Abstract

Hyperbolic metamaterials (HMMs) show great promise in photonics applications because their unconventional open isofrequency surface permits enlargement of wavenumbers without limitation. Although optical behaviors in HMMs can be macroscopically described by theoretical calculations with the effective medium approximation (EMA), neglect of microscopic phenomena in each layer leads to discrepancies from exact numerical results. We clarify the origin of bulk propagating waves in HMMs and we show that they can be classified into two modes: long- and short-range surface-plasmon-based coupled modes (LRSP and SRSP, respectively). Especially, we find that the ratio of the number of LRSP and SRSP couplings dominates the property of each propagation mode. This plasmonic interpretation bridges the gap between the EMA and numerical exact solutions, thereby facilitating studies on HMM applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (1)

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
[Crossref]

2015 (3)

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photonics Rev. 9(3), 345–353 (2015).
[Crossref] [PubMed]

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

V. E. Babicheva, M. Y. Shalaginov, S. Ishii, A. Boltasseva, and A. V. Kildishev, “Long-range plasmonic waveguides with hyperbolic cladding,” Opt. Express 23(24), 31109–31119 (2015).
[Crossref] [PubMed]

2014 (3)

S. V. Zhukovsky, A. A. Orlov, V. E. Babicheva, A. V. Lavrinenko, and J. E. Sipe, “Photonic-band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials,” Phys. Rev. A 90(1), 013801 (2014).
[Crossref]

S. Ishii, M. Y. Shalaginov, V. E. Babicheva, A. Boltasseva, and A. V. Kildishev, “Plasmonic waveguides cladded by hyperbolic metamaterials,” Opt. Lett. 39(16), 4663–4666 (2014).
[Crossref] [PubMed]

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

2013 (4)

M. Noginov, M. Lapine, V. Podolskiy, and Y. Kivshar, “Focus issue: hyperbolic metamaterials,” Opt. Express 21(12), 14895–14897 (2013).
[Crossref] [PubMed]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[Crossref]

M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
[Crossref]

S. V. Zhukovsky, O. Kidwai, and J. E. Sipe, “Physical nature of volume plasmon polaritons in hyperbolic metamaterials,” Opt. Express 21(12), 14982–14987 (2013).
[Crossref] [PubMed]

2012 (4)

Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012).
[Crossref] [PubMed]

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. B 86(11), 115420 (2012).
[Crossref]

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

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 452502 (2012).
[Crossref]

2011 (2)

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

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. B 84(11), 115438 (2011).
[Crossref]

2010 (1)

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

2008 (1)

J. L. Zhang, H. T. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures,” Appl. Phys. Lett. 92(5), 053104 (2008).
[Crossref]

2007 (2)

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

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (4)

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

S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanostructures,” Phys. Rev. B 72(8), 085117 (2005).
[Crossref]

J. Takahara, F. Kusunoki, and T. Kobayashi, “Guiding of two-dimensional optical waves in a negative dielectric gap having a dielectric core,” Proc. SPIE 5928, 1–10 (2005).
[Crossref]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).

2004 (1)

B. Wang and G. P. Wang, “Metal heterowaveguides for nanometric focusing of light,” Appl. Phys. Lett. 85(16), 3599–3601 (2004).
[Crossref]

2003 (3)

P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. 37(4), 259–263 (2003).
[Crossref]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[Crossref] [PubMed]

K. Tanaka and M. Tanaka, “Simulations of nanometric optical circuits based on surface plasmon polariton gap waveguide,” Appl. Phys. Lett. 82(8), 1158–1160 (2003).
[Crossref]

2001 (1)

I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
[Crossref]

2000 (2)

P. Berini, “Plasmon-polariton modes guided by a metal film of finite width bounded by different dielectrics,” Opt. Express 7(10), 329–335 (2000).
[Crossref] [PubMed]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B 61(15), 10484–10503 (2000).
[Crossref]

1999 (1)

1985 (1)

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and V. E. Kravtsov, “Notes on crystal optics of superlattices,” Solid State Commun. 55(1), 85–90 (1985).
[Crossref]

Alekseyev, L. V.

Aryaee Panah, M. E.

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
[Crossref]

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 (2007).
[Crossref]

Babicheva, V. E.

Belov, P.

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
[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. B 86(11), 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. B 84(11), 115438 (2011).
[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(4), 259–263 (2003).
[Crossref]

Berini, P.

Bodganov, A.

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
[Crossref]

Boltasseva, A.

Caldwell, J. D.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[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. B 86(11), 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. B 84(11), 115438 (2011).
[Crossref]

Chen, Y.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Cortes, C. L.

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 452502 (2012).
[Crossref]

Dickson, W.

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photonics Rev. 9(3), 345–353 (2015).
[Crossref] [PubMed]

Dmitriev, P.

