## Anisotropy and particle-size effects in nanostructured plasmonic metamaterials

Optics Express, Vol. 17, Issue 24, pp. 22012-22022 (2009)

http://dx.doi.org/10.1364/OE.17.022012

Acrobat PDF (1690 KB)

### 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

## 1. Introduction

1. 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]

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

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

4. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. **85**, 3966–3969 (
2000). [CrossRef] [PubMed]

5. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science **312**, 1780–1782 (
2006). [CrossRef] [PubMed]

6. 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]

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

8. L. M. Liz-Marzán, “Tailoring surface plasmon through the morphology and assembly of metal nanoparticles,” Langmuir **22**, 32–41 (
2006). [CrossRef]

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). [CrossRef]

10. D. Zanchet, M. S. Moreno, and D. Ugarte, “Anomalous packing in thin nanoparticle supercrystals,” Phys. Rev. Lett. **82**, 5277–5280 (
1999). [CrossRef]

11. 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]

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

13. T. Ung, L. M. Liz-Marzán, and P. Mulvaney, “Optical properties of thin films of Au@SiO_{2} particles,” J. Phys. Chem. B **105**, 3441–3452 (
2001). [CrossRef]

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

15. H. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexópoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A **16**, 1682–1699 (
1999). [CrossRef]

16. 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]

17. 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]

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

19. 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]

21. J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London, Ser. A203, 385 ( 1904); 205, 237 ( 1906). [CrossRef]

*f*as the only parameters. When the shape of the constituent materials is not spherical, Bruggeman’s theory [22] can yield reasonable results in some cases, although it is generally better to use MG theory for spherical objects. However, both of these models fail at large filling fractions [23, 24

24. R. Sainidou and F. J. García de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express **16**, 4499–4506 (
2008). [CrossRef] [PubMed]

25. 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]

26. P. Sheng, “Theory for the dielectric function of granular composite media,” Phys. Rev. Lett. **45**, 60–63 (
1980). [CrossRef]

27. 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]

28. 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]

29. D. R. McKenzie and R. C. McPhedran, “Exact modeling of cubic lattice permittivity and conductivity,” Nature **265**, 128–129 (
1977). [CrossRef]

30. 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]

31. 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]

32. 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]

33. 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]

34. 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]

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

37. 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]

*R*and

_{s}*R*for s- and p-polarized light using a rigorous layer-KKR method [38

_{p}38. N. Stefanou, V. Yannopapas, and A. Modinos, Comput. Phys. Commun.113, 49 ( 1998); 132, 189 ( 2000). [CrossRef]

39. 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]

*l*≤

*l*

_{max}. The results presented below have converged for

*l*

_{max}=12. The number of plane waves that mediate the inter-layer interaction is also limited to a finite set around the specular reflection direction, with good convergence achieved for ~300 waves in the calculations here reported. This method has been previously used to study isotropic nanoparticle-based metamaterials [40

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

*R*and

_{s}*R*obtained by means of the layer-KKR method are then compared to those of a homogeneous anisotropic semi-infinite material, which we obtain from a straightforward generalization of the methods of Ref. [20]:

_{p}*ε*

^{eff}and

*µ*

^{eff}are the electric permittivity and the magnetic permeability of the homogenous medium, the symbols ‖ and ⊥ refer to the response to field components parallel and perpendicular to the surface of the material (i.e., to the particle planes), respectively, and

*θ*is the angle of incidence. By construction, the material response is isotropic with respect to the surface plane directions.

*α, β*, and

*γ*are obtained by comparing Eqs. (1) and (2) to the numerical results of the layer-KKR method for several angles of incidence. Although no magnetic response is expected in the system under consideration for small particle size, we show below that inclusion of an effective

*µ*

^{eff}is important to describe retardation effects due to the finite size of the particles compared to the wavelength. In what follows, tabulated complex, frequency-dependent dielectric functions of gold, silver [41

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

43. 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]

## 2. Anisotropy due to lattice structure

*d*=6 nm in the calculations that follow, and we neglect nonlocal effects, which should be significant for these small dimensions [44, 45

45. F. J. García 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]

*f*(i.e., the metal volume fraction) constant. This is done by simultaneously modifying the radius of the spheres

*r*(and thus their volume) and the distance between layers

*d*(see Fig. 1). The effective dielectric constant is then calculated for fixed

_{z}*f*and for a range

*d*values.

