Abstract

We show how to achieve a giant permittivity combined with negligible losses in both the visible and the near-IR for composites made of alternating layers of plasmonic and gain materials as the electric field is directed normally to the layers. The effects of nonlocality are taken into account that makes the method quite realistic. Solving the dispersion equation for eigenmodes of an infinite layered composite, we show that both propagating and nonpropagating modes can be excited, that leads to the realization of a giant nonlocal permittivity. Both phase and group velocities for the propagating eigenmode have been calculated showing that slow light can be achieved in the system under study. The results obtained open new possibilities for designing nanolaser, slow-light, superresolution imaging devices, etc.

© 2015 Optical Society of America

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    [Crossref]
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    [Crossref]
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2014 (3)

2013 (5)

2012 (3)

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
[Crossref]

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

2011 (5)

T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
[Crossref]

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (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, 25242–25254 (2011).
[Crossref]

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

2010 (1)

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

2009 (5)

J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
[Crossref]

R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
[Crossref]

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
[Crossref]

2008 (2)

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

2007 (5)

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

R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
[Crossref]

N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
[Crossref]

A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
[Crossref]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

2006 (2)

M.A. Noginov, G. Zhu, M. Bahoura, J. Adegoke, C.E. Small, B.A. Ritzo, V.P. Drachev, and V.M. Shalaev, “Enhancement of surface plasmons in an Ag aggregate by optical gain in a dielectric medium,” Opt. Lett. 31, 3022–3024 (2006).
[Crossref] [PubMed]

P.A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[Crossref]

2004 (1)

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

2003 (2)

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

2002 (1)

A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
[Crossref]

1991 (1)

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
[Crossref]

1981 (1)

G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
[Crossref]

1980 (1)

D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

1979 (1)

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

1976 (1)

H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
[Crossref]

1956 (1)

S.M. Rytov, “Electromagnetic properties of laminated medium,” Sov. Phys. JETP 2, 466–475 (1956).

1952 (1)

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
[Crossref]

Adegoke, J.

Agranovich, V.M.

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
[Crossref]

Akozbek, N.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Alu, A.

A. Alu, “First-ptinciples homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

Anantha Ramakrishna, S.

J.B. Pendry and S. Anantha Ramakrishna, “Refining the perfect lens,” Physica B 338, 329–332 (2003).
[Crossref]

S. Anantha Ramakrishna and J.B. Pendry, “Removal of absorption and increase in resolution in a near-field lens via optical gain,” Phys. Rev. B 67, 201101 (2003).
[Crossref]

Anderson, Z.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Avrutsky, I.

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

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nature Photon. 2, 465–473 (2008).
[Crossref]

Bahoura, M.

Belov, P.A.

R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
[Crossref]

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

P.A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[Crossref]

P.A. Belov, “Subwavelength imaging by extremely anisotropic media,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S.A. Maier, eds. (Wiley, 2013).
[Crossref]

Bergman, D.J.

D.J. Bergman, “Exactly solvable microscopic geometries and rigorous bounds for the complex dielectric constant of a two-component composite material,” Phys. Rev. Lett. 44, 1285–1287 (1980).
[Crossref]

Berini, P.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

P. Berini, “Loss compensation and amplification of surface plasmon polaritons,” in Active Plasmonics and Tuneable Plasmonic Metamaterials, A.V. Zayats and S. A. Maier, eds. (Wiley, 2013).
[Crossref]

Bloemer, M.J.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Bondarenko, O.

Boyd, R.W.

R.W. Boyd, “Slow and fast light: fundamentals and applications,” J. Mod. Opt. 56, 1908–1915 (2009).
[Crossref]

Briggs, D.P.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Brijs, B.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Brongersma, S.H.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Cappeddu, M.G.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Centini, M.

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

Chebykin, A.V.

A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Chern, R.L.

Choi, M.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
[Crossref] [PubMed]

Christy, R.W.

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1979).
[Crossref]

Ciattori, A.

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
[Crossref]

Concharenko, A.V.

D’Orazio, A.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

de Ceglia, D.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
[Crossref]

D. de Ceglia, M.A. Vincenti, M.G. Cappeddu, M. Centini, N. Akozbek, A. D’Orazio, J.W. Haus, M.J. Bloemer, and M. Scalora, “Tailoring metallodielectric structures for superresolution and superguiding applications in the visible and near-ir ranges,” Phys. Rev. A 77, 033848 (2008).
[Crossref]

De Leon, I.

P. Berini and I. De Leon, “Surface plasmon-polariton amplifiers and lasers,” Natute Photon. 6, 16–24 (2012).
[Crossref]

Dewalt, C.J.

Di Falco, A.

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B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
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R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
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W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
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T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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Kidwai, O.

Kildishev, A.V.

