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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 14 — Jul. 2, 2012
  • pp: 15100–15105
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Spatial dispersion of multilayer fishnet metamaterials

Sergey S. Kruk, David A. Powell, Alexander Minovich, Dragomir N. Neshev, and Yuri S. Kivshar  »View Author Affiliations


Optics Express, Vol. 20, Issue 14, pp. 15100-15105 (2012)
http://dx.doi.org/10.1364/OE.20.015100


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Abstract

We study the anisotropic properties of multilayer fishnet optical metamaterials and describe topological transitions between the elliptic and hyperbolic dispersion regimes. In contrast to other hyperbolic media, multilayer fishnet metamaterials may have negative components not only in the effective permittivity tensor but also in the effective permeability tensor, thus allowing the realization of magnetic hyperbolic and generalized indefinite media.

© 2012 OSA

1. Introduction

For the optical applications of anisotropic media, it is useful to classify them according to the shape of their iso-frequency contours. The properties of anisotropic dielectric media are described by a diagonal permittivity tensor with positive elements, yielding elliptical iso-frequency contours. For metals below their plasma frequency all components of the permittivity tensor are negative, the dispersion equations do not have real solutions, and all waves are evanescent. The dispersion types of non-gyrotropic natural materials are basically confined to these two types, limiting the range of functionalities which can be achieved.

However, the use of metamaterials substantially expands this classification, by also allowing the permeability to be engineered at optical wavelengths. In particular, the indefinite medium discussed by Smith and Schurig [1

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

] is an artificial metamaterial structure which supports propagating waves with extremely large wavevectors and diverging density of states. Such a medium is described by the effective tensors of electric and/or magnetic susceptibilities where some of the components are negative, and the corresponding dispersion is hyperbolic. Importantly, such a hyperbolic medium is not only a hypothetical idea but can be practically realized.

In the simplest case, electric hyperbolic media appear as effective media in the theories describing the averaged characteristics of alternating dielectric and metallic layers and lattices created by metallic wires, or plasmonic crystals of nanorods [2

2. 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, 075103 (2006). [CrossRef]

6

6. L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B 85, 165408 (2012). [CrossRef]

]. The presence of this hyperbolic dispersion opens a number of novel applications of metamaterials, including subwavelength imaging, subwavelength cavities, control and enhancement of spontaneous emission, thermal superconductivity, etc. [2

2. 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, 075103 (2006). [CrossRef]

, 3

3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006). [CrossRef]

, 7

7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336, 205–209 (2012). [CrossRef] [PubMed]

, 8

8. E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444v1.

]. However, so far the realization of magnetic hyperbolic media or generalized indefinite media with both electric and magnetic responses has not been discussed or demonstrated.

In order to achieve magnetic hyperbolic media with a highly anisotropic μ-tensor, here we suggest utilizing multilayer optical fishnet metamaterials with artificial magnetism [9

9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]

]. Such structures have previously been shown to exhibit complex anisotropic behavior in the mesoscopic regime [10

10. M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl. 10(3), 263–270 (2012). [CrossRef]

, 11

11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. 12, 063037 (2010). [CrossRef]

]. We choose these metamaterials due to their low losses, bulk properties and double-negative response in a wide spectral region. By studying the optical properties of the fishnet structures for oblique light incidence and different polarizations, we derive their dispersion relations and show that these relations exhibit hyperbolic dispersion in the negative index spectral range. Our results shed new light on optical emission in fishnet metamaterials and open new possibilities for imaging and emission control.

2. Optical properties of multilayer fishnet metamaterials

We consider the fishnet metamaterial with the unit cell shown in Fig. 1(a), consisting of metal (Ag) and dielectric (MgF2) layers. Figs. 1(b) and 1(c) show the single-layer fishnet consisting of two Ag layers, separated by a MgF2 layer and multilayer fishnet structures respectively. We numerically study their optical properties using CST Microwave Studio. For the permittivity of Ag we use experimental data from Ref. [12

12. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

], while for MgF2 we use fixed permittivity of 1.90. Without lack of generality we assume the electric vector E⃗ to be in the xz coordinate plane.

Fig. 1 (a) A unit cell with a=500 nm, b=351 nm, c=100 nm, d=45 nm, e=30 nm. (b) Single functional layer fishnet metamaterial. (c) Ten functional layer fishnet metamaterial. (d–f) Transmission, n and FOM for the single-layer fishnet (dashed curves) and 22-layer fishnet (solid curves). (g–i) Transmission, Real part of n and FOM for the fishnets versus wavelength and number of functional layers.

Figure 1(d) shows the transmission spectra for the single-layer and 22-layers fishnets consisting of 23 Ag layers separated by 22 MgF2 layers. Figure 1(g) also shows the transmission versus wavelength and increasing number of layers. An important conclusion from these simulations is that the addition of more functional layers doesn’t decrease dramatically the overall transmission through the metamaterial. As such the multilayer fishnet has been favored as a good example of a bulk metamaterial [9

9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]

].

