Abstract

We study the optical properties of metamaterials formed by layers of metallic nanoparticles. The effective optical constants of these materials are retrieved from the calculated angle-dependent Fresnel reflection coefficients for s and p incident-light polarization. We investigate the degree of anisotropy in the effective permittivity as a function of inter-layer spacing, particle size, filling fraction of the metal, and particle shape. For layers of spherical particles periodically arranged in a hexagonal lattice, the anisotropy disappears for the three inter-layer spacings corresponding to simple cubic (sc), bcc, and fcc volume symmetry. For non-spherical particles, an isotropic response can be still obtained with other values of the inter-layer spacing. Finally, we provide a quantitative answer to the question of how many layers are needed to form an effectively homogeneous metamaterial slab. Surprisingly, only one layer can be enough, except in the spectral range close to the particle plasmon resonances.

© 2009 Optical Society of America

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  9. A. Taleb, V. Russier, A. Courty, and M. P. Pileni, "Collective optical properties of silver nanoparticles organized in two-dimensional superlattices," Phys. Rev. B 59, 13,350-13,358 (1999).
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    [CrossRef]
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    [CrossRef]

2008 (3)

F. J. Garcıa de Abajo, "Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides," J. Phys. Chem. C 112, 17,983-17,987 (2008).
[CrossRef]

A. Boltasseva and V. M. Shalaev, "Fabrication of optical negative-index metamaterials: Recent advances and outlook," Metamaterials 2, 1-17 (2008).
[CrossRef]

R. Sainidou and F. J. Garcıa de Abajo, "Plasmon guided modes in nanoparticle metamaterials," Opt. Express 16, 4499-4506 (2008).
[CrossRef] [PubMed]

2007 (2)

C. R. Simovski and S. A. Tretyakov, "Local constitutive parameters of metamaterials from an effective-medium perspective," Phys. Rev. B 75, 195,111 (2007).
[CrossRef]

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

2006 (3)

L. M. Liz-Marzan, "Tailoring surface plasmon through the morphology and assembly of metal nanoparticles," Langmuir 22, 32-41 (2006).
[CrossRef]

J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006).
[CrossRef] [PubMed]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

2005 (1)

S. Riikonen, I. Romero, and F. J. Garcıa de Abajo, "Plasmon tunability in metallodielectric metamaterials," Phys. Rev. B 71, 235,104 (2005).
[CrossRef]

2004 (4)

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and negative refractive index," Science 305, 788-792 (2004).
[CrossRef] [PubMed]

H. Fan, K. Yang, D. M. Boye, T. Sigmon, K. J. Malloy, H. Xu, G. P. Lopez, and C. J. Brinker, "Self-assymbly of ordered, robust, three-dimensional gold nanocrystal/silica arrays," Science 304, 567 (2004).
[CrossRef] [PubMed]

P. Xu and Z. Li, "Study of frequency band gaps in metal-dielectric composite materials," J. Phys. D 37, 1718-1724 (2004).
[CrossRef]

2002 (3)

F. J. Garcıa de Abajo and A. Howie, "Retarded field calculation of electron energy loss in inhomogeneous dielectrics," Phys. Rev. B 65, 115,418 (2002).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, "Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients," Phys. Rev. B 65, 195,104 (2002).
[CrossRef]

K. W. Whites and F. Wu, "Effects of particle shape on the effective permittivity of composite materials with measurements for lattices of cubes," IEEE Trans. Microw. Theory Tech. 50, 1723-1729 (2002).
[CrossRef]

2001 (6)

F. Wu and K. W. Whites, "Quasi-static effective permittivity of periodic composites containing complex shaped dielectric particles," IEEE Trans. Antennas Propag. 49, 1174-1182 (2001).
[CrossRef]

Z. Wang, C. T. Chan, W. Zhang, N. Ming, and P. Sheng, "Three-dimensional self -assembly of metal nanoparticles: Possible photonic crystal with a complete gap below the plasma frequency," Phys. Rev. B 64, 113,108 (2001).
[CrossRef]

A. A. Zakhidov, R. H. Baughman, I. I. Khayrullin, I. A. Udad, M. Kozlov, N. Eradat, V. Z. Vardeny, M. Sigalas, and R. Biswas, "Three-dimensionally periodic conductive nanostructures: network versus cermet topologies for metallic PGB," Synthetic Metals 116, 419-426 (2001).
[CrossRef]

