Abstract

We investigate electromagnetic wave propagation in multilayered metal-dielectric hyperbolic metamaterials (HMMs). We demonstrate that high-k propagating waves in HMMs are volume plasmon polaritons. The volume plasmon polariton band is formed by coupling of short-range surface plasmon polariton excitations in the individual metal layers.

© 2013 Optical Society of America

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  1. M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
    [Crossref]
  2. M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  4. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
    [Crossref]
  5. Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
    [Crossref]
  6. Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
    [Crossref]
  7. D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
    [Crossref]
  8. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
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  9. A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
    [Crossref]
  10. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
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  13. A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
    [Crossref] [PubMed]
  14. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
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  15. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009).
    [Crossref]
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    [Crossref]
  21. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
    [Crossref]
  22. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
    [Crossref] [PubMed]
  23. I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
    [Crossref]
  24. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
    [Crossref]
  25. X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
    [Crossref]
  26. J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
    [Crossref]
  27. I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
    [Crossref]
  28. S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
    [Crossref]
  29. E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
    [Crossref] [PubMed]
  30. H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
    [Crossref]
  31. R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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  35. T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
    [Crossref]

2013 (1)

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

2012 (6)

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
[Crossref]

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012).
[Crossref] [PubMed]

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
[Crossref] [PubMed]

2011 (7)

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011).
[Crossref] [PubMed]

2010 (4)

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

2009 (4)

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

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
[Crossref] [PubMed]

2007 (1)

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

2006 (4)

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

2005 (1)

2004 (1)

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

2003 (1)

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

2001 (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

1981 (1)

Acosta, M. F.

Alekseyev, L. V.

Avrutsky, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Barnakov, Y.

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6.

Barnakov, Yu. A.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Basharin, A. A.

Belov, P.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Belov, P. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Boltasseva, A.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Bonner, C. E.

Caglayan, H.

Cakmak, A. O.

Capolino, F.

Chebykin, A.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Colak, E.

Cortes, C. L.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Drachev, V. P.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Dryden, D.

Economou, E. N.

Elser, J.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Elson, J. M.

Eriksson, T. S.

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Fang, A.

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

Feng, S.

Foteinopoulou, S.

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

García de Abajo, J.

Genanakis, G.

Granqvist, C. G.

Gray, S.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Han, S.

Hjortsberg, A.

Iorsh, I.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Ishii, S.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref]

Jacob, Z.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Kafesaki, M.

Katsarakis, N.

Kidwai, O.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Kildishev, A. V.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref]

Kim, J.-Y.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Kivshar, Yu.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Kivshar, Yu. S.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Kolinko, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[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]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Lee, S.

Lee, T.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Li, H.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6.

Liu, H.

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Maria, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Maslovski, S.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Mavidis, Ch.

Mayy, M.

Merino, R. I.

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Mock, J. J.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Naik, G.V.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Narimanov, E.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6.

Narimanov, E. E.

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

Narimanov, E.E.

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Nataraj, G.

Newman, W.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Ni, X.

Niklasson, G. A.

Noginov, M. A.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6.

Nuzzo, R.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Orenstein, M.

Orera, V. M.

Orlov, A.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Orlov, A. A.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

Overfelt, P. L.

Ozbay, E.

Pendry, J.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Poddubny, A.

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Poddubny, A. N.

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Podolskiy, V.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Ramakrishna, S.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Reyes-Coronado, A.

Rogers, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Rosenblatt, G.

Ruppin, R.

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

Rye, P.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Salakhutdinov, I.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Schilling, J.

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Schurig, D.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Shalaev, V. M.

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref]

Shalaev, V.M.

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Sipe, J. E.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Smith, D. R.

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

Smolyaninov, I. I.

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

Soukoulis, C. M.

Stewart, M.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Stewart, W.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Thoreson, M. D.

Truong, T.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Tumkur, T.

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Villa, A. D.

