Abstract

The feature of spatial dispersion in periodic layered metamaterials is theoretically investigated. An effective medium model is proposed to derive the nonlocal effective permittivity tensor, which exhibits drastic variations in the wave vector domain. Strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. In particular, the nonlocal effect gives rise to additional waves that are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments. Spatial dispersion is also manifest on the parabolic-like dispersion, a non-standard type of dispersion in the medium. The associated negative refraction and backward wave occur even when the effective permittivity components are all positive, which is considered a property not available in the local medium.

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  28. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
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    [CrossRef]
  30. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
    [CrossRef]
  31. L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).
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    [CrossRef]
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    [CrossRef]
  34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
    [CrossRef]
  35. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
    [CrossRef]
  36. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
    [CrossRef]
  37. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
    [CrossRef] [PubMed]
  38. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
    [CrossRef]
  39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969).
    [CrossRef]
  40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003).
    [CrossRef]
  41. J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London203, 385 (1904).
    [CrossRef]
  42. R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968).
    [CrossRef]

2012 (1)

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

2011 (4)

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

2010 (2)

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010).
[CrossRef] [PubMed]

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

2009 (3)

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
[CrossRef]

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009).
[CrossRef] [PubMed]

2008 (2)

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter20, 325211 (2008).
[CrossRef]

F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

2007 (1)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

2006 (3)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
[CrossRef]

M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp.49, 1029 (2006).
[CrossRef]

2004 (1)

C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E70, 046616 (2004).
[CrossRef]

2003 (2)

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

2001 (1)

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm.190, 205–209 (2001).
[CrossRef]

1996 (1)

D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett.76, 4713–4716 (1996).
[CrossRef] [PubMed]

1993 (1)

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B48, 15372–15389 (1993).
[CrossRef]

1989 (1)

1984 (1)

R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B29, 2232–2237 (1984).
[CrossRef]

1976 (1)

M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B14, 3384–3393 (1976).
[CrossRef]

1975 (1)

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B11, 2871–2876 (1975).
[CrossRef]

1973 (1)

A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B7, 2787–2810 (1973).
[CrossRef]

1970 (1)

A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B2, 835–850 (1970).
[CrossRef]

1969 (2)

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969).
[CrossRef]

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969).
[CrossRef]

1968 (3)

R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968).
[CrossRef]

A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett.21, 85–88 (1968).
[CrossRef]

K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev.172, 607–624 (1968).
[CrossRef]

1963 (1)

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963).
[CrossRef]

1958 (2)

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev.112, 1555–1567 (1958).
[CrossRef]

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids5, 11–22 (1958).
[CrossRef]

1948 (1)

G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci.195, 336–364 (1948).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London203, 385 (1904).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984).
[CrossRef]

Atkinson, R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Belov, P. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E70, 046616 (2004).
[CrossRef]

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Bishop, M. F.

M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B14, 3384–3393 (1976).
[CrossRef]

Braun, J.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Brillouin, L.

L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

Chan, C. T.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

Chebykin, A. V.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

Chen, B.

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B48, 15372–15389 (1993).
[CrossRef]

de Waele, R.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

Dressel, M.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969).
[CrossRef]

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Evans, P. R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Fan, X.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

Fang, A.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
[CrossRef]

Fuchs, R.

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969).
[CrossRef]

K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev.172, 607–624 (1968).
[CrossRef]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

Giessen, H.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984).
[CrossRef]

Gompf, B.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Gray, S. K.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010).
[CrossRef] [PubMed]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009).
[CrossRef] [PubMed]

Harrison, M. J.

A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B2, 835–850 (1970).
[CrossRef]

A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett.21, 85–88 (1968).
[CrossRef]

Hendren, W. R.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Hopfield, J. J.

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963).
[CrossRef]

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev.112, 1555–1567 (1958).
[CrossRef]

Hornreich, R. M.

R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968).
[CrossRef]

Hubner, U.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Jauho, A.-P.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Jones, W. E.

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969).
[CrossRef]

Kivshar, Y. S.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Kliewer, K. L.

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969).
[CrossRef]

K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev.172, 607–624 (1968).
[CrossRef]

Kocherga, O. D.