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
[Crossref]

Ellis, C. T.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

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 (2007).
[Crossref]

Elson, J.

Elson, J. M.

S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanostructures,” Phys. Rev. B 72(8), 085117 (2005).
[Crossref]

Enoch, S.

J. L. Zhang, H. T. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures,” Appl. Phys. Lett. 92(5), 053104 (2008).
[Crossref]

Feng, S.

S. Feng, J. M. Elson, and P. L. Overfelt, “Transparent photonic band in metallodielectric nanostructures,” Phys. Rev. B 72(8), 085117 (2005).
[Crossref]

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

Ferrari, L.

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

Fogler, M. M.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

Francescato, Y.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

Giannini, V.

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J. L. Zhang, H. T. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures,” Appl. Phys. Lett. 92(5), 053104 (2008).
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O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
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Kim, J. Y.

Z. Jacob, J. Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
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L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
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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. B 84(11), 115438 (2011).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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Z. Jacob, J. Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
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Narimanov, E. E.

Z. Jacob, J. Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B 100(1), 215–218 (2010).
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Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterial substrates,” Adv. Optoelectron. 2012, 452502 (2012).
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I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
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Novoselov, K. S.

J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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Orlov, A. A.

S. V. Zhukovsky, A. A. Orlov, V. E. Babicheva, A. V. Lavrinenko, and J. E. Sipe, “Photonic-band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials,” Phys. Rev. A 90(1), 013801 (2014).
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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. B 86(11), 115420 (2012).
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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. B 84(11), 115438 (2011).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–957 (2013).
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M. Noginov, M. Lapine, V. Podolskiy, and Y. Kivshar, “Focus issue: hyperbolic metamaterials,” Opt. Express 21(12), 14895–14897 (2013).
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V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).

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O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
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Salakhutdinov, I.

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

O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
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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. B 86(11), 115420 (2012).
[Crossref]

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S. V. Zhukovsky, A. A. Orlov, V. E. Babicheva, A. V. Lavrinenko, and J. E. Sipe, “Photonic-band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials,” Phys. Rev. A 90(1), 013801 (2014).
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S. V. Zhukovsky, O. Kidwai, and J. E. Sipe, “Physical nature of volume plasmon polaritons in hyperbolic metamaterials,” Opt. Express 21(12), 14982–14987 (2013).
[Crossref] [PubMed]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
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D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

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J. Takahara, F. Kusunoki, and T. Kobayashi, “Guiding of two-dimensional optical waves in a negative dielectric gap having a dielectric core,” Proc. SPIE 5928, 1–10 (2005).
[Crossref]

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O. Takayama, E. Shkondin, A. Bodganov, M. E. Aryaee Panah, K. Golenitskii, P. Dmitriev, T. Repan, R. Malureanu, P. Belov, F. Jensen, and A. V. Lavrinenko, “Midinfrared surface waves on a high aspect ratio nanotrench platform,” ACS Photonics 4(11), 2899–2907 (2017).
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M. S. Tame, K. R. McEnery, Ş. K. Özdemir, J. Lee, S. A. Maier, and M. S. Kim, “Quantum plasmonics,” Nat. Phys. 9(6), 329–340 (2013).
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J. L. Zhang, H. T. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures,” Appl. Phys. Lett. 92(5), 053104 (2008).
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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media—Media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. 31(2), 129–133 (2001).
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Vasilantonakis, N.

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photonics Rev. 9(3), 345–353 (2015).
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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. B 84(11), 115438 (2011).
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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
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L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photonics Rev. 9(3), 345–353 (2015).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
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Zayats, A. V.

N. Vasilantonakis, M. E. Nasir, W. Dickson, G. A. Wurtz, and A. V. Zayats, “Bulk plasmon-polaritons in hyperbolic nanorod metamaterial waveguides,” Laser Photonics Rev. 9(3), 345–353 (2015).
[Crossref] [PubMed]

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J. L. Zhang, H. T. Jiang, S. Enoch, G. Tayeb, B. Gralak, and M. Lequime, “Two-dimensional complete band gaps in one-dimensional metal-dielectric periodic structures,” Appl. Phys. Lett. 92(5), 053104 (2008).
[Crossref]

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L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

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S. V. Zhukovsky, A. A. Orlov, V. E. Babicheva, A. V. Lavrinenko, and J. E. Sipe, “Photonic-band-gap engineering for volume plasmon polaritons in multiscale multilayer hyperbolic metamaterials,” Phys. Rev. A 90(1), 013801 (2014).
[Crossref]

S. V. Zhukovsky, O. Kidwai, and J. E. Sipe, “Physical nature of volume plasmon polaritons in hyperbolic metamaterials,” Opt. Express 21(12), 14982–14987 (2013).
[Crossref] [PubMed]