M. Gadenne, V. Podolskiy, P. Gadenne, P. Sheng, and V. M. Shalaev, "Plasmon-enhanced absorption by optical phonons in metal-dielectric composites," Europhys. Lett. 53, 364-370 (2001).
[CrossRef]

F. Caruso, M. Spasova, V. S. no Maceira, and L. M. Liz-Marzan, "Multilayer assemblies of silica-encapsulated gold nanoparticles on decomposable colloid templates," Adv. Mater. 13, 1090-1094 (2001).
[CrossRef]

T. Ung, L. M. Liz-Marzan, and P. Mulvaney, "Optical properties of thin films of Au@SiO2 particles," J. Phys. Chem. B 105, 3441-3452 (2001).
[CrossRef]

2000 (3)

J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

S. B. Jones and S. P. Friedman, "Particle shape effects on the effective permittivity of anisotropic or isotropic media consisiting of aligned or randomly oriented ellipsoidal particles," Water Resour. Res. 36, 2821-2833 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, "MULTEM 2: A new version of the program for transmission and band-structure calculations of photonic crystals," Comput. Phys. Commun. 132, 189-196 (2000).
[CrossRef]

1999 (3)

A. Taleb, V. Russier, A. Courty, and M. P. Pileni, "Collective optical properties of silver nanoparticles organized in two-dimensional superlattices," Phys. Rev. B 59, 13,350-13,358 (1999).
[CrossRef]

D. Zanchet, M. S. Moreno, and D. Ugarte, "Anomalous packing in thin nanoparticle supercrystals," Phys. Rev. Lett. 82, 5277-5280 (1999).
[CrossRef]

H. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexopoulos, "Effective response functions for photonic bandgap materials," J. Opt. Soc. Am. A 16, 1682-1699 (1999).
[CrossRef]

1998 (1)

F. J. Garcıa de Abajo and A. Howie, "Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics," Phys. Rev. Lett. 80, 5180-5183 (1998).
[CrossRef]

1996 (1)

R. Fuchs, R. G. Barrera, and J. L. Carrillo, "Spectral representations of the electron energy loss in composite media," Phys. Rev. B 54, 12,824-12,834 (1996).
[CrossRef]

1995 (2)

J. S. Ahn, K. H. Kim, T. W. Noh, D. H. Riu, K. H. Boo, and H. E. Kim, "Effective-medium theories for spheroidal particles randomly oriented on a plane: Application to the optical properties of a SiC whisker-Al2O3 composite," Phys. Rev. B 52, 15,244-15,252 (1995).

R. G. Barrera and R. Fuchs, "Theory of electron energy loss in a random system of spheres," Phys. Rev. B 52, 3256-3273 (1995).
[CrossRef]

1991 (1)

M. F. MacMilland, R. P. Devaty, and J. V. Mantese, "Infrared properties of Pt/Al2O3 cermet films," Phys. Rev. B 43, 13,838-13,845 (1991).

1990 (1)

A. H. Sihvola and I. V. Lindell, "Chiral Maxwell-Garnett mixing formula," Electron. Lett. 26, 118-119 (1990).
[CrossRef]

1981 (1)

G. W. Milton, "Bounds and exact theories for the transport properties of inhomogeneous media," Appl. Phys. A-Mater. Sci. Process. 26, 207-220 (1981).
[CrossRef]

1980 (1)

P. Sheng, "Theory for the dielectric function of granular composite media," Phys. Rev. Lett. 45, 60-63 (1980).
[CrossRef]

1979 (1)

D. J. Bergman, "Dielectric constant of a two-component granular composite: A practical scheme for calculating the pole spectrum," Phys. Rev. B 19, 2359-2368 (1979).
[CrossRef]

1978 (2)

R. C. McPhedran and D. R. McKenzie, "The conductivity of lattices of spheres. I. The simple cubic lattices," Proc. R. Soc. London, Ser. A 359, 45-63 (1978).
[CrossRef]

D. R. McKenzie, R. C. McPhedran, and G. H. Derrick, "The conductivity of lattices of spheres. II. The body centred and face centred cubic lattices," Proc. R. Soc. London, Ser. A 362, 211-232 (1978).
[CrossRef]

1977 (1)

D. R. McKenzie and R. C. McPhedran, "Exact modeling of cubic lattice permittivity and conductivity," Nature 265, 128-129 (1977).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1935 (1)

D. A. G. Bruggeman, "Calculation of various physics constants in heterogenous substances. Dielectricity constants and conductivity of mixed bodies from isotropic substances," Ann. Phys. (Leipzig) 24, 636-664 (1935).

