Abstract

In general, optical nanomaterials composed of noncentrosymmetric nanoscatterers are bifacial, meaning that two counter-propagating waves inside the material behave differently. Thus far a practical theory for the description of such materials has been missing. Herein, we present a theory that connects the design of the bifacial nanomaterial’s “atoms” with the refractive index and wave impedance of the medium. We also introduce generalized Fresnel coefficients and investigate the role of electromagnetic multipoles on the bifaciality. We find that in any material two counter-propagating waves must experience the same refractive index, but their impedances can differ. The model is demonstrated in practice by the design of a nanomaterial slab with one of its facets being optically reflective, while the other being totally non-reflective.

© 2013 OSA

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2013 (6)

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt.15, 023001 (2013).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Multipole polarizability of a nanodimer in optical waves,” J. Europ. Opt. Soc. Rap. Public.8, 13009 (2013).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical nanomaterials,” arXiv:1303.6432 (2013).

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Y. Zeng, H.-T. Chen, and D. A. R. Dalvit, “The role of magnetic dipoles and non-zero-order Bragg waves in metamaterial perfect absorbers,” Opt. Express21, 3540–3546 (2013).
[CrossRef] [PubMed]

2012 (8)

P. Grahn, A. Shevchenko, and M. Kaivola, “Electric dipole-free interaction of visible light with pairs of subwavelength-size silver particles,” Phys. Rev. B86, 035419 (2012).
[CrossRef]

H.-T. Chen, “Interference theory of metamaterial perfect absorbers,” Opt. Express20, 7165–7172 (2012).
[CrossRef] [PubMed]

Z. Li, K. Aydin, and E. Ozbay, “Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions,” Phot. Nano. Fund. Appl.10, 329–336 (2012).
[CrossRef]

B. Gompf, B. Krausz, B. Frank, and M. Dressel, “k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B86, 075462 (2012).
[CrossRef]

T. Pakizeh, “Unidirectional radiation of a magnetic dipole coupled to an ultracompact nanoantenna at visible wavelengths,” J. Opt. Soc. Am. B29, 2446–2452 (2012).
[CrossRef]

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
[CrossRef]

S. H. Alavi Lavasani and T. Pakizeh, “Color-switched directional ultracompact optical nanoantennas,” J. Opt. Soc. Am. B29, 1361–1366 (2012).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

2011 (4)

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
[CrossRef] [PubMed]

2010 (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

2009 (6)

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E79, 026610 (2009).
[CrossRef]

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
[CrossRef]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

E. Pshenay-Severin, U. Hübner, C. Menzel, C. Helgert, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Double-element metamaterial with negative index at near-infrared wavelengths,” Opt. Lett.34, 1678–1680 (2009).
[CrossRef] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett.9, 2343–2349 (2009).
[CrossRef] [PubMed]

2008 (4)

T. Pakizeh, A. Dmitriev, M. S. Abrishamian, N. Granpayeh, and M. Käll, “Structural asymmetry and induced optical magnetism in plasmonic nanosandwiches,” J. Opt. Soc. Am. B25, 659–667 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

J. H. Lee, Y.-W. Song, K. H. Hwang, J. G. Lee, J. Ha, and D.-S. Zang, “Optically bifacial thin-film wire-grid polarizers with nano-patterns of a graded metal-dielectric composite layer,” Opt. Express16, 16867–16876 (2008).
[PubMed]

2007 (2)

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B75, 075119 (2007).
[CrossRef]

H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express15, 1076–1083 (2007).
[CrossRef] [PubMed]

2005 (1)

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

2004 (1)

M. Sarrazin and J. P. Vigneron, “Nonreciprocal optical reflection from a bidimensional array of subwavelength holes in a metallic film,” Phys. Rev. B70, 193409 (2004).
[CrossRef]

2003 (1)

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
[CrossRef]

2000 (1)

E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum–dielectric interface,” Proc. R. Soc. Lond. A456, 1193–1215 (2000).
[CrossRef]

1996 (1)

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Abrishamian, M. S.