S. I. Pekar and O. D. Kocherga, Crystal Optics and Additional Light Waves (Benjamin/Cummings, 1983).

Koschny, T.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
[CrossRef]

Kuipers, L.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).

Lee, J. C. W.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).

Maradudin, A. A.

M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B14, 3384–3393 (1976).
[CrossRef]

A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B7, 2787–2810 (1973).
[CrossRef]

Marques, R.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Maslovski, S. I.

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Maxwell-Garnett, J. C.

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London203, 385 (1904).
[CrossRef]

McMahon, J. M.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010).
[CrossRef] [PubMed]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009).
[CrossRef] [PubMed]

Melnyk, A. R.

A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B2, 835–850 (1970).
[CrossRef]

A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett.21, 85–88 (1968).
[CrossRef]

Mills, D. L.

A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B7, 2787–2810 (1973).
[CrossRef]

Mortensen, N. A.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Murphy, A.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Nefedov, I. S.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Nelson, D. F.

D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett.76, 4713–4716 (1996).
[CrossRef] [PubMed]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B48, 15372–15389 (1993).
[CrossRef]

Orlov, A. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Pekar, S. I.

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids5, 11–22 (1958).
[CrossRef]

S. I. Pekar and O. D. Kocherga, Crystal Optics and Additional Light Waves (Benjamin/Cummings, 1983).

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
[CrossRef]

Pitaevski, L. P.

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).

Podolskiy, V. A.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Pollard, R. J.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Polman, A.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

Raza, S.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Reuter, G. E. H.

G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci.195, 336–364 (1948).
[CrossRef]

Ruppin, R.

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm.190, 205–209 (2001).
[CrossRef]

R. Ruppin, “Optical properties of a spatially dispersive cylinder,” J. Opt. Soc. Am. B6, 1559–1563 (1989).
[CrossRef]

R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B29, 2232–2237 (1984).
[CrossRef]

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B11, 2871–2876 (1975).
[CrossRef]

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

Schatz, G. C.

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010).
[CrossRef] [PubMed]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009).
[CrossRef] [PubMed]

Shtrikman, S.

R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968).
[CrossRef]

Silveirinha, M.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Simovski, C. R.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E70, 046616 (2004).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Sondheimer, E. H.

G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci.195, 336–364 (1948).
[CrossRef]

Soukoulis, C. M.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
[CrossRef]

Thomas, D. G.

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963).
[CrossRef]

Toscano, G.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Tretyakov, S. A.

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
[CrossRef]

Verhagen, E.

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

Vladimir, M. A.

M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp.49, 1029 (2006).
[CrossRef]

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

Vozianova, A. V.

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

Wang, G. P.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

Weiss, T.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
[CrossRef]

Wubs, M.

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

Wurtz, G. A.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Yannopapas, V.

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter20, 325211 (2008).
[CrossRef]

Yu, N. G.

M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp.49, 1029 (2006).
[CrossRef]

Zayats, A. V.

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007).
[CrossRef]

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

J. Phys. Chem. C (1)

F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C112, 17983–17987 (2008).
[CrossRef]

J. Phys. Chem. Solids (1)

S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids5, 11–22 (1958).
[CrossRef]

J. Phys.-Condes. Matter (1)

V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter20, 325211 (2008).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett.37, 259–263 (2003).
[CrossRef]

Nano Lett. (1)

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett.10, 3473–3481 (2010).
[CrossRef] [PubMed]

Opt. Comm. (1)

R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm.190, 205–209 (2001).
[CrossRef]

Philos. Trans. R. Soc. London (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London203, 385 (1904).
[CrossRef]

Phys. Rev. (6)

R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev.171, 1065–1074 (1968).
[CrossRef]

J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963).
[CrossRef]

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev.182, 539–554 (1969).
[CrossRef]

J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev.112, 1555–1567 (1958).
[CrossRef]

W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev.178, 1201–1203 (1969).
[CrossRef]

K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev.172, 607–624 (1968).
[CrossRef]

Phys. Rev. B (13)

A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B2, 835–850 (1970).
[CrossRef]

R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B11, 2871–2876 (1975).
[CrossRef]

A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B7, 2787–2810 (1973).
[CrossRef]

S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B84, 121412 (2011).
[CrossRef]

P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B67, 113103 (2003).
[CrossRef]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011).
[CrossRef]

