Abstract

Spatial dispersion is an intriguing property of essentially all nanostructured optical media. In particular, it makes optical waves with equal frequencies and polarizations have different wavelengths, if they propagate in different directions. This can offer new approaches to control light radiation and propagation. Spatially dispersive nanomaterials, such as metamaterials, are often treated in terms of wave parameters, such as refractive index and impedance retrieved from reflection and transmission coefficients of the material at each incidence angle. Usually, however, the waves are approximated as transverse, which simplifies the description, but yields wrong results, if spatial dispersion or optical anisotropy is significant. In this work, we present a method to calculate the wave parameters of a general spatially dispersive and optically anisotropic medium without applying such an approximation. The method allows one to evaluate the true impedances and field vectors of the effective waves, obtaining thus the true light intensity and energy propagation direction in the medium. The equations are applied to several examples of spatially dispersive and anisotropic materials. The method introduces new insights into optics of nanostructured media and extends the design of such media towards optical phenomena involving significant spatial dispersion.

© 2017 Optical Society of America

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References

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  2. C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
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  3. C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
    [Crossref]
  4. T. Paul, C. Menzel, W. Smigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
    [Crossref]
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  9. V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
    [Crossref]
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    [Crossref]
  11. V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
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  16. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
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    [Crossref]
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    [Crossref]
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2016 (2)

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
[Crossref] [PubMed]

2015 (3)

A. Shevchenko, P. Grahn, and M. Kaivola, “Spatially dispersive functional optical metamaterials,’ J. Nanophoton. 9, 093097 (2015).
[Crossref]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

2014 (1)

A. Shevchenko, P. Grahn, and M. Kaivola, “Internally twisted spatially dispersive optical metamaterials,” J. Nanophot. 8, 083074 (2014).
[Crossref]

2013 (3)

P. Grahn, A. Shevchenko, and M. Kaivola, “Theoretical description of bifacial optical nanomaterials,” Opt. Express 21, 23471–23485 (2013).
[Crossref] [PubMed]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

2012 (1)

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. B 86, 075462 (2012).
[Crossref]

2011 (1)

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

2010 (3)

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

2009 (3)

M. G. Silveirinha, “Poynting vector, heating rate, and stored energy in structured materials: A first-principle derivation,” Phys. Rev. B 80, 235120 (2009).
[Crossref]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[Crossref]

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

2008 (3)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439 (2008).
[Crossref] [PubMed]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

2007 (1)

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

2006 (2)

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

2001 (1)

E. B. Graham and R. E. Raab, “The role of the macroscopic surface density of bound electric dipole moment in reflection,” Proc. R. Soc. Lond. A 457, 471 (2001).
[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. A 456, 1193 (2000).
[Crossref]

1999 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Bartal, G.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439 (2008).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

Casse, B.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Chandra, S. K. R.

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

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. B 86, 075462 (2012).
[Crossref]

DuBroff, R. E.

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (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. B 86, 075462 (2012).
[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. B 86, 075462 (2012).
[Crossref]

Graham, E. B.

E. B. Graham and R. E. Raab, “The role of the macroscopic surface density of bound electric dipole moment in reflection,” Proc. R. Soc. Lond. A 457, 471 (2001).
[Crossref]

E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum-dielectric interface,” Proc. R. Soc. Lond. A 456, 1193 (2000).
[Crossref]

Grahn, P.

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Spatially dispersive functional optical metamaterials,’ J. Nanophoton. 9, 093097 (2015).
[Crossref]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Internally twisted spatially dispersive optical metamaterials,” J. Nanophot. 8, 083074 (2014).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Theoretical description of bifacial optical nanomaterials,” Opt. Express 21, 23471–23485 (2013).
[Crossref] [PubMed]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013).
[Crossref]

Gultepe, E.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Hecht, B.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2006).
[Crossref]

Huang, Y.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

Kaivola, M.

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
[Crossref] [PubMed]

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Spatially dispersive functional optical metamaterials,’ J. Nanophoton. 9, 093097 (2015).
[Crossref]

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Internally twisted spatially dispersive optical metamaterials,” J. Nanophot. 8, 083074 (2014).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Theoretical description of bifacial optical nanomaterials,” Opt. Express 21, 23471–23485 (2013).
[Crossref] [PubMed]

Karrila, A.