Alaee, R.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Alavi Lavasani, S. H.

Albrektsen, O.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Altewischer, E.

Antosiewicz, T. J.

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
[CrossRef]

Apell, S. P.

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
[CrossRef]

Aydin, K.

Z. Li, K. Aydin, and E. Ozbay, “Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions,” Phot. Nano. Fund. Appl.10, 329–336 (2012).
[CrossRef]

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E79, 026610 (2009).
[CrossRef]

Banas, A.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Banas, K.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Boltasseva, A.

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Cai, W.

Chen, H.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Chen, H.-T.

Chen, S.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

Chettiar, U. K.

Chipouline, A.

Choi, M.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Curto, A. G.

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Dalvit, D. A. R.

Dmitriev, A.

Drachev, V. P.

Dressel, M.

B. Gompf, B. Krausz, B. Frank, and M. Dressel, “k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B86, 075462 (2012).
[CrossRef]

Dudley, D. G.

D. G. Dudley, Mathematical Foundations for Electromagnetic Theory (IEEE Press, New York, 1994).
[CrossRef]

Engheta, N.

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B75, 075119 (2007).
[CrossRef]

Evlyukhin, A. B.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Fedotov, V. A.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

Fleischhauer, M.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

Frank, B.

B. Gompf, B. Krausz, B. Frank, and M. Dressel, “k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B86, 075462 (2012).
[CrossRef]

Giessen, H.

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
[CrossRef] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

Gompf, B.

B. Gompf, B. Krausz, B. Frank, and M. Dressel, “k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B86, 075462 (2012).
[CrossRef]

Graham, E. B.

E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum–dielectric interface,” Proc. R. Soc. Lond. A456, 1193–1215 (2000).
[CrossRef]

E. B. Graham and R. E. Raab, “Reflection from noncentrosymmetric uniaxial crystals: a multipole approach,” J. Opt. Soc. Am. A13, 1239–1248 (1996).
[CrossRef]

Grahn, P.

P. Grahn, A. Shevchenko, and M. Kaivola, “Multipole polarizability of a nanodimer in optical waves,” J. Europ. Opt. Soc. Rap. Public.8, 13009 (2013).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical nanomaterials,” arXiv:1303.6432 (2013).

P. Grahn, A. Shevchenko, and M. Kaivola, “Electric dipole-free interaction of visible light with pairs of subwavelength-size silver particles,” Phys. Rev. B86, 035419 (2012).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

Granpayeh, N.

Ha, J.

Helgert, C.

Hentschel, M.

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
[CrossRef] [PubMed]

Hübner, U.

Hwang, K. H.

Johansson, P.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6, 4370–4379 (1972).
[CrossRef]

Kaelberer, T.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

Kaivola, M.

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical nanomaterials,” arXiv:1303.6432 (2013).

P. Grahn, A. Shevchenko, and M. Kaivola, “Multipole polarizability of a nanodimer in optical waves,” J. Europ. Opt. Soc. Rap. Public.8, 13009 (2013).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electric dipole-free interaction of visible light with pairs of subwavelength-size silver particles,” Phys. Rev. B86, 035419 (2012).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

Käll, M.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett.9, 2343–2349 (2009).
[CrossRef] [PubMed]

T. Pakizeh, A. Dmitriev, M. S. Abrishamian, N. Granpayeh, and M. Käll, “Structural asymmetry and induced optical magnetism in plasmonic nanosandwiches,” J. Opt. Soc. Am. B25, 659–667 (2008).
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Kang, K.-Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Kang, S. B.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Kästel, J.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

Kildishev, A. V.

Kim, Y.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Koschny, Th.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
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B. Gompf, B. Krausz, B. Frank, and M. Dressel, “k-dependent optics of nanostructures: Spatial dispersion of metallic nanorings and split-ring resonators,” Phys. Rev. B86, 075462 (2012).
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A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
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Kwak, M. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Langguth, L.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
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Langhammer, C.