M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B14, 3384–3393 (1976).
[CrossRef]

R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B29, 2232–2237 (1984).
[CrossRef]

B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B48, 15372–15389 (1993).
[CrossRef]

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011).
[CrossRef]

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012).
[CrossRef]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B74, 115116 (2006).
[CrossRef]

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009).
[CrossRef]

Phys. Rev. E (1)

C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E70, 046616 (2004).
[CrossRef]

Phys. Rev. Lett. (7)

R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett.102, 127405 (2009).
[CrossRef] [PubMed]

J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett.103, 097403 (2009).
[CrossRef] [PubMed]

D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett.76, 4713–4716 (1996).
[CrossRef] [PubMed]

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett.106, 185501 (2011).
[CrossRef] [PubMed]

A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett.21, 85–88 (1968).
[CrossRef]

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett.97, 073901 (2006).
[CrossRef] [PubMed]

E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett.105, 223901 (2010).
[CrossRef]

Phys. Usp. (1)

M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp.49, 1029 (2006).
[CrossRef]

Proc. R. Soc. A-Math. Phys. Eng. Sci. (1)

G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci.195, 336–364 (1948).
[CrossRef]

Other (4)

L. Brillouin, Wave Propagation in Periodic Structures (Dover, 1953).

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984).
[CrossRef]

S. I. Pekar and O. D. Kocherga, Crystal Optics and Additional Light Waves (Benjamin/Cummings, 1983).

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Figures (11)

Fig. 1
Fig. 1

Schematic diagram of the periodic layered structure consisting of two alternating materials.

Fig. 2
Fig. 2

Variations of (a) the pole frequencies ω1 and ω2 and (b) the cutoff frequency ω3 with respect to the dielectric fraction f1 (= 1 − f2) in the unit cell.

Fig. 3
Fig. 3

In-plane effective permittivity (a) ε z eff and (b) ε x eff for the periodic metal-dielectric layered structure with ε1 = 3, ε 2 = 1 ω p 2 / ω 2, a/λp = 0.2, and f1 = 0.75 (f2 = 0.25).

Fig. 4
Fig. 4

In-plane effective permittivity (a) ε z eff and (b) ε x eff for the periodic metal-dielectric layered structure with ε1 = 3, ε 2 = 1 ω p 2 / ω 2, a/λp = 0.2, and f1 = 0.25 (f2 = 0.75).

Fig. 5
Fig. 5

Equifrequency contours at (a) ω/ωp = 0.39 and (b) ω/ωp = 0.6 for the periodic metal-dielectric layered structure with ε1 = 3, ε 2 = 1 ω p 2 / ω 2, f1 = 0.75 (f2 = 0.25), and a/λp = 0.2. Gray circles are equifrequency contours in vacuum. Gray and black arrows are wave vectors in vacuum and the layered structure, respectively, at θ = 40°.

Fig. 6
Fig. 6

Electric field and surface charge patterns for the eigenmodes associated with the parabolic-like dispersion depicted in Fig. 5 for (a) ω/ωp = 0.39 (bonding mode) and (b) ω/ωp = 0.6 (antibonding mode). Red and blue colors correspond to positive and negative charges, respectively.

Fig. 7
Fig. 7

Effective permittivity ε x eff at (a) ω/ωp = 0.39 and (b) ω/ωp = 0.6 for the periodic metal-dielectric layered structure with the same parameters as in Fig. 5.

Fig. 8
Fig. 8

Equifrequency contours at (a) ω/ωp = 0.38 and (b) ω/ωp = 0.62 for the periodic metal-dielectric layered structure with ε1 = 3, ε 2 = 1 ω p 2 / ω 2, f1 = 0.25 (f2 = 0.75), and a/λp = 0.2. Gray circles are equifrequency contours in vacuum. Gray and black arrows are wave vectors in vacuum and the layered structure, respectively, at θ = 40°.

Fig. 9
Fig. 9

Electric field and surface charge patterns for the eigenmodes associated with the parabolic-like dispersion depicted in Fig. 8 for (a) ω/ωp = 0.38 (antibonding mode) and (b) ω/ωp = 0.62 (bonding mode). Red and blue colors correspond to positive and negative charges, respectively.