Next, we extract the effective wavevector k by using the inverted Fresnel formula [13

13. 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, 195104 (2002). [CrossRef]

, 14

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).

]:
k=±1hcos1(1r2+t22t)+2πmh,
(1)
where h is the thickness of the structure, r and t are the complex reflection and transmission coefficients and m is an integer branch number. The criteria for choosing the correct sign and branch number can be found in Refs. [13

13. 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, 195104 (2002). [CrossRef]

, 14

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).

]. We define the effective refractive index as n = k/k0, where k0 is the wavenumber in free space; and the figure of merit FOM = Re[n]/Im[n]. Figures 1(e,f) show the calculated spectra for n and FOM for the single-layer and 22-layer structures. It is seen that for the 22-layer structure the spectral region with negative Re[n] becomes broader, absorption Im[n] becomes smaller and FOM increases dramatically. Alternatively, Figs. 1(h,i) show n and FOM versus wavelength and number of layers. Similar to Ref. [15

15. J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B. 80, 035109 (2009). [CrossRef]

] we see that after approximately ten functional layers the optical properties of the fishnet metamaterial are essentially independent of the number of layers.

Figure 2(a) shows the profile of the Ex component at wavelength 1.25 μm through the yz plane of the 22 layer structure. Mode analysis of these fields confirm the values of k and n obtained by Eq. (1). However, to differentiate between the forward and backward-wave regimes at normal incidence, it is necessary to view the time variation of the fields, which is shown in Media 1. The left column in the Media 1 corresponds to a wavelength of 0.86 μm and positive values of Re[k] and Re[n]. As we can see, for this case the phase velocity inside the metamaterial has a forward direction. The right column of the Media 1 corresponds to a wavelength of 1.25 μm and negative values of Re[k] and Re[n]. In this case we can indeed observe a backward direction of phase velocity inside the structure, corresponding to the negative index of the metamaterial. The upper and middle rows correspond to electric Ex and magnetic Hy transverse components, respectively. The lower row corresponds to the longitudinal component of electric field Ez, which represents the displacement current through the dielectric layers induced by the magnetic field. Importantly, the Ez component is strong only in the negative-index spectral region and is a signature of the artificial magnetic response of the metamaterial. The effective circuits of current and displacement current cover several layers of the structure, which leads to the qualitative difference between multilayer and single-layer fishnet metamaterials [9

9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]

].

Fig. 2 Propagation of the electromagnetic wave through the multilayer fishnet metamaterial in yz cross-section. (a) Normal incidence at wavelength 1.25 μm (see Media 1). (b) Oblique incidence at 45° angle, 1.20 μm wavelength (see Media 2). Color arrows show the direction of the phase velocity.

3. Response at oblique incidence

Fig. 3 (a,e) TE and TM polarization of incident wave. (b,f) Transmission of the ten-layer fishnet versus wavelength and angle of incidence for TE and TM. (c,f) Isofrequency surfaces for wavelengths 1.03 μm (red curves), 1.09 μm (green curve) and 1.17 μm (blue curve) for TE and TM polarizations. (d,h) Real part of neff for normal incidence (red), 30° angle of incidence (green) and 60° angle of incidence (blue) for TE and TM. Solid curves correspond to the absolute values of neff, dashed curves correspond to the neff with sign chosen as proposed in [13, 14].

Next, we extract the wavevector k⃗ inside the metamaterial. For the kz component we use Eq. (1) while the boundary conditions ensure that kx and ky are continuous at the interface. Figure 3(c) shows dispersion Re[kz] versus ky above the light cone for three different wavelengths for the TE-polarization. At wavelength 1.03 μm (red curve) Re[kz] > 0 and the structure exhibits ordinary elliptic dispersion. At wavelength 1.17 μm (blue curve) Re[kz] < 0 and the structure exhibits hyperbolic dispersion [1

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

, 18

18. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef] [PubMed]

]. At wavelength 1.09 μm (green curve) Re[kz] = 0 and the structure is in a regime of topological transition between elliptic and hyperbolic dispersions [7

7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336, 205–209 (2012). [CrossRef] [PubMed]

], corresponding to a uniaxial version of epsilon-near-zero metamaterials [19

19. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). [CrossRef] [PubMed]

].

In Fig. 3(g) we plot the isofrequency contours for the TM case. In comparison to the TE case, we see that the dispersion is much more complex [11

11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. 12, 063037 (2010). [CrossRef]

]. The blue curve, corresponding to the backward-wave case, is elliptical for small kx, but as kx increases it crosses through zero and becomes a forward-wave. This zero kz feature corresponds to excitation of the gap plasmon, which is responsible for the effective magnetic response of the metamaterial. As seen in the transmission plot [Fig. 3(f)] this mode has strong dispersion for TM incident polarization. The change of incident angle shifts this plasmon resonance, causing the negative index condition to be broken and resulting in a change of sign in kz. For the zero-index wavelength of 1.09 μm the dispersion exhibits an X-shaped contour, as shown by the green curve, while for the positive index wavelength of 1.03 μm the dispersion is twin V-shaped contours. In all these cases the contours are diverging for larger kx, indicating that this media should exhibit number of indefinite medium properties, such as high density of states and strong Purcell factor enhancement.