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
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R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
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C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
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T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

E. Pshenay-Severin, U. Hübner, C. Menzel, C. Helgert, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Double-element metamaterial with negative index at near-infrared wavelengths,” Opt. Lett.34, 1678–1680 (2009).
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C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
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C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

Lee, J. G.

Lee, J. H.

Lee, S. H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Lee, Y.-H.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
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Li, Z.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt.15, 023001 (2013).
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Z. Li, K. Aydin, and E. Ozbay, “Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions,” Phot. Nano. Fund. Appl.10, 329–336 (2012).
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Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E79, 026610 (2009).
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Liu, N.

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
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N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
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Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
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Menzel, C.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
[CrossRef]

E. Pshenay-Severin, U. Hübner, C. Menzel, C. Helgert, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Double-element metamaterial with negative index at near-infrared wavelengths,” Opt. Lett.34, 1678–1680 (2009).
[CrossRef] [PubMed]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
[CrossRef]

Miljkovic, V. D.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

Min, B.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Moser, H. O.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Mutlu, M.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt.15, 023001 (2013).
[CrossRef]

Nielsen, M. G.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Ozbay, E.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt.15, 023001 (2013).
[CrossRef]

Z. Li, K. Aydin, and E. Ozbay, “Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions,” Phot. Nano. Fund. Appl.10, 329–336 (2012).
[CrossRef]

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E79, 026610 (2009).
[CrossRef]

Pakizeh, T.

Papasimakis, N.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

Park, N.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Paul, T.

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
[CrossRef]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

Paul Alivisatos, A.

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
[CrossRef] [PubMed]

Pertsch, T.

Pfau, T.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

Pors, A.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Pshenay-Severin, E.

Quidant, R.

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
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E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum–dielectric interface,” Proc. R. Soc. Lond. A456, 1193–1215 (2000).
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E. B. Graham and R. E. Raab, “Reflection from noncentrosymmetric uniaxial crystals: a multipole approach,” J. Opt. Soc. Am. A13, 1239–1248 (1996).
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Rockstuhl, C.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
[CrossRef]

E. Pshenay-Severin, U. Hübner, C. Menzel, C. Helgert, A. Chipouline, C. Rockstuhl, A. Tünnermann, F. Lederer, and T. Pertsch, “Double-element metamaterial with negative index at near-infrared wavelengths,” Opt. Lett.34, 1678–1680 (2009).
[CrossRef] [PubMed]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New Jersey, 2007), 2nd ed.

Sarrazin, M.

M. Sarrazin and J. P. Vigneron, “Nonreciprocal optical reflection from a bidimensional array of subwavelength holes in a metallic film,” Phys. Rev. B70, 193409 (2004).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
[CrossRef]

Shalaev, V. M.

Shegai, T.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

Shevchenko, A.

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical nanomaterials,” arXiv:1303.6432 (2013).

P. Grahn, A. Shevchenko, and M. Kaivola, “Multipole polarizability of a nanodimer in optical waves,” J. Europ. Opt. Soc. Rap. Public.8, 13009 (2013).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electric dipole-free interaction of visible light with pairs of subwavelength-size silver particles,” Phys. Rev. B86, 035419 (2012).
[CrossRef]

Shin, J.

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Silveirinha, M.

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B75, 075119 (2007).
[CrossRef]

Smith, D. R.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
[CrossRef]

Song, Y.-W.

Soukoulis, C. M.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
[CrossRef]

Taminiau, T. H.

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of photonics (Wiley, New Jersey, 2007), 2nd ed.

Tsai, D. P.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

Tünnermann, A.

van Exter, M. P.

van Hulst, N. F.

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Vier, D. C.

D. R. Smith, D. C. Vier, Th. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E71, 036617 (2005).
[CrossRef]

Vigneron, J. P.

M. Sarrazin and J. P. Vigneron, “Nonreciprocal optical reflection from a bidimensional array of subwavelength holes in a metallic film,” Phys. Rev. B70, 193409 (2004).
[CrossRef]

Volpe, G.

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Wadell, C.