O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Effective-medium approach to planar multilayer hyperbolic metamaterials: Strengths and limitations,” Phys. Rev. A 85(5), 053842 (2012).
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ACS Photonics (1)

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Figures (9)

Fig. 1
Fig. 1 Schematic view of the structure studied and its effective permittivity. (a) Schematic representation of a planar optical waveguide with a HMM core. The direction of propagation of light is shown by the red arrow. (b) A cross-section of (a). (c) Derived real parts of the permittivity tensor components of the multilayered structure as a function of the metal filling ratio f. The structure consisted of Au and SiO2, and had an operating wavelength λ0 = 1550 nm. Inset: Schematic isofrequency surfaces of this multilayer structure in wavenumber space.
Fig. 2
Fig. 2 Propagation modes calculated by the EMA at λ0 = 1550 nm. (a) neff of the zero- and first-order modes as a function of the normalized HMM thickness, calculated by using the EMA. Inset: the cross-section of an HMM-core optical waveguide. (b) The electric Ex and magnetic Hy field distributions of the zero- and first-order modes at h = 165 nm and f = 0.5.
Fig. 3
Fig. 3 neff vs. HMM core thickness calculated by the FEM for f = 0.5 at λ0 = 1550 nm. The core thickness was varied by changing Nm with a constant period of 30 nm.
Fig. 4
Fig. 4 The electric and magnetic field distributions of propagation modes in a planar HMM, calculated by the FEM at λ0 = 1550 nm. (a) Ex and (b) Hy field distributions of the lower-index modes. (c) Ex and (d) Hy field distributions of the higher-index modes. These are the waveguide cross-section. The total thickness of the core h = 165 nm, which is the subwavelength size (h0 ≈0.1) at a period of 10 nm at f = 0.5.
Fig. 5
Fig. 5 Schematic illustration of the magnetic field profiles Hy for various plasmonic coupled modes. (a) LRSP, SRSP, and GSP. (b) LG mode and SG mode. The white area is dielectric and the gray area is metal.
Fig. 6
Fig. 6 The transition and generation process of propagation modes in a planar HMM. (a) Schematic illustration of this calculation system of decreasing the SiO2 layer thickness d and increasing Nm. (b) The transition of neff of propagation modes at λ0 = 1550 nm. The thickness of the Au layers is 15 nm and d is varied from 2 μm to 15 nm. neff of LRSP-like and SRSP-like modes gradually approach the same value, whereas the neff of other modes (r, s) increase with increasing Nm.
Fig. 7
Fig. 7 Hy field profile of LRSP-like, SRSP-like, and SG-unit-cell-like mode (0, 0) at Nm = 2 and d = 150 nm
Fig. 8
Fig. 8 neff vs. period at λ0 = 1550 nm calculated by the FEM. (a) neff of LG0 and LG1 with respect to the period at f = 0.5. (b) neff of SG0 and SG1 with respect to the period with fixed Nm = 11 and f = 0.5. neff of GSP (orange dashed–dotted line), SRSP (green dashed line), and LRSP (purple dotted line) are also plotted as functions of the SiO2 gap and the Au film thickness, which are half the size of a period.
Fig. 9
Fig. 9 Calculated results for Bloch waves in a bulk HMMs. (a) Schematic illustration of a bulk HMM. (b) neff of LG0 and SG0 modes at Nm = 101 and Bloch waves in bulk HMMs with respect to the period at λ0 = 1550 nm. The solid lines are analytically calculated Bloch waves kB = 0 (green line) and kB = π/(dd + dm) (orange line) in bulk HMMs. Diamonds and circles are the neff of the LG0 and SG0 modes calculated by the FEM, respectively.

Equations (5)

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ε =f ε m +(1f) ε d , ε // = ε m ε d f ε d +(1f) ε m ,
2 H y x 2 = γ H 2 H y ( 0xh ), 2 H y x 2 = γ c 2 H y ( x0,hx ), γ H = ( β 2 ε // ω 2 c 2 ) ε , γ c = β 2 ω 2 c 2 ε c ,
[ sin( γ H h 2 )+ γ H ε c γ c ε cos( γ H h 2 ) ][ sin( γ H h 2 ) γ c ε γ H ε c cos( γ H h 2 ) ]=0.
cosh( γ m d m )cosh( γ d d d )+ 1 2 ( γ d ε m γ m ε d + γ m ε d γ d ε m )sinh( γ m d m )sinh( γ d d d ) =cos[ k B ( d d + d m )], γ d = β 2 ω 2 c 2 ε d , γ m = β 2 ω 2 c 2 ε m ,
γ d ε d tanh( γ d d d 2 )+ γ m ε m tanh( γ m d m 2 )=0( k B =0 ), γ d ε d tanh( γ d d d 2 )+ γ m ε m coth( γ m d m 2 )=0( k B = π d d + d m ).

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