We now consider how well these isofrequency contours are described by a scalar refractive index. We start from the formula neff cos(θ) = kz/k0, where θ is the angle of refraction, and then define the real and the imaginary parts as [20

20. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959).

]:
Re[n]=±a+a24b2;Im[n]=Re[kzk0]Im[kzk0]/Re[n],
(2)
where
a=Re[kzk0]2+Im[kzk0]2sin(ϕ)2;b=Re[kzk0]2Im[kzk0]2.
(3)
Here ϕ is the angle of incidence.

Figures 3(d) and 3(h) show Re[n] for three different angles of incidence for the TE and TM polarizations, respectively. The red curve corresponds to Re[n] at normal incidence. The dashed green and blue curves correspond to Re[n] at oblique incidence with the sign chosen from the sign of kz to yield results consistent with the normal incidence case. The solid green and blue curves correspond to |Re[n]|. As one can see, for oblique incidence the Re[n] curves become discontinuous while the |Re[n]| curves are continuous. For the TE case, the break in Re[n] corresponds to the topological transition between elliptic and hyperbolic dispersion. For the case of normal incidence along a high symmetry direction, the sign of Re[n] defines whether the group and phase velocity are parallel or antiparallel. But for the case of oblique incidence, the phase and group velocities are not in general collinear, therefore the relationship between them cannot be adequately characterized using the sign of the refractive index. Thus even an angle-dependent scalar refractive index is not generally meaningful in such media [14

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).

].

4. Effective parameters

Having demonstrated the different dispersion regimes, and shown the inadequacy of an angle-dependent scalar index, we now consider how well the structure is described by effective parameters. The symmetry of the structure ensures us that it does not exhibit gyrotropy or magnetoelectric coupling, therefore it can be described with two tensors permittivity ε̂ and permeability μ̂ having zero non-diagonal elements. All diagonal components are functions of frequency ω due to the resonant behavior of the metamaterial, in addition, they are complex due to high losses. Our coordinate system is chosen in such a way that the main axes of the tensors coincide with x, y and z coordinate axes. Furthermore μz = 1 as the structure does not exhibit artificial magnetism for the Hz component. As we consider E⃗ to be in the xz plane, the components εy and μx do not contribute to the response in our simulations.

Dispersion relations for the real parts of the k⃗ can be derived from Maxwell’s equations:
TE:ky2εxμz+kz2εxμy=ω2c2;TM:kx2εzμy+kz2εxμy=ω2c2.
(4)
For the TE case, the dispersion curves obtained with Eq. (4) fit numerical data if εx = 0.45, μy = 0.18 for the elliptical dispersion [Fig. 3(c), red curve] and εx = −0.20, μy = −0.48 for the hyperbolic dispersion [Fig. 3(c), blue curve]. For the TM case, we can describe the dispersion curves plotted in Fig. 3(g) with Eq. (4) only if we assume εx to be dependent on kx. Therefore, to describe our numerical data the permittivity and permeability must take the form:
ε^=(εx(ω,kx)000εy000εz(ω));μ^=(μx000μy(ω)0001).
(5)
Importantly, the hyperbolic dispersion observed in the negative index band has contributions from both of these tensors, in contrast to hyperbolic media reported to date where all components of μ̂ are 1. Thus the multilayered fishnet is a general form of indefinite media incorporating both the electric and magnetic responses. Similar to the earlier studies of the electric hyperbolic media [21

21. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011). [CrossRef]

] the components of the nonlocal permittivity tensor depend on the wavevector. However, we note that this strong optical nonlocality of metal-dielectric structures can be engineered by changing the ratio between the thicknesses of metal and dielectric layers [22

22. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]

].

5. Conclusions

We have analyzed the anisotropic dispersion properties of multilayer fishnet metamaterials, including the dependence of their optical properties on the number of layers. We have revealed that such structures can be employed for the observation of nontrivial effects associated with negative refraction and backward waves, and they can model generalized indefinite media with both ε̂ and μ̂ tensors having negative components. The multi-functional properties of multilayer fishnet metamaterials including their magnetic hyperbolic dispersion open a number of novel applications, including control and enhancement of spontaneous emission.

Acknowledgments

The authors acknowledge financial support from the Australian Research Council through the Discovery Project and Centre of Excellence schemes.

References and links

1.

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

2.

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, 075103 (2006). [CrossRef]

3.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74, 075103 (2006). [CrossRef]

4.

M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009). [CrossRef]

5.

A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B 79, 245127 (2009). [CrossRef]

6.

L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B 85, 165408 (2012). [CrossRef]

7.

H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336, 205–209 (2012). [CrossRef] [PubMed]

8.

E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444v1.

9.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008). [CrossRef] [PubMed]

10.

M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl. 10(3), 263–270 (2012). [CrossRef]

11.

M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