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
[CrossRef]

Weiss, T.

N. Liu, M. Hentschel, T. Weiss, A. Paul Alivisatos, and H. Giessen, “Three-dimensional plasmon rulers,” Science332, 1407–1410 (2011).
[CrossRef] [PubMed]

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

Willatzen, M.

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

Woerdman, J. P.

Xu, S.

R. Alaee, C. Menzel, A. Banas, K. Banas, S. Xu, H. Chen, H. O. Moser, F. Lederer, and C. Rockstuhl, “Propagation of electromagnetic fields in bulk terahertz metamaterials: A combined experimental and theoretical study,” Phys. Rev. B87, 075110 (2013).
[CrossRef]

Yuan, H.-K.

Zang, D.-S.

Zeng, Y.

Zengin, G.

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

Zheludev, N. I.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science330, 1510–1512 (2010).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for quasiplanar chiral metamaterials,” Appl. Phys. Lett.93, 233106 (2008).
[CrossRef]

J. Europ. Opt. Soc. Rap. Public. (1)

P. Grahn, A. Shevchenko, and M. Kaivola, “Multipole polarizability of a nanodimer in optical waves,” J. Europ. Opt. Soc. Rap. Public.8, 13009 (2013).
[CrossRef]

J. Opt. (1)

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt.15, 023001 (2013).
[CrossRef]

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

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

J. Phys. Chem. C (1)

T. J. Antosiewicz, S. P. Apell, C. Wadell, and C. Langhammer, “Absorption enhancement in lossy transition metal elements of plasmonic nanosandwiches,” J. Phys. Chem. C116, 20522–20529 (2012).
[CrossRef]

Nano Lett. (1)

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett.9, 2343–2349 (2009).
[CrossRef] [PubMed]

Nat. Commun. (2)

T. Shegai, S. Chen, V. D. Miljković, G. Zengin, P. Johansson, and M. Käll, “A bimetallic nanoantenna for directional colour routing,” Nat. Commun.2, 481 (2011).
[CrossRef] [PubMed]

A. G. Curto, T. H. Taminiau, G. Volpe, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Multipolar radiation of quantum emitters with nanowire optical antennas,” Nat. Commun.4, 1750 (2013).
[CrossRef] [PubMed]

Nature (1)

M. Choi, S. H. Lee, Y. Kim, S. B. Kang, J. Shin, M. H. Kwak, K.-Y. Kang, Y.-H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature470, 369–373 (2011).
[CrossRef] [PubMed]

Nature Mater. (1)

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nature Mater.8, 758–762 (2009).
[CrossRef]

New J. Phys. (2)

S. I. Bozhevolnyi, A. B. Evlyukhin, A. Pors, M. G. Nielsen, M. Willatzen, and O. Albrektsen, “Optical transparency by detuned electrical dipoles,” New J. Phys.13, 023034 (2011).
[CrossRef]

P. Grahn, A. Shevchenko, and M. Kaivola, “Electromagnetic multipole theory for optical nanomaterials,” New J. Phys.14, 093033 (2012).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phot. Nano. Fund. Appl. (1)

Z. Li, K. Aydin, and E. Ozbay, “Retrieval of effective parameters for bianisotropic metamaterials with omega shaped metallic inclusions,” Phot. Nano. Fund. Appl.10, 329–336 (2012).
[CrossRef]

Phys. Rev. B (10)

M. Sarrazin and J. P. Vigneron, “Nonreciprocal optical reflection from a bidimensional array of subwavelength holes in a metallic film,” Phys. Rev. B70, 193409 (2004).
[CrossRef]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B65, 195104 (2002).
[CrossRef]

C. Rockstuhl, C. Menzel, T. Paul, and F. Lederer, “Optical activity in chiral media composed of three-dimensional metallic meta-atoms,” Phys. Rev. B79, 035321 (2009).
[CrossRef]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B79, 115430 (2009).
[CrossRef]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B75, 075119 (2007).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Illustration of a three-dimensional nanomaterial composed of periodically arranged nanoscatterers. Light propagating in the nanomaterial can be described in terms of plane waves reflected back and forth by successive crystal planes. Each such plane is characterized by a transmission coefficient τ and reflection coefficient ρ.