Kim, Y.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
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A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
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P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

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M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
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N.M. Lawandy, “Subwavelength lasers,” Appl. Phys. Lett. 90, 143104 (2007).
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Lee, S.

Lee, S.H.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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Lee, Y.H.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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Li, H.

H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
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A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
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Loidl, A.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

Lomakin, V.

Lunkenheimer, P.

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

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T. Mackay and A. Lakhtakia, “Negative refraction, negative phase velocity, and counterposition in bianisotropic materials ans metamaterials,” Phys. Rev. B 79, 235121 (2009).
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T.G. Mackay and A. Lakhtakia, “Towards a realization of Schwarzschild-(anti-)de Sitter spacetime as a particulate metamaterial,” Phys. Rev. B 83, 195424 (2011).
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W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
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A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
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G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
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Min, B.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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Mizrahi, A.

Moitra, P.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Molesky, S.

Mortensen, N.A.

N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013).
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Ni, X.

Noginov, M.A.

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A.V. Chebykin, A.A. Orlov, C.R. Simovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
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Page, A.F.

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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Palmans, R.

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
[Crossref]

Park, N.

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
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Podivilov, E.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
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Podolsky, V.

I. Avrutsky, I. Salakhutdinov, J. Elser, and V. Podolsky, “Highly confined optical modes in nanoscale metal-dielectric multilayers,” Phys. Rev. B 75, 241402 (2007).
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Pukhov, A.A.

A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
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P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

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F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
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W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
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P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

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Rizza, C.

C. Rizza, A. Di Falco, and A. Ciattori, “Gain assisted nanocomposite multilayers with near zero permittivity at visible frequencies,” Appl. Phys. Lett. 99, 221107 (2011).
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M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
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R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
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R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
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M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
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J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
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M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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J. Shin, J.T. Shen, and S. Fan, “Three-dimensional metamaterials with an ultrahigh effective refractive index over a broad band width,” Phys. Rev. Lett. 102, 093903 (2009).
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Simovski, C.R.

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Slutsky, B.

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F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
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B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
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R.S. Savelev, I.V. Shadrivov, P.A. Belov, N.N. Rosanov, S.V. Fedorov, A.A. Sukhorukov, and Y.S. Kivshar, “Loss compensation in metal-dielectric layered metamaterials,” Phys. Rev. B 87, 115139 (2013).
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Valentine, J.

P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

Venger, E.F.

Vincenti, M.A.

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
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A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
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T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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P. Moitra, Y. Yang, Z. Anderson, I.I. Kravchenko, D.P. Briggs, and J. Valentine, “Realization of an all-dielectric zero-index optical metamaterial,” Nature Photon.7, 791–795 (20013).

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W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
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A.V. Dorofeenko, A.A. Zyablovsky, A.A. Pukhov, A.A. Lisyansky, and A.P. Vinogradov, “Light propagation in composite materials with gain layers,” Physics Uspekhi 55, 1080–1097 (2012).
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Adv. Opt. Photon. (1)

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A.V. Goncharenko, “Comment on ”Subwavelength lasers” [Appl. Phys. Lett. 90, 143104 (2007)],” Appl. Phys. Lett. 91, 246101 (2007).
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Eur. Phys. J. Special Topics (1)

P. Lunkenheimer, S. Krohns, S. Riegg, S.G. Ebbinghaus, A. Reller, and A. Loidl, “Colossal dielectric constants in transition-metal oxides,” Eur. Phys. J. Special Topics 180, 161–189 (2010).

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G.W. Milton, “Bounds on the complex permittivity of a two-component composite material,” J. Appl. Phys. 52, 5286–5293 (1981).
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A.P. Vinogradov and A.V. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. 94, 482–488 (2002).
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R. Pierre and B. Gralak, “Appropriate truncation for photonic crystals,” J. Mod. Opt. 55, 1759–1770 (2007).
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H. Li, “Index of alkali halides and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 5, 329–528 (1976).
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Macromolecules (1)

W. Groh and A. Zimmermann, “What is the lowest refractive index of an organic polymer?” Macromolecules 24, 6660–6663 (1991).
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Metamaterials (1)

V.M. Agranovich and Yu.N. Gartstein, “Electrodynamics of metamaterials and the Landau-Lifshitz approach to the magnetic permeability,” Metamaterials 9, 1–9 (2009).
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Microel. Eng. (1)

W. Zhang, S.H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K. Maex, “Influence of the electron mean free path on the resistivity of thin metal films,” Microel. Eng. 76, 146–152 (2004).
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Nature (1)

M. Choi, S.H. Lee, Y. Kim, S.B. Kang, J. Shin, M.H. Kwak, K.Y. Kang, Y.H. Lee, N. Park, and B. Min, “A terahertz metamaterial with unnaturally high refractive index,” Nature 470, 369–374 (2011).
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Nature Commun. (1)

