Abstract

On the basis of the formalism of the Boltzmann kinetic equation for the distribution function of the conduction electrons, the photonic band structure of binary dielectric-metal superlattice is theoretically studied. Using the constitutive nonlocal relation between the electrical current density and the electric field inside the metallic layer, the dispersion equation for photonic eigenmodes in the periodic stack is analytically expressed in terms of the surface impedances at the interfaces of the metal and dielectric layers. In the case of very thin metallic layers, the optic spectrum for the superlattice exhibits narrow pass bands as a result of the strong contrast between the impedances of the dielectric and the metal. The narrow pass bands are attributed to Fabry-Perot resonances in the relatively-thick dielectric layer. The metal nonlocality is well pronounced in the infrared and, therefore, the nonlocal effect upon the photonic band structure of the superlattice can be strong when the Fabry-Perot resonance bands are in that frequency range. Our results for the photonic spectrum have been compared with those obtained within the local Drude-Lorentz model. Noticeably differences not only in the the magnitude, but also in the sign of the real part of the Bloch wave number in the Fabry-Perot resonance bands, have been found.

© 2014 Optical Society of America

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References

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  1. A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).
  2. E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.
  3. A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
    [CrossRef]
  4. V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
    [CrossRef]
  5. D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
    [CrossRef]
  6. P. Markoš, C. M. Soukoulis, Wave Propagation] From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).
  7. J. Elser, V. A. Podolskiy, I. Salakhutdinov, I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
    [CrossRef]
  8. A. Orlov, I. Iorsh, P. Belov, Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21, 1593–1598 (2013).
    [CrossRef] [PubMed]

2013

2009

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

2007

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

2004

D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[CrossRef]

1997

V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Abrikosov, A. A.

A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).

Avrutsky, I.

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

Belov, P.

Díaz-Monge, F.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

Elser, J.

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

Iorsh, I.

Kaner, E. A.

E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.

Kivshar, Y.

Krokhin, A. A.

E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Makarov, N. M.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Markoš, P.

P. Markoš, C. M. Soukoulis, Wave Propagation] From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Orlov, A.

Paredes-Juárez, A.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

Pérez-Rodríguez, F.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

Podolskiy, V. A.

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

Ramos-Mendieta, F.

D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[CrossRef]

Salakhutdinov, I.

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

Soto-Puebla, D.

D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[CrossRef]

Soukoulis, C. M.

P. Markoš, C. M. Soukoulis, Wave Propagation] From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

Xiao, M.

D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[CrossRef]

Appl. Phys. Lett.

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

JETP Lett.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009).
[CrossRef]

Opt. Express

Phys. Lett. A

D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[CrossRef]

Phys. Rev. B

V. Kuzmiak, A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[CrossRef]

Other

A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).

E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.

P. Markoš, C. M. Soukoulis, Wave Propagation] From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).

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Figures (3)

Fig. 1
Fig. 1

A sketch of the stack structure.

Fig. 2
Fig. 2

Frequency dependence of the real (a) and imaginary (b and c) parts of the surface impedances ζ0 and ζd calculated within the nonlocal (Boltzmann) and Drude-Lorentz (DL) models.

Fig. 3
Fig. 3

Optic spectrum for a vacuum-aluminum superlattice, predicted by the nonlocal (Boltzmann) and local (Drude-Lorentz) formalisms. Here, γ = κd.

Equations (14)

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E a n ( x ) = A n + exp [ i k a ( x x a n ) ] + A n exp [ i k a ( x x a n ) ]
E b n ( x ) = i k μ b d b { H b n ( x a n + 1 ) s = cos [ k s ( x a n + 1 x ) ] k s 2 k 2 ε ( k s ) μ b H b n ( x b n ) s = cos [ k s ( x x b n ) ] k s 2 k 2 ε ( k s ) μ b }
ε ( k s ) = 1 ω p 2 ω ( ω + i ν ) 𝒦 ( k s l ω ) ,
𝒦 ( k s l ω ) = 3 2 0 1 ( 1 n x 2 ) d n x 1 + ( k s l ω n x ) 2 = 3 2 { ( 1 k s l ω + 1 ( k s l ω ) 3 ) arctan ( k s l ω ) 1 ( k s l ω ) 2 } .
𝒦 ( k s l ω ) 1 ( k s l ω ) 2 / 5 , ( k s | l ω | ) 2 1 ;
𝒦 ( k s l ω ) 3 π / 4 | k s | l ω , | k s l ω | 1 .
( A n + 1 + A n + 1 ) = ( Q 11 Q 12 Q 21 Q 22 ) ( A n + A n ) .
Q 11 = ( Z a ζ 0 ) 2 ζ d 2 2 Z a ζ d exp ( i φ a ) ,
Q 12 = ( Z a 2 ζ 0 2 + ζ d 2 ) 2 Z a ζ d exp ( i φ a ) ,
Q 21 = Z a 2 ζ 0 2 + ζ d 2 2 Z a ζ d exp ( i φ a ) ,
Q 22 = ( Z a + ζ 0 ) 2 ζ d 2 2 Z a ζ d exp ( i φ a ) .
ζ 0 = i k μ b d b s = 1 k s 2 k 2 ε ( k s ) μ b , ζ d = i k μ b d b s = cos ( k s d b ) k s 2 k 2 ε ( k s ) μ b .
ζ 0 ( D L ) = i Z b cos φ b sin φ b , ζ d ( D L ) = i Z b 1 sin φ b ,
cos ( κ d ) = ζ 0 ζ d cos φ a i Z a 2 + ζ 0 2 ζ d 2 2 Z a ζ d sin φ a .

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