Fig. 2
Fig. 2

Illustration of transmission and reflection of a plane wave propagating inside a nanomaterial. The vertical lines represent planes of nanoscatterers within the nanomaterial and the red arrows stand for the wave vectors. In (a) and (b) the field propagates between the planes at an angle of θ and π + θ, respectively.

Fig. 3
Fig. 3

Illustration of a plane wave incident on a boundary between two bifacial materials. In material 1, the incident and reflected waves see the same refractive index n, but different wave impedances that are, respectively, denoted by η 1 R and η 1 L.

Fig. 4
Fig. 4

(a) Illustration of a gold disc nanodimer used as an artificial atom in each unit cell of the considered bifacial nanomaterial. (b) Illustration of a two-dimensional nanodimer array spanning a plane within the nanomaterial. (c) Normal-incidence intensity transmission T and reflection R of the array as functions of the wavelength λ0 in vacuum. The angles of 0 and π correspond to illumination from the side of the smaller and larger discs, respectively.

Fig. 5
Fig. 5

Normalized intensity, I/I0, of the plane wave radiated in the backward direction by the electric dipoles (black solid lines) and current quadrupoles (green dotted lines) excited in the nanodimer array of Fig. 4(b). I0 is the intensity of the incident wave that propagates in the direction of (a) θ = 0 and (b) θ = π. The actual reflection coefficient of the array is determined by the interference between the waves created by these two multipoles.

Fig. 6
Fig. 6

(a) Refractive index and (b) wave impedance spectra of the nanodimer nanomaterial for TE polarized light. The real parts (solid lines) and imaginary parts (dashed lines) of the quantities are shown separately. Different propagation angles θ in the host dielectric are marked with different colors. The wave impedances of vacuum and glass (refractive index 1.5) are shown in (b) by the horizontal black dashed lines.

Fig. 7
Fig. 7

(a) Refractive index and (b) wave impedance spectra of the nanodimer nanomaterial for TM polarized light. The real parts (solid lines) and imaginary parts (dashed lines) of the quantities are shown separately. Different propagation angles θ in the host dielectric are marked with different colors. The wave impedances of vacuum and glass (refractive index 1.5) are shown in (b) by the horizontal black dashed lines.

Fig. 8
Fig. 8

Normal-incidence intensity transmission T and reflection R of a nanomaterial slab composed of 5 layers of nanodimers embedded in glass. The slab is located in vacuum. The lines show the results obtained by using Eqs. (21)(26) with the wave parameters obtained from a single two-dimensional array in glass. The stars show the results of direct numerical calculation.

Equations (35)