T. Pickering, J.M. Hamm, A.F. Page, S. Wuestner, and O. Hess, “Cavity-free plasmonic nanolasing enabled by dispersionless stopped light,” Nature Commun. 5, 5972 (2014).
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Opt. Express (5)

Opt. Lett. (1)

Photon. Nanostr. - Fundam. Appl. (1)

N.A. Mortensen, “Nonlocal formalism for nanoplasmonics: phenomenological and semi-classical considerations,” Photon. Nanostr. - Fundam. Appl. 11, 303–309 (2013).
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Phys. Rev. (1)

F.W. Reynolds and G.R. Stilwell, “Mean free paths of electrons in evaporated metal films,” Phys. Rev. 88, 418–419 (1952).
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Phys. Rev. A (2)

M.A. Vincenti, D. de Ceglia, V. Rondinone, A. Ladisa, A. D’Orazio, M.J. Bloemer, and M. Scalora, “Loss compensation in metal-dielectric structures in negative-refraction and super-resolving regimes,” Phys. Rev. A 80, 053807 (2009).
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Phys. Rev. B (10)

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Other (5)

To avoid confusion, it should be noted that Wiener dealt with the conductivity. Because the conductivity is proportional to the imaginary part of the permittivity, his lower bound (for the real conductivity) in fact corresponds to the upper bound for the permittivity.

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Figures (7)

Fig. 1
Fig. 1 Sketh of the unit cell under consideration.
Fig. 2
Fig. 2 The lower Wiener bound (solid curves) computed for a two-component multilayered composite with ε1 = −6 + 0.2i and different values of ε2: ε 2 A = 2.2 0.03 i (blue curve), ε 2 B = 2.2 0.04 i (green curve), ε 2 C = 2.2 0.05 i (red curve), and ε 2 D = 2.2 0.15 i (orange curve). The arrows show the values of ε eff 0: ε eff 0 = 7.88 , 12.05 , 19.795, and −13.86 at ε″2 =−0.03, −0.04, −0.05, and −0.15, respectively. The dashed lines show the upper Wiener bound.
Fig. 3
Fig. 3 The nonlocal effective permittivity vs ε″2 calculated with the use of Eqs. (9) and (3) for two eigenmodes for d1 = 20 nm (blue dotted curves) and d1 = 40 nm (red dashed curves). The black curve shows the effective permittivity calculated with the use of Eq. (4) (local approximation).
Fig. 4
Fig. 4 The real and imaginary parts of k vs f for four eigenmodes at k0d1 = 0.38, ε1 = −20 + 0.45i, and ε2 = 1.82 − 0.04i. The arrow shows the position of zero of Imk.
Fig. 5
Fig. 5 The same as in Fig. 4 but at ε2 = 1.82 − 0.05i. The arrows show the positions of zero of Rek.
Fig. 6
Fig. 6 The effective permittivity vs ε″2 calculated according to Eq. (9) for the propagating eigenmode at different values of the thickness d1.
Fig. 7
Fig. 7 The phase and group velocities vs ε″2 for the propagating eigenmode.

Equations (13)

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ε lb = ε eff = [ f ε 1 + f 2 ε 2 ] 1 ,
ε ub = ε eff = f ε 1 + f 2 ε 2 .
f = f 0 = ε 2 | ε 1 | 2 ε 2 | ε 1 | 2 ε 1 | ε 2 | 2 .
ε eff 0 ε eff ( f 0 ) = ε 2 | ε 1 | 2 ε 1 | ε 2 | 2 ε 2 ε 1 ε 1 ε 2 .
ε 1 / ε 1 = ε 2 / ε 2 .
ε 1 | ε 2 | 2 ε 2 | ε 1 | 2 2 ε 2 ( ε 2 ε 1 ε 1 ε 2 ) = 0 .
ε 2 max = ε 1 ε 21 ( ε 1 ε 21 ) 2 + ε 2 2 | ε 1 | 2 ε 21
ε eff 0 max ε eff 0 ( ε 2 max ) = ε 2 2 ε 2 2 + 2 ε 1 ε 2 | ε 1 | 2 ε 2 ε 1 .
ε ˜ eff ( k ) = k 2 / k 0 2 ,
cos ( k 1 d 1 ) cos ( k 2 d 2 ) γ sin ( k 1 d 1 ) sin ( k 2 d 2 ) = 1
cot ( k 2 d f 2 ) ( f k 1 + γ f 2 k 2 ) + cot ( k 1 d f ) ( f 2 k 2 + γ f k 1 ) + γ d = 0 ,
ε 2 = ε + σ ω 0 2 ω 2 i Δ ω ω .
d ε 2 d k 0 = c d ε 2 d ω = σ c Δ ω ω 0 ( 2 Δ ω + i ω 0 ) = c ε 2 ( 2 Δ ω + i ω 0 ) = ε 2 k 0 ( 2 δ + i )

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