 6414–6419 (2007).
    [Crossref]
  4. D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
    [Crossref]
  5. Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
    [Crossref]
  6. K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
    [Crossref]
  7. E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
    [Crossref]
  8. 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(16), 2501–2503 (2009).
    [Crossref]
  9. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
    [Crossref]
  10. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
    [Crossref]
  11. J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express 18(2), 1059–1069 (2010).
    [Crossref]
  12. Z. Y. Yang, M. Zhao, P. X. Lu, and Y. F. Lu, “Ultrabroadband optical circular polarizers consisting of double-helical nanowire structures,” Opt. Lett. 35(15), 2588–2590 (2010).
    [Crossref]
  13. N. N. Potravkin, E. B. Cherepetskaya, I. Perezhogin, and V. Makarov, “Ultrashort elliptically polarized laser pulse interaction with helical photonic metamaterial,” Opt. Mater. Express 4(10), 2090–2101 (2014).
    [Crossref]
  14. A. Reyna and C. de Araújo, “High-order optical nonlinearities in plasmonic nanocomposites – a review,” Adv. Opt. Photonics 9(4), 720–774 (2017).
    [Crossref]
  15. N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
    [Crossref]
  16. L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
    [Crossref]
  17. M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
    [Crossref]
  18. R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
    [Crossref]
  19. R. Kullock, A. Hille, A. Haußmann, S. Grafström, and L. M. Eng, “SHG simulations of plasmonic nanoparticles using curved elements,” Opt. Express 19(15), 14426–14436 (2011).
    [Crossref]
  20. T. G. Dominique Barchiesi, “Fitting the optical constants of gold, silver, chromium, titanium, and aluminum in the visible bandwidth,” J. Nanophotonics 8(1), 083097 (2014).
    [Crossref]
  21. P. Šolín, K. Segeth, and I. Doležel, Higher-order Finite Element Methods (Chapman & Hall/CRC, 2004).
  22. M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” J. Comput. Phys. 232(1), 189–213 (2013).
    [Crossref]
  23. www.csimsoft.com/trelis .
  24. www.software.intel.com/en-us/intel-mkl .

2018 (2)

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
[Crossref]

2017 (1)

A. Reyna and C. de Araújo, “High-order optical nonlinearities in plasmonic nanocomposites – a review,” Adv. Opt. Photonics 9(4), 720–774 (2017).
[Crossref]

2016 (1)

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

2014 (3)

R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
[Crossref]

T. G. Dominique Barchiesi, “Fitting the optical constants of gold, silver, chromium, titanium, and aluminum in the visible bandwidth,” J. Nanophotonics 8(1), 083097 (2014).
[Crossref]

N. N. Potravkin, E. B. Cherepetskaya, I. Perezhogin, and V. Makarov, “Ultrashort elliptically polarized laser pulse interaction with helical photonic metamaterial,” Opt. Mater. Express 4(10), 2090–2101 (2014).
[Crossref]

2013 (1)

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” J. Comput. Phys. 232(1), 189–213 (2013).
[Crossref]

2012 (1)

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref]

2011 (2)

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

R. Kullock, A. Hille, A. Haußmann, S. Grafström, and L. M. Eng, “SHG simulations of plasmonic nanoparticles using curved elements,” Opt. Express 19(15), 14426–14436 (2011).
[Crossref]

2010 (2)

2009 (3)

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(16), 2501–2503 (2009).
[Crossref]

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

2007 (2)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Y. Huang, Y. Zhou, and S.-T. Wu, “Broadband circular polarizer using stacked chiral polymer films,” Opt. Express 15(10), 6414–6419 (2007).
[Crossref]

1995 (1)

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Alam, M. Z.

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
[Crossref]

Alù, A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref]

Arakawa, Y.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

Belkin, M. A.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref]

Bergot, M.

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” J. Comput. Phys. 232(1), 189–213 (2013).
[Crossref]

Boltasseva, A.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Boyd, R. W.

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
[Crossref]

R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
[Crossref]

Broer, D. J.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Burger, S.

Caspani, L.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Chen, Y.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Cherepetskaya, E. B.

Clerici, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

de Araújo, C.