Voroshilov, P. M.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

Vozianova, A.

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

Webb, K. J.

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

Wiltshire, M.

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

Yao, J.

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

Zhu, G.

M. A. Noginov, H. Li, Yu. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Zhukovsky, S. V.

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
[Crossref]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. 36(13), 2530–2532 (2011).
[Crossref] [PubMed]

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Jacob, J.-Y. Kim, G.V. Naik, A. Boltasseva, E.E. Narimanov, and V.M. Shalaev, “Engineering the photonic density of states with metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
[Crossref]

Appl. Phys. Lett. (3)

Z. Jacob, I. I. Smolyaninov, and E.E. Narimanov, “Broadband Purcell effect: Radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
[Crossref]

D. R. Smith, D. Schurig, J. J. Mock, P. Kolinko, and P. Rye, “Partial focusing of radiation by a slab of indefinite media,” Appl. Phys. Lett. 84(13), 2244–2246 (2004).
[Crossref]

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

J. Chem. Phys. (1)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

J. Mod. Opt. (1)

S. Ramakrishna, J. Pendry, M. Wiltshire, and W. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419 (2003).

J. Opt. (1)

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

J. Phys.: Condens. Matter (1)

R. Ruppin, “Surface polaritons of a left-handed material slab,” J. Phys.: Condens. Matter 13(9), 1811 (2001).
[Crossref]

Laser Photon. Rev. (1)

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photon. Rev. 7(2), 265–271 (2013).
[Crossref]

Nanotechnol. (1)

T. Truong, J. Maria, J. Yao, M. Stewart, T. Lee, S. Gray, R. Nuzzo, and J. Rogers, “Nanopost plasmonic crystals,” Nanotechnol. 20(43), 434011 (2009).
[Crossref]

Opt. Express (7)

S. Feng, J. M. Elson, and P. L. Overfelt, “Optical properties of multilayer metal-dielectric nanofilms with all-evanescent modes,” Opt. Express 13(11), 4113–4124 (2005).
[Crossref] [PubMed]

G. Rosenblatt and M. Orenstein, “Competing coupled gaps and slabs for plasmonic metamaterial analysis,” Opt. Express 19(21), 20372–20385 (2011).
[Crossref] [PubMed]

E. Colak, H. Caglayan, A. O. Cakmak, A. D. Villa, F. Capolino, and E. Ozbay, “Frequency dependent steering with backward leaky waves via photonic crystal interface layer,” Opt. Express 17(12), 9879–9890 (2009).
[Crossref] [PubMed]

X. Ni, S. Ishii, M. D. Thoreson, V. M. Shalaev, S. Han, S. Lee, and A. V. Kildishev, “Loss-compensated and active hyperbolic metamaterials,” Opt. Express 19(25), 25242–25254 (2011).
[Crossref]

A. A. Basharin, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Backward wave radiation from negative permittivity waveguides and its use for THz subwavelength imaging,” Opt. Express 20(12), 12752–12760 (2012).
[Crossref] [PubMed]

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Genanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20(13), 14663–14682 (2012).
[Crossref] [PubMed]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Lett. A (1)

I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Yu. Kivshar, “Spontaneous emission enhancement in metal-dielectric metamaterials,” Phys. Lett. A 376(3), 185–187 (2012).
[Crossref]

Phys. Rev. A (2)

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic meta-materials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
[Crossref]

A. N. Poddubny, P. A. Belov, and Yu. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Phys. Rev. B (7)

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84(3), 035128 (2011).
[Crossref]

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74(11), 115116 (2006).
[Crossref]

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

A. Chebykin, A. Orlov, A. Vozianova, S. Maslovski, Yu. Kivshar, and P. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84(11), 115438 (2011).
[Crossref]

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84(4), 045424 (2011).
[Crossref]