_{z}*d*, 3D cubic lattices are obtained, with the layers under consideration oriented perpendicular to the 〈111〉 direction. More precisely, for

_{z}*d*=

_{z}*d*/2√6,

*d*=

_{z}*d*/√6, and

*d*=2

_{z}*d*/√6 we obtain a body-centered cubic lattice (bcc), a simple cubic lattice (sc), and a face-centered cubic lattice (fcc). In these particular instances, the system becomes isotropic. For other inter-layer distances, the response of the system is anisotropic.

*ε*

^{eff}}, the so-called loss function, which can be directly measured using electron energy-loss spectroscopy [46] and exhibits a pronounced peak near the vanishing of the real part of the dielectric function (Re{

*ε*

^{eff}}=0), signaling the existence of a bulk plasmon resonance. The loss function is isotropic for sc and fcc lattices, but it is anisotropic for values of

*d*corresponding to non-cubic lattices. This is the case of the

_{z}*d*=1.31 lattice considered in Fig. 2, in which the degeneracy of bulk plasmons is broken for directions parallel and perpendicular to the layers (the difference in plasmon wavelength is ~10nm).

_{z}/d*d*values and represented the plasmon resonance wavelengths as a function of

_{z}*d*for both field orientations in Fig. 3. The values of

_{z}/d*d*corresponding to the three cubic lattices are marked by vertical dashed lines. In these three cases, both the parallel and perpendicular components of the dielectric tensor exhibit plasmon resonance peaks at the same wavelength. Plasmons associated to parallel or perpendicular polarization with respect to the layer planes have different wavelengths except for these three crossing points of cubic symmetry. Two different filling fractions of the metal are considered in Fig. 3 (30% and 40%). The behavior is similar in both cases, but the higher filling fraction leads to lower plasmon wavelengths, which is due to stronger inter-particle interaction [40

_{z}/d40. S. Riikonen, I. Romero, and F. J. García de Abajo, “Plasmon tunability in metallodielectric metamaterials,” Phys. Rev. B **71**, 235,104 (
2005). [CrossRef]

## 3. Anisotropy due to particle shape

47. F. J. García 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]

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

*ε*

^{eff}} for layered materials as a function of the aspect ratio of the particles

*r*

_{⊥}/

*r*

_{‖}, where

*r*

_{‖}and

*r*

_{⊥}are the ellipsoids semi-axes along directions parallel and perpendicular to the particle layers, and the ellipsoids are taken to be axially symmetric with respect to the layer normal direction. The geometrical parameters are varied in such a way that the metal filling fraction is kept constant. Three different values of

*d*are considered, leading to two different plasmon resonances for electric-field polarization either parallel or perpendicular to the layers. However, these plasmons are degenerate for specific values of the ellipsoids aspect ratio that depend on

_{z}/d*d*. The fcc lattice predicts full degeneracy for spheres (

_{z}/d*r*

_{⊥}=

*r*

_{‖}), but isotropic behavior is observed in non-cubic lattices for oblate ellipsoids (

*r*

_{⊥}<

*r*

_{‖}) at short inter-layer spacings and for prolate ellipsoids (

*r*

_{⊥}>

*r*

_{‖}) in the opposite case. The anisotropy due to the lattice is compensated by the anisotropy produced by the particle shape in the crossing points at which degeneracy occurs.

## 4. Limits to homogenization

*µ*

^{eff}=1 for the particle dimensions and material properties under consideration. Therefore, we expect to obtain

*γ*=1 from our simulations [see Eq. (5)]. Furthermore, we also expect that cubic lattices yield

*α*

^{2}

*β/γ*=1 [see Eqs. (3), (4), and (5)]. However, these conditions break down for relatively large particles, so that the effective-homogeneous-mediumassumption is no longer valid.