A. Reyna and C. de Araújo, “High-order optical nonlinearities in plasmonic nanocomposites – a review,” Adv. Opt. Photonics 9(4), 720–774 (2017).
[Crossref]

de Gennes, P.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

Decker, M.

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(16), 2501–2503 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 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,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Doležel, I.

P. Šolín, K. Segeth, and I. Doležel, Higher-order Finite Element Methods (Chapman & Hall/CRC, 2004).

Dominique Barchiesi, T. G.

T. G. Dominique Barchiesi, “Fitting the optical constants of gold, silver, chromium, titanium, and aluminum in the visible bandwidth,” J. Nanophotonics 8(1), 083097 (2014).
[Crossref]

Duruflé, M.

M. Bergot and M. Duruflé, “High-order optimal edge elements for pyramids, prisms and hexahedra,” J. Comput. Phys. 232(1), 189–213 (2013).
[Crossref]

Eng, L. M.

Faccio, D.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Falco, A.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Fedotov, V. A.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Ferrera, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Gabitov, I. R.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27

Gansel, J. K.

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express 18(2), 1059–1069 (2010).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

Grafström, S.

Haußmann, A.

Hille, A.

Huang, Y.

Iwamoto, S.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Kaipurath, R. P. M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Kim, J.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Kinsey, N.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Konishi, K.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Kriegler, C. E.

Kullock, R.

Kumagai, N.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Kuwata-Gonokami, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Lei, D.

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

Leon, I. D.

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
[Crossref]

R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
[Crossref]

Li, G.

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

Linden, S.

J. K. Gansel, M. Wegener, S. Burger, and S. Linden, “Gold helix photonic metamaterials: A numerical parameter study,” Opt. Express 18(2), 1059–1069 (2010).
[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(16), 2501–2503 (2009).
[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Litchinitser, N. M.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27

Liu, X.-X.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Lu, P. X.

Lu, Y. F.

Lub, J.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Maimistov, A. I.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27

Makarov, V.

Mol, G. N.

D. J. Broer, J. Lub, and G. N. Mol, “Wide-band reflective polarizers from cholesteric polymer networks with a pitch gradient,” Nature 378(6556), 467–469 (1995).
[Crossref]

Nomura, M.

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

Panoiu, N.

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

Perezhogin, I.

Pietrzyk, M.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Plum, E.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Potravkin, N. N.

Prost, J.

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

Reyna, A.

A. Reyna and C. de Araújo, “High-order optical nonlinearities in plasmonic nanocomposites – a review,” Adv. Opt. Photonics 9(4), 720–774 (2017).
[Crossref]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

Roger, T.

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Ruther, M.

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

Schulz, S.

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
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Segeth, K.

P. Šolín, K. Segeth, and I. Doležel, Higher-order Finite Element Methods (Chapman & Hall/CRC, 2004).

Sha, W.

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

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L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

Shalaev, V. M.

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27

Shi, Z.

R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
[Crossref]

Šolín, P.

P. Šolín, K. Segeth, and I. Doležel, Higher-order Finite Element Methods (Chapman & Hall/CRC, 2004).

Soukoulis, C. M.

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Tsai, D. P.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

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M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
[Crossref]

von Freymann, G.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
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[Crossref]

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
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Wu, S.-T.

Yang, Z. Y.

Zhao, M.

Zhao, Y.

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref]

Zheludev, N. I.

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Zhou, J.

Zhou, Y.

Adv. Mater. (1)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Adv. Opt. Photonics (1)

A. Reyna and C. de Araújo, “High-order optical nonlinearities in plasmonic nanocomposites – a review,” Adv. Opt. Photonics 9(4), 720–774 (2017).
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[Crossref]

J. Opt. (1)

N. Panoiu, W. Sha, D. Lei, and G. Li, “Nonlinear optics in plasmonic nanostructures,” J. Opt. 20(8), 083001 (2018).
[Crossref]

Nat. Commun. (1)

Y. Zhao, M. A. Belkin, and A. Alù, “Twisted optical metamaterials for planarized ultrathin broadband circular polarizers,” Nat. Commun. 3(1), 870 (2012).
[Crossref]

Nat. Photonics (1)