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Kivijärvi, V.

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
[Crossref] [PubMed]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

Koledintseva, M. Y.

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

Krausz, 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. B 86, 075462 (2012).
[Crossref]

Lalanne, P.

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

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lederer, F.

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

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lindfors, K.

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

Liu, Y.

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439 (2008).
[Crossref] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Liu, Z.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Lu, W.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Lu, Z.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Menon, L.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Menzel, C.

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

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

Murakowski, J.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Novotny, L.

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2006).
[Crossref]

Nyman, M.

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
[Crossref] [PubMed]

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

Paul, T.

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

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

Pertsch, T.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

Prather, D.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Raab, R. E.

E. B. Graham and R. E. Raab, “The role of the macroscopic surface density of bound electric dipole moment in reflection,” Proc. R. Soc. Lond. A 457, 471 (2001).
[Crossref]

E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum-dielectric interface,” Proc. R. Soc. Lond. A 456, 1193 (2000).
[Crossref]

Rockstuhl, C.

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

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Schneider, G.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Schuetz, C.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Schwartz, R. W.

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

Shevchenko, A.

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “Optical-image transfer through a diffraction-compensating metamaterial,” Opt. Express 24, 9806 (2016).
[Crossref] [PubMed]

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Spatially dispersive functional optical metamaterials,’ J. Nanophoton. 9, 093097 (2015).
[Crossref]

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

A. Shevchenko, P. Grahn, and M. Kaivola, “Internally twisted spatially dispersive optical metamaterials,” J. Nanophot. 8, 083074 (2014).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Theoretical description of bifacial optical nanomaterials,” Opt. Express 21, 23471–23485 (2013).
[Crossref] [PubMed]

Shi, S.

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Sihvola, A.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE, 1999).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Poynting vector, heating rate, and stored energy in structured materials: A first-principle derivation,” Phys. Rev. B 80, 235120 (2009).
[Crossref]

Simovski, C. R.

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

Smigaj, W.

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

Soukoulis, C. M.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

Sridhar, S.

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

Stacy, A. M.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Stratton, J.

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941)

Sun, C.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

Tretyakov, S.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

Wang, Y.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Yao, J.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Zhang, X.

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Y. Liu, G. Bartal, and X. Zhang, “All-angle negative refraction and imaging in a bulk medium made of metallic nanowires in the visible region,” Opt. Express 16, 15439 (2008).
[Crossref] [PubMed]

Adv. Mater. (1)

T. Paul, C. Menzel, C. Rockstuhl, and F. Lederer, “Advanced optical metamaterials,” Adv. Mater. 22, 2354 (2010).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

B. Casse, W. Lu, Y. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using threedimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212 (1999).
[Crossref]

J. Nanophot. (1)

A. Shevchenko, P. Grahn, and M. Kaivola, “Internally twisted spatially dispersive optical metamaterials,” J. Nanophot. 8, 083074 (2014).
[Crossref]

J. Nanophoton. (1)

A. Shevchenko, P. Grahn, and M. Kaivola, “Spatially dispersive functional optical metamaterials,’ J. Nanophoton. 9, 093097 (2015).
[Crossref]

J. Opt. (1)

V. Kivijärvi, M. Nyman, A. Shevchenko, and M. Kaivola, “An optical metamaterial with simultaneously suppressed optical diffraction and surface reflection,” J. Opt. 18, 035103 (2016).
[Crossref]

Metamaterials (1)

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

Nat. Photon. (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photon. 7, 958 (2013).
[Crossref]

New J. Phys. (2)

V. Kivijärvi, M. Nyman, A. Karrila, P. Grahn, A. Shevchenko, and M. Kaivola, “Interaction of metamaterials with optical beams,” New J. Phys. 17, 063019 (2015).
[Crossref]

P. Grahn, A. Shevchenko, and M. Kaivola, “Interferometric description of optical metamaterials,” New J. Phys. 15(11), 113044 (2013).
[Crossref]

Opt. Express (3)

Phys. Rev. Appl. (1)