H. Liu and K. J. Webb, “Leaky wave radiation from planar anisotropic metamaterial slabs,” Phys. Rev. B 81(20), 201404 (2010).
[Crossref]

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolskiy, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75(24), 241402(R) (2007).
[Crossref]

Phys. Rev. E (1)

J. Schilling, “Uniaxial metallo-dielectric metamaterials with scalar positive permeability,” Phys. Rev. E 74(4), 046618 (2006).
[Crossref]

Phys. Rev. Lett. (1)

I. I. Smolyaninov and E. E. Narimanov, “Metric signature transitions in optical metamaterials,” Phys. Rev. Lett. 105(6), 067402 (2010).
[Crossref] [PubMed]

Other (3)

E. Narimanov, M. A. Noginov, H. Li, and Y. Barnakov, “Darker than Black: Radiation-absorbing Metamaterial,” in Quantum Electronics and Laser Science Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper QPDA6.

E. E. Narimanov and I. I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444 (2011).

A. Yariv and P. Yeh, Optical Waves in Crystals (New York: Wiley, 1983).

Cited By

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Figures (2)
Fig. 1
Fig. 1 (a) Isofrequency surfaces in the dispersion relation ( k x 2 + k y 2 ) / ε z + k z 2 / ε x , y = ω 2 / c 2 for conventional anisotropic medium (εx,y,z > 0) and indefinite medium (εx,y < 0 and εz > 0). (b) An infinite multilayer HMM with wave vector decomposition used in the paper (which is schematic in that w and k can be complex). (c) Comparison between the actual multilayer dispersion relation [Eq. (4)] for dm + dd =13.7 and 27.4 nm with the effective-medium dispersion relation for ρ = 0.17. The materials have εm = −17.2 (gold [18]) and εd = 2.59 (alumina [19]) for λ = 715 nm [5]. (d) Isofrequency surfaces for the actual dispersion relation (bottom) vs. effective medium approximation (top).

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Fig. 2
Fig. 2 Comparison between Eq. (4) (solid) and the coupled-SRSPP approximation given by Eq. (7) (dashed) for dm + dd =13.7 nm and ρ = 0.17, 0.25, and 0.5 for (a) the pole expansion as in Eq. (6) and (b) the pole expansion without the added constant terms (the vertical marks showing the location of κspc/ω for the respective filling fractions); (c) same as (a) for ρ = 0.12 (the effective medium approximation is indistinguishable from the inner branch of the exact relation). The materials are the same as in Fig. 1(c).

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Equations (7)

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

M 1 = 1 T m [ T m 2 − R m 2 R m − R m 1 ] [ e i w d d d 0 0 e − i w d d d ] .
R m = r d m + t d m r m d t m d e 2 i w m d m [ 1 − r m d 2 e 2 i w m d m ] − 1 ,                       T m = t d m t m d e i w m d m [ 1 − r m d 2 e 2 i w m d m ] − 1 .
r m d = w m ε d − w d ε m w m ε d + w d ε m ,     r d m = w d ε m − w m ε d w d ε m + w m ε d ,                 t m d = 2 w m ε m ε d w m ε d + w d ε m ,             t d m = 2 w d ε d ε m w d ε m + w m ε d .
Tr M 1 2 = cos [ k B ( d m + d d ) ] = cos ( w m d m ) cos ( w d d d ) − [ ( η + η − 1 ) / 2 ] sin ( w m d m ) sin ( w d d d ) ,
1 − r m d 2 e 2 i w m d m = 0
R m = − τ m / ( κ − κ s p ) − ( τ m / κ s p ) ,               T m 2 − R m 2 = − 2 τ m 2 / [ κ s p ( κ − κ s p ) ] − ( τ m / κ s p ) 2 .
cos [ k B ( d m + d d ) ] ≈ [ 1 − ( κ − κ s p ) / ( 2 τ m ) ] e i w d d d + [ ( κ − κ s p ) / ( 2 τ m ) ] e − i w d d d .

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