Ahn, J. S.

J. S. Ahn, K. H. Kim, T. W. Noh, D. H. Riu, K. H. Boo, and H. E. Kim, "Effective-medium theories for spheroidal particles randomly oriented on a plane: Application to the optical properties of a SiC whisker-Al2O3 composite," Phys. Rev. B 52, 15,244-15,252 (1995).

Alexopoulos, N. G.

Barrera, R. G.

R. Fuchs, R. G. Barrera, and J. L. Carrillo, "Spectral representations of the electron energy loss in composite media," Phys. Rev. B 54, 12,824-12,834 (1996).
[CrossRef]

R. G. Barrera and R. Fuchs, "Theory of electron energy loss in a random system of spheres," Phys. Rev. B 52, 3256-3273 (1995).
[CrossRef]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, "Terahertz magnetic response from artificial materials," Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Baughman, R. H.

A. A. Zakhidov, R. H. Baughman, I. I. Khayrullin, I. A. Udad, M. Kozlov, N. Eradat, V. Z. Vardeny, M. Sigalas, and R. Biswas, "Three-dimensionally periodic conductive nanostructures: network versus cermet topologies for metallic PGB," Synthetic Metals 116, 419-426 (2001).
[CrossRef]

Bergman, D. J.

D. J. Bergman, "Dielectric constant of a two-component granular composite: A practical scheme for calculating the pole spectrum," Phys. Rev. B 19, 2359-2368 (1979).
[CrossRef]

Biswas, R.

A. A. Zakhidov, R. H. Baughman, I. I. Khayrullin, I. A. Udad, M. Kozlov, N. Eradat, V. Z. Vardeny, M. Sigalas, and R. Biswas, "Three-dimensionally periodic conductive nanostructures: network versus cermet topologies for metallic PGB," Synthetic Metals 116, 419-426 (2001).
[CrossRef]

Boltasseva, A.

A. Boltasseva and V. M. Shalaev, "Fabrication of optical negative-index metamaterials: Recent advances and outlook," Metamaterials 2, 1-17 (2008).
[CrossRef]

Boo, K. H.

J. S. Ahn, K. H. Kim, T. W. Noh, D. H. Riu, K. H. Boo, and H. E. Kim, "Effective-medium theories for spheroidal particles randomly oriented on a plane: Application to the optical properties of a SiC whisker-Al2O3 composite," Phys. Rev. B 52, 15,244-15,252 (1995).

Boye, D. M.

H. Fan, K. Yang, D. M. Boye, T. Sigmon, K. J. Malloy, H. Xu, G. P. Lopez, and C. J. Brinker, "Self-assymbly of ordered, robust, three-dimensional gold nanocrystal/silica arrays," Science 304, 567 (2004).
[CrossRef] [PubMed]

Bruggeman, D. A. G.

D. A. G. Bruggeman, "Calculation of various physics constants in heterogenous substances. Dielectricity constants and conductivity of mixed bodies from isotropic substances," Ann. Phys. (Leipzig) 24, 636-664 (1935).

Cai, W. S.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Carrillo, J. L.

R. Fuchs, R. G. Barrera, and J. L. Carrillo, "Spectral representations of the electron energy loss in composite media," Phys. Rev. B 54, 12,824-12,834 (1996).
[CrossRef]

Caruso, F.

F. Caruso, M. Spasova, V. S. no Maceira, and L. M. Liz-Marzan, "Multilayer assemblies of silica-encapsulated gold nanoparticles on decomposable colloid templates," Adv. Mater. 13, 1090-1094 (2001).
[CrossRef]

Chan, C. T.

Z. Wang, C. T. Chan, W. Zhang, N. Ming, and P. Sheng, "Three-dimensional self -assembly of metal nanoparticles: Possible photonic crystal with a complete gap below the plasma frequency," Phys. Rev. B 64, 113,108 (2001).
[CrossRef]

Chettiar, U. K.

W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Contopanagos, H.

Courty, A.

A. Taleb, V. Russier, A. Courty, and M. P. Pileni, "Collective optical properties of silver nanoparticles organized in two-dimensional superlattices," Phys. Rev. B 59, 13,350-13,358 (1999).
[CrossRef]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006).
[CrossRef] [PubMed]

Derrick, G. H.