A. Shevchenko, V. Kivijärvi, P. Grahn, M. Kaivola, and K. Lindfors, “Bifacial metasurface with quadrupole optical response,” Phys. Rev. Appl. 4, 014019 (2015).
[Crossref]

Phys. Rev. B (7)

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
[Crossref]

M. G. Silveirinha, “Poynting vector, heating rate, and stored energy in structured materials: A first-principle derivation,” Phys. Rev. B 80, 235120 (2009).
[Crossref]

T. Paul, C. Rockstuhl, C. Menzel, and F. Lederer, “Anomalous refraction, diffraction, and imaging in metamaterials,” Phys. Rev. B 79, 115430 (2009).
[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. B 86, 075462 (2012).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, and F. Lederer, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).
[Crossref]

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B 81, 035320 (2010).
[Crossref]

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

Phys. Rev. Lett. (1)

Z. Lu, S. Shi, J. Murakowski, G. Schneider, C. Schuetz, and D. Prather, “Experimental demonstration of selfcollimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref]

Proc. R. Soc. Lond. A (2)

E. B. Graham and R. E. Raab, “Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum-dielectric interface,” Proc. R. Soc. Lond. A 456, 1193 (2000).
[Crossref]

E. B. Graham and R. E. Raab, “The role of the macroscopic surface density of bound electric dipole moment in reflection,” Proc. R. Soc. Lond. A 457, 471 (2001).
[Crossref]

Progr. Electromagn. Res. (1)

M. Y. Koledintseva, S. K. R. Chandra, R. E. DuBroff, and R. W. Schwartz, “Modeling of dielectric mixtures containing conducting inclusions with statistically distributed aspect ratio,” Progr. Electromagn. Res. 66, 213 (2006).
[Crossref]

Science (1)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[Crossref] [PubMed]

Other (4)

J. Stratton, Electromagnetic Theory (McGraw-Hill, 1941)

L. Novotny and B. Hecht, Principles of nano-optics (Cambridge University, 2006).
[Crossref]

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEEE, 1999).
[Crossref]

Cited By

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Figures (8)

Fig. 1
Fig. 1

Reflection and transmission of (a) a TE and (b) a TM wave at a boundary between two spatially dispersive media. The electric and magnetic field vectors are not necessarily transverse with respect to the wave vector (shown by the black arrows) and the incidence and reflection angles can differ.

Fig. 2
Fig. 2

Reflection and transmission of plane waves by a slab of a spatially dispersive material. The arrows show the directions of the wave vectors. Parameters ρij and τij are the tangential reflection and transmission coefficients that for the electric field are given by Eqs. (3) and (4) and for the magnetic field by Eqs. (5) and (6). They describe the tangential components of the waves propagating from medium i to medium j at each surface of the slab independently of the polarization of the incidence wave.

Fig. 3
Fig. 3

Schematic presentation of isofrequency contours of n(θ) in the xz-plane for (a) TE and (b) TM polarizations. The contours are the cross sections of the corresponding isofrequency surfaces r = Re{n(θ, ϕ)}, and the vectors E and H are tangential to these surfaces.

Fig. 4
Fig. 4

The correction factor C(θ) (blue line) for the impedance of a TM polarized wave due to a non-transverse character of the electric field in (a) an elliptic/hyperbolic material with no = 1.5 and ne = 2 and (b) a diffraction-compensating material described by Eq. (34). If the waves were transverse also in E, C(θ) would be equal to 1, as shown by the circular dashed lines.

Fig. 5
Fig. 5

A single layer of a metamaterial composed of silver nanorods in glass. The rods are 130 nm long and 30 nm thick. They are tilted in the plane of the layer (xz-plane) by an angle of 45°. The material is composed of many such layers distributed periodically in the vertical y-direction. The lattice constants of the material along the x-, y-, and z-axes are Λx = Λy = 130 nm and Λz = 200 nm.