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f ( θ ) = τ ( θ ) exp ( i k z s Λ z ) ,
g ( θ ) = ρ ( θ ) exp ( i k z s Λ z ) ,
U j = f ( θ ) U j 1 + g ( π θ ) U j ,
U j = g ( θ ) U j + f ( θ ) U j + 1 .
U j + 1 + U j 1 a U j = 0 ,
a = f ( θ ) + f ( θ ) 1 [ 1 g ( θ ) g ( π θ ) ] .
U j + 1 = U j exp ( i k z Λ z ) ,
k z Λ z = ± arccos ( a / 2 ) + 2 π m ,
n ( θ ) = ± k 0 1 [ ( k 0 n s sin θ ) 2 + k z 2 ] 1 / 2 ,
E ( z ) = U j exp ( i k z s z ) + U j exp ( i k z s z ) = U j { exp ( i k z s z ) + g ( π θ ) 1 [ 1 f ( θ ) exp ( i k z Λ z ) ] exp ( i k z s z ) } .
H ( z ) = ξ U j exp ( i k z s z ) U j exp ( i k z s z ) η s = ξ U j η s { exp ( i k z s z ) g ( π θ ) 1 [ 1 f ( θ ) exp ( i k z Λ z ) ] exp ( i k z s z ) } ,
ξ = { cos θ for TE polarization , 1 / cos θ for TM polarization .
E ( z ) H ( z ) = η s ξ g ( π θ ) + [ 1 f ( θ ) exp ( i k z Λ z ) ] g ( π θ ) [ 1 f ( θ ) exp ( i k z Λ z ) ] .
cos θ eff = ± [ 1 n s 2 n ( θ ) 2 sin 2 θ ] 1 / 2 .
η TE ( θ ) = η s cos θ eff cos θ g ( π θ ) + [ 1 f ( θ ) exp ( i k z Λ z ) ] g ( π θ ) [ 1 f ( θ ) exp ( i k z Λ z ) ] .
η TM ( θ ) = η s cos θ cos θ eff g ( π θ ) + [ 1 f ( θ ) exp ( i k z Λ z ) ] g ( π θ ) [ 1 f ( θ ) exp ( i k z Λ z ) ] .
E j ( r ) = [ E j TE x ^ + E j TM ( y ^ k z j k j z ^ k y k j ) ] exp ( i k j r ) ,
H j ( r ) = [ E j TE ( y ^ k z j k j z ^ k y k j ) E j TM x ^ ] exp ( i k j r ) η j R .
E r 1 ( r ) = [ E r 1 TE x ^ + E r 1 TM ( y ^ k z 1 k 1 + z ^ k y k 1 ) ] exp ( i k r 1 r ) ,
H r 1 ( r ) = [ E r 1 TM x ^ E r 1 TE ( y ^ k z 1 k 1 + z ^ k y k 1 ) ] exp ( i k r 1 r ) η 1 L .
τ 12 TE = k z 1 / ( k 1 η 1 R ) + k z 1 / ( k 1 η 1 L ) k z 1 / ( k 1 η 1 L ) + k z 2 / ( k 2 η 2 R ) ,
τ 12 TM = k z 1 / ( k 1 η 1 R ) + k z 1 / ( k 1 η 1 L ) k z 1 / ( k 1 η 2 R ) + k z 2 / ( k 2 η 1 L ) ,
ρ 12 TE = k z 1 / ( k 1 η 1 R ) k z 2 / ( k 2 η 2 R ) k z 1 / ( k 1 η 1 L ) + k z 2 / ( k 2 η 2 R ) ,
ρ 12 TM = k z 2 / ( k 2 η 1 R ) k z 1 / ( k 1 η 2 R ) k z 1 / ( k 1 η 2 R ) + k z 2 / ( k 2 η 1 L ) ,
t = exp ( i k z 2 D ) τ 12 τ 23 1 ρ 21 ρ 23 exp ( 2 i k z 2 D ) ,
r = ρ 12 + τ 12 exp ( 2 i k z 2 D ) ρ 23 τ 21 1 ρ 21 ρ 23 exp ( 2 i k z 2 D ) .
J x ( r ) = i ω u , v ( p x Q x z d d z ) δ ( r u Λ x ^ v Λ y ^ ) ,
J x ( z ) = i ω Λ 2 ( p x Q x z d d z ) δ ( z ) .
( d 2 d z 2 + k s 2 ) A x ( z ) = μ s J x ( z ) ,
A x ( z ) = k s 2 ω ε s Λ 2 [ p x i k s Q x z sign ( z ) ] exp ( i k s | z | ) .
E sca ( r ) = i ω [ A ( r ) + 1 k s 2 A ( r ) ] .
ρ = i k s 2 ε s Λ 2 [ α x + i k s β x z ] .
τ = 1 + i k s 2 ε s Λ 2 [ α x i k s β x z ] .
p x = 6 π i ε s E 0 k s 3 [ a E ( 1 , 1 ) a E ( 1 , 1 ) 2 ] ,
Q x z = π ε s E 0 k s 4 [ 3 a M ( 1 , 1 ) + 3 a M ( 1 , 1 ) 5 a E ( 2 , 1 ) + 5 a E ( 2 , 1 ) ] ,

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