Fig. 10
Fig. 10

Effective permittivity ε x eff at (a) ω/ωp = 0.38 and (b) ω/ωp = 0.62 for the periodic metal-dielectric layered structure with the same parameters as in Fig. 8.

Fig. 11
Fig. 11

Out-of-plane effective permittivity ε y eff for the periodic metal-dielectric layered structure with ε1 = 3, ε 2 = 1 ω p 2 / ω 2, a/λp = 0.2, and f1 = 0.5 (f2 = 0.5) for (a) ω/ωp = 0.4 and (b) ω/ωp = 0.6. Red curves correspond to ε y eff = 0.

Equations (24)

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

cos ( k x a ) = cos ( q 1 a 1 ) cos ( q 2 a 2 ) 1 2 ( ε 2 q 1 ε 1 q 2 + ε 1 q 2 ε 2 q 1 ) sin ( q 1 a 1 ) sin ( q 2 a 2 ) ,
cos ( k x a ) = cos ( q 1 a 1 ) cos ( q 2 a 2 ) 1 2 ( q 1 q 2 + q 2 q 1 ) sin ( q 1 a 1 ) sin ( q 2 a 2 )
ε _ eff = [ ε x eff 0 0 0 ε y eff 0 0 0 ε z eff ] ,
k x 2 ε z eff + k z 2 ε x eff = k 0 2 ,
k x 2 + k z 2 = ε y eff k 0 2
ε z eff = ε z 0 α 12 k 0 2 a 2 1 1 12 k x 2 a 2 ,
ε x eff = ε x 0 α 12 k 0 2 a 2 1 + ε x 0 ε z 0 ( β 12 k z 2 a 2 γ 6 k 0 2 a 2 ) ,
ε y eff = ε y 0 ( 1 + 1 6 k z 2 a 2 ) + a 2 12 k 0 2 ( k x 4 k z 4 ) α 12 k 0 2 a 2 ,
ε y 0 = ε z 0 = f 1 ε 1 + f 2 ε 2 ,
ε x 0 = ε 1 ε 2 f 2 ε 1 + f 1 ε 2 ,
α = [ f 1 2 ε 1 + ( 1 f 1 2 ) ε 2 ] [ ( 1 f 2 2 ) ε 1 + f 2 2 ε 2 ] ,
β = 1 ε 1 ε 2 [ ( 1 2 f 1 f 2 ) ε 1 + 2 f 1 f 2 ε 2 ] [ 2 f 1 f 2 ε 1 + ( 1 2 f 1 f 2 ) ε 2 ] ,
γ = 1 ε 1 ε 2 [ f 1 3 f 2 ε 1 3 + f 1 ( 1 2 f 1 2 f 2 + f 2 3 ) ε 1 2 ε 2 + f 2 ( 1 2 f 1 f 2 2 + f 1 3 ) ε 1 ε 2 2 + f 1 f 2 3 ε 2 3 ] .
ε z 0 = ε y 0 = ( f 1 ε 1 + f 2 ) ( 1 ω 0 2 ω 2 ) ,
ε x 0 = ( ε f 2 ε 1 + f 1 ) ( ω 2 ω p 2 ω 2 ω 2 ) ,
ω 0 = ω p ε 1 ( f 1 / f 2 ) + 1 ,
ω = ω p ε 1 ( f 2 / f 1 ) + 1
k z 4 a 4 + 2 β ( 6 ε z 0 ε x 0 γ k 0 2 a 2 ) k z 2 a 2 + 1 β ( 12 k x 2 a 2 k x 4 a 4 12 ε z 0 k 0 2 a 2 + α k 0 4 a 4 ) = 0 .
S = 1 2 Re [ E × H * ] ω 4 ε i j k E i E j * .
d k z d k x = k x ε z eff ( 1 1 2 k x ε z eff ε z eff k x ) k z ε x eff ( 1 1 2 k z ε x eff ε x eff k z ) .
FOM i = 1 2 k j ε i eff ε i eff k j ,
x 2 ( 1 ε x 2 ) a 2 + y 2 b 2 = 1 ,
x 2 ( 1 ε x 2 ) a 2 + y 2 b 2 = 1 .
x 2 ( 1 + ε x 2 ) + y 2 ( 1 ε y 2 ) = r 2 ,

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