Fig. 6
Fig. 6

The real (blue) and imaginary (red) parts of n and η for TM-polarized waves in a metamaterial composed of tilted silver nanorods in glass calculated using the approach of this paper ((a) and (b)) and an effective medium approximation of Refs. [13] and [25] ((c) and (d)); θ is the propagation angle in the material and η is normalized to that in vacuum. The imaginary parts of the quantities are multiplied by 100. The red dashed line shows negative Im{η}. The dashed brown line shows the impedance of glass. The grey sectors cover the angles inaccessible from glass. The angles too close to θ = ±90° are also excluded due to increased numerical errors.

Fig. 7
Fig. 7

Propagation of an optical beam in a metamaterial of Fig. 5. The beam intensity distribution is shown in (a) and the phase distribution in (b). The beam propagates upwards towards an interface with glass (shown by the horizontal white line). The wavefronts are parallel to the interface, corresponding to θ = 0, but the energy transfer direction shown by the blue dashed line in (a) is given by an angle β = -25° as calculated from Eq. (28).

Fig. 8
Fig. 8

Generation of light in a slab of a metamaterial of Fig. 5 by a planar distribution of electric dipoles in the middle of the slab. The position of the dipoles is shown by the horizontal black line. For x-directed dipoles, the beam intensity profile is shown in (a) and the phase distribution in (b). The metamaterial-glass interfaces are shown by the horizontal white lines. The energy transfer directions predicted by Eq. (28) are shown in (a) by the blue dashed lines.

Equations (34)

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

E i + E r = E t
H i H r = H t
ρ E = E r / E i ,
τ E = E t / E i ,
ρ H = H r / H i ,
τ H = H t / H i ,
η = E / H ,
ρ E = 1 η i 1 η t 1 η t + 1 η r ,
τ E = 1 η i + 1 η r 1 η t + 1 η r ,
ρ H = η t η i η t + η r ,
τ H = η i + η i η t + η r .
t f t ( θ i ) = τ 12 τ 23 exp ( i k f z d ) 1 ρ 21 ρ 23 exp ( i [ k f z + k b z ] d ) ,
r f r ( θ i ) = ρ 12 + ρ 23 τ 21 τ 12 exp ( i [ ( k f z + k b z ) ] d ) 1 ρ 21 ρ 23 exp ( i [ ( k f z + k b z ) ] d ) ,
t b t ( π θ i ) = τ 21 τ 32 exp ( i k b z d ) 1 ρ 21 ρ 23 exp ( i [ ( k f z + k b z ) ] d ) t f ,
r b r ( π θ i ) = ρ 32 + ρ 21 τ 32 τ 23 exp ( i [ ( k f z + k b z ) ] d ) 1 ρ 21 ρ 23 exp ( i [ ( k f z + k b z ) ] d ) r f .
k f z = [ i ln ( a ± a 2 b ) + 2 π m ] / d ,
ξ f = 1 t f exp ( i k b z d ) + r f 1 t f exp ( i k b z d ) r f ,
k b z = [ i ln ( a a 2 b ) + 2 π m ] / d ,
ξ b = 1 t b exp ( i k f z d ) + r b 1 t b exp ( i k f z d ) r b ,
a = 1 + t f t b r f r b 2 t f ,
b = t b t f .
k 0 n f = ± k 2 + k f z 2 ,
k 0 n b = ± k 2 + k b z 2 ,
sin θ = n s n f sin θ i ,
tan α = H z H = n n sin θ ( n / θ ) cos θ cos θ ( n / θ ) sin θ ,
n TE = ξ n s | cos α | = ξ η s | n cos θ + ( n / θ ) sin θ | n 2 + ( n / θ ) 2 ,
η s = η s | cos θ i |
tan β = E z E = n sin θ ( n / θ ) cos θ n cos θ + ( n / θ ) sin θ ,
η TM = ξ η s | cos β | = ξ η s n 2 + ( n / θ ) 2 | n cos θ + ( n / θ ) sin θ | ,
η s = η s | cos θ i | .
sin | Δ γ | = | Im { n } n sin γ | ,
C ( θ ) = | cos θ / cos β | = | cos θ | n 2 + ( n / θ ) 2 | n cos θ + ( n / θ ) sin θ |
n ( θ ) = n o n e n e 2 cos 2 θ + n o 2 sin 2 θ ,
n ( θ ) = n 0 cos θ ,