*α*

^{2}

*β/γ*-1 (right panel) are represented for silver spheres of different size (

*r*=5×10

^{-4}nm,

*r*=5nm, and

*r*=20nm), arranged in a fcc lattice with a metal filling fraction of 40%. We expect this system to exhibit isotropic response, which it certainly gives outside the plasmon resonances (in particular, for wavelengths below 350 nm) or for very small particles (electrostatic limit). However, there is a significant departure from isotropy even for

*r*=5nm particles at wavelengths around 400 nm (plasmon region). These anomalies are matched by large deviations from unity of the quantity

*α*

^{2}

*β/γ*-1 in the plasmon region for all particle sizes, and over all the spectral region under consideration for

*r*=20nm particles. Thus, in contrast to what can be expected, we conclude that even relatively small particles (e.g., 40 nm particles at a wavelength of 400 nm) are problematic when trying to define a homogeneous medium picture. The electrostatic effective dielectric function (obtained in the small-particle limit) yields a response that can be safely used for small particles (e.g., the 10 nm particles considered in next section), at least when calculating the reflectance and transmittance of finite particle films. However, the actual Fresnel coefficients, including their phase information, becomes problematic even at small particle sizes. Finally, it should be mentioned that Eqs. (1) and (2) work extremely well compared to the reflection coefficients calculated from the layer-KRR approach for a (wavelength-dependent) single choice of parameters

*α, β*, and

*γ*valid for all angles of incidence, although these coefficients do not satisfy the expected identity α2β/γ=1. Besides the problems noted with homogenization, there is also the possibility that the surface of our metamaterials does require a more careful description than that provided by the abrupt termination of an effective homogeneous film.

## 5. How many layers are needed to form a metamaterial

^{25}(opaque film) particle layers (solid curves) arranged in a close-packed (fcc) structure with metal filling fraction of 40%. We have considered two different sizes of the spheres (radius

*r*=5nm and

*r*=20nm). We compare these results with the reflectance and transmittance of homogeneous films made of an effective material (broken curves). The effective dielectric function of the homogeneous material is calculated as explained above for a semi-infinite metamaterial of the same metal filling fraction in the electrostatic limit (

*r*→0). The thickness of the homogeneous film is taken to be the number of layers times the inter-layer spacing

*d*(even for a single layer, the value of

_{z}*d*corresponding to the fcc(111) structure is the appropriate one; we have verified this by calculating the reflectance for

_{z}*f*=10% (not shown), in which case

*r*≪

*d*, so that the actual choice of thickness in the effective homogeneous film is not so obvious). Interestingly, the homogeneous-film model works rather well even for a single layer, although there are some discrepancies within the spectral region of the plasmon resonances, and the model does not work above ~400nm for spheres of radius

_{z}*r*=20nm. This was expected, since we are using the electrostatic effective dielectric function for the homogeneousmedium, so that retardation effects become already important for 40 nm particles, as discussed in Sec. 4.

*r*=5nm and fixed wavelength

*λ*=360nm. The effective-homogeneous-film model works extremely well down to the single layer over a wide range of angles of incidence.

## 6. Conclusion

## Acknowledgments

## References and links

1. | 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 |

2. | D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science |

3. | A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials |

4. | J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. |

5. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

6. | 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 |

7. | W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

8. | L. M. Liz-Marzán, “Tailoring surface plasmon through the morphology and assembly of metal nanoparticles,” Langmuir |

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 |

10. | D. Zanchet, M. S. Moreno, and D. Ugarte, “Anomalous packing in thin nanoparticle supercrystals,” Phys. Rev. Lett. |

11. | M. Gadenne, V. Podolskiy, P. Gadenne, P. Sheng, and V. M. Shalaev, “Plasmon-enhanced absorption by optical phonons in metal-dielectric composites,” Europhys. Lett. |

12. | F. Caruso, M. Spasova, V. S. no Maceira, and L. M. Liz-Marzán, “Multilayer assemblies of silica-encapsulated gold nanoparticles on decomposable colloid templates,” Adv. Mater. |