M. Z. Alam, S. Schulz, J. Upham, I. D. Leon, and R. W. Boyd, “Large optical nonlinearity of nanoantennas coupled to an epsilon-near-zero material,” Nat. Photonics 12(2), 79–83 (2018).
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Nature (1)

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[Crossref]

Opt. Commun. (1)

R. W. Boyd, Z. Shi, and I. D. Leon, “The third-order nonlinear optical susceptibility of gold,” Opt. Commun. 326, 74–79 (2014).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Opt. Mater. Express (1)

Phys. Rev. Lett. (3)

L. Caspani, R. P. M. Kaipurath, M. Clerici, M. Ferrera, T. Roger, J. Kim, N. Kinsey, M. Pietrzyk, A. Falco, V. Shalaev, A. Boltasseva, and D. Faccio, “Enhanced nonlinear refractive index in $\epsilon$ϵ-near-zero materials,” Phys. Rev. Lett. 116(23), 233901 (2016).
[Crossref]

K. Konishi, M. Nomura, N. Kumagai, S. Iwamoto, Y. Arakawa, and M. Kuwata-Gonokami, “Circularly polarized light emission from semiconductor planar chiral nanostructures,” Phys. Rev. Lett. 106(5), 057402 (2011).
[Crossref]

E. Plum, X.-X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: Optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[Crossref]

Science (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref]

Other (5)

N. M. Litchinitser, I. R. Gabitov, A. I. Maimistov, and V. M. Shalaev, “Innovation and intellectual property rights,” in Tutorials in Metamaterials, M. A. Noginov and V. A. Podolskiy, eds. (CRC Press, Boca Raton, 2012), pp. 1–27

P. de Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon Press, 1993).

P. Šolín, K. Segeth, and I. Doležel, Higher-order Finite Element Methods (Chapman & Hall/CRC, 2004).

www.csimsoft.com/trelis .

www.software.intel.com/en-us/intel-mkl .

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Figures (5)

Fig. 1.
Fig. 1. Sketch of the basic element of the metamaterial (a), two variants of its capturing by first-order HEX08 (b) and second-order HEX20 (c) elements and the types of elements used for hybrid unstructured mesh generation (d). The incident circularly polarized wave with wavevector $\mathbf {k}$ is schematically shown in (a).
Fig. 2.
Fig. 2. Dependency of transmission coefficients of right-hand (red) and left-hand (blue) CP normally incident waves on their wavelength in the case of linear optical response of the metamaterial. The basic element is a right-hand helix of $N=1$ (a), $2$ (b) and $3$ (c) turns.
Fig. 3.
Fig. 3. Distribution of normalized absolute value of electric field strength vector on the surface of a metal helix with linear optical response and turn number $N=1$ (a and c) and $N=2$ (b and d) in the cases of left-hand (a and b) and right-hand (c and d) CP normally incident monochromatic light with wavelength $\lambda =700\textrm {nm}$. Black curves show the conduction current lines on the surface of the metal.
Fig. 4.
Fig. 4. Dependency of transmission coefficients of right-hand (red) and left-hand (blue) CP waves on their wavelength in the cases of linear and nonlinear (defocusing) optical response of the metal helices with turn number $N=3$. Solid curves correspond to linear response, dashed curves — $E_0^{2}|\textrm {Re}\;\chi _s| = 0.25$ (a), $0$ (b) and $E_0^{2}|\textrm {Im}\;\chi _s| = 0$ (a), $0.25$ (b), dotted curves — $E_0^{2}|\textrm {Re}\;\chi _s| = 0.5$ (a), $0$ (b) and $E_0^{2}|\textrm {Im}\;\chi _s| = 0$ (a), $0.5$ (b).
Fig. 5.
Fig. 5. Dependency of transmission coefficients of left-hand CP wave on the product $E_0^{2}|\textrm {Re}\;\chi _s|$ (a) and $E_0^{2}|\textrm {Im}\;\chi _s|$ (b) for three different wavelengths.

Equations (2)

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ε ( ω ) = ε ω D 2 / [ ω ( ω i γ D ) ] ω L 2 Δ ε / [ ω 2 ω L 2 i ω γ L ] .
rot rot E ( ω 2 / c 2 ) [ ( ε + 6 π χ 2 | E | 2 ) E + 3 π χ 1 ( E E ) E ] = 0.

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