D. R. McKenzie, R. C. McPhedran, and G. H. Derrick, "The conductivity of lattices of spheres. II. The body centred and face centred cubic lattices," Proc. R. Soc. London, Ser. A 362, 211-232 (1978).
[CrossRef]

Devaty, R. P.

M. F. MacMilland, R. P. Devaty, and J. V. Mantese, "Infrared properties of Pt/Al2O3 cermet films," Phys. Rev. B 43, 13,838-13,845 (1991).

Eradat, N.

A. A. Zakhidov, R. H. Baughman, I. I. Khayrullin, I. A. Udad, M. Kozlov, N. Eradat, V. Z. Vardeny, M. Sigalas, and R. Biswas, "Three-dimensionally periodic conductive nanostructures: network versus cermet topologies for metallic PGB," Synthetic Metals 116, 419-426 (2001).
[CrossRef]

Fan, H.

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Figures (8)

Fig. 1.
Fig. 1.

Schematic view of the system under study, consisting of planar periodic arrays of spherical or ellipsoidal nanoparticles with hexagonal symmetry. Two geometrical parameters characterize the lattice of the structure: the inter-layer spacing dz and the inter-particle spacing within each layer d. Consecutive layers are laterally offset a distance d/√3 along the (1,1) layer direction. A close-packed structure is obtained for dz =2d/√6.

Fig. 2.
Fig. 2.

Spectral optical response of layered systems formed by spherical silver nanoparticles for fixed spacing within each layer and fixed filling fraction of the metal (f=30%). Various values of the spacing between consecutive layers are considered: dz =d/√6≈0.41d (isotropic simple-cubic lattice), dz =2d/√6≈0.82d (isotropic fcc lattice), and dz =1.31d (anisotropic lattice with different response to electric fields parallel and perpendicular to the particle layers).

Fig. 3.
Fig. 3.

Dependence of the bulk-plasmon resonance wavelength on inter-layer spacing dz for spherical silver particles arranged as described in Fig. 1. We represent the two plasmons associated to polarization parallel and perpendicular to the particle layers, respectively. These plasmons are degenerate for values of dz corresponding to isotropic cubic lattices. Two different values of the filling fraction of the metal have been considered. (a) Silver spheres in vacuum. (b) Gold spheres in silica.

Fig. 4.
Fig. 4.

Dependence of the bulk-plasmon resonance wavelength on the aspect ratio r /r of ellipsoidal silver particles arranged in layered structures as depicted in Fig. 1. The filling fraction of the metal is 40% in all cases. Three different values of dz are considered. Plasmon energies corresponding to electric polarization parallel (broken curves) and perpendicular (solid curves) with respect to the particle layers are displayed. The two plasmons are degenerate for specific values of the aspect ratio. This occurs for spheres in a fcc lattice, but non-spherical particles can produce isotropic response in non-cubic lattices (see crossing points of solid and broken curves).

Fig. 5.
Fig. 5.

Left panels: effective dielectric function of a fcc lattice of silver spheres for different sizes of the particles (radius r=5×10-4 nm, r=5nm, and r=20nm, see labels). Right panels: representation of α 2 β/γ-1 [see Eqs. (3), (4), and (5)]. The filling fraction of the metal is 40% in all cases.

Fig. 6.
Fig. 6.

Normal-incidence reflectance for systems of 1, 2, 4, and ∞ layers of silver nanospheres (solid curves) compared to the reflectance of the equivalent homogeneous layers with effective dielectric function (broken curves). The layers are arranged in a fcc lattice and the sphere radius is r=5 nm (left) and r=20 nm (right). The thickness of the homogeneous layers is taken to be equal to the number of spheres times the inter-layer distance dz .

Fig. 7.
Fig. 7.

Normal-incidence transmittance for the same systems as in Fig. 6.

Fig. 8.
Fig. 8.

Dependence of the reflectance on angle of incidence at fixed wavelength λ=360nm for p-polarized light on the same systems as in Fig. 6.

Equations (5)

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R s = cos θ α γ sin 2 θ cos + α γ sin 2 θ
R p = cos θ α 1 β sin 2 θ cos θ + α 1 β sin 2 θ ,
α = ε eff μ eff ,
β = 1 ε eff ε eff ,
γ = 1 μ eff μ eff ,

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