13. | T. Ung, L. M. Liz-Marzán, and P. Mulvaney, “Optical properties of thin films of Au@SiO |

14. | H. Fan, K. Yang, D. M. Boye, T. Sigmon, K. J. Malloy, H. Xu, G. P. López, and C. J. Brinker, “Self-assymbly of ordered, robust, three-dimensional gold nanocrystal/silica arrays,” Science |

15. | H. Contopanagos, C. A. Kyriazidou, W. M. Merrill, and N. G. Alexópoulos, “Effective response functions for photonic bandgap materials,” J. Opt. Soc. Am. A |

16. | 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 |

17. | 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 |

18. | P. Xu and Z. Li, “Study of frequency band gaps in metal-dielectric composite materials,” J. Phys. D |

19. | C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B |

20. | J. D. Jackson, |

21. | J. C. Maxwell-Garnett, Philos. Trans. R. Soc. London, Ser. A203, 385 ( 1904); 205, 237 ( 1906). [CrossRef] |

22. | 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) |

23. | M. F. MacMilland, R. P. Devaty, and J. V. Mantese, “Infrared properties of Pt/Al |

24. | R. Sainidou and F. J. García de Abajo, “Plasmon guided modes in nanoparticle metamaterials,” Opt. Express |

25. | D. J. Bergman, “Dielectric constant of a two-component granular composite: A practical scheme for calculating the pole spectrum,” Phys. Rev. B |

26. | P. Sheng, “Theory for the dielectric function of granular composite media,” Phys. Rev. Lett. |

27. | G. W. Milton, “Bounds and exact theories for the transport properties of inhomogeneous media,” Appl. Phys. A-Mater. Sci. Process. |

28. | R. Fuchs, R. G. Barrera, and J. L. Carrillo, “Spectral representations of the electron energy loss in composite media,” Phys. Rev. B |

29. | D. R. McKenzie and R. C. McPhedran, “Exact modeling of cubic lattice permittivity and conductivity,” Nature |

30. | R. C. McPhedran and D. R. McKenzie, “The conductivity of lattices of spheres. I. The simple cubic lattices,” Proc. R. Soc. London, Ser. A |

31. | 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 |

32. | F. Wu and K. W. Whites, “Quasi-static effective permittivity of periodic composites containing complex shaped dielectric particles,” IEEE Trans. Antennas Propag. |

33. | 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. |

34. | 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. |

35. | A. H. Sihvola and I. V. Lindell, “Chiral Maxwell-Garnett mixing formula,” Electron. Lett. |

36. | 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-Al |

37. | R. G. Barrera and R. Fuchs, “Theory of electron energy loss in a random system of spheres,” Phys. Rev. B |

38. | N. Stefanou, V. Yannopapas, and A. Modinos, Comput. Phys. Commun.113, 49 ( 1998); 132, 189 ( 2000). [CrossRef] |

39. | 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. |

40. | S. Riikonen, I. Romero, and F. J. García de Abajo, “Plasmon tunability in metallodielectric metamaterials,” Phys. Rev. B |

41. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

42. | E. D. Palik, |

43. | 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 |

44. | U. Kreibig and M. Vollmer, |

45. | F. J. García de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C |

46. | R. F. Egerton, |

47. | F. J. García de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. |

48. | F. J. García de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(160.3918) Materials : Metamaterials

(160.4236) Materials : Nanomaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: August 24, 2009

Revised Manuscript: October 16, 2009

Manuscript Accepted: November 4, 2009

Published: November 17, 2009

**Citation**

Isabel Romero and F. Javier García de Abajo, "Anisotropy and particle-size effects in
nanostructured plasmonic metamaterials," Opt. Express **17**, 22012-22022 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-22012

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### References

- 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]
- A. Boltasseva and V. M. Shalaev, "Fabrication of optical negative-index metamaterials: Recent advances and outlook," Metamaterials 2, 1-17 (2008). [CrossRef]
- J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
- 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]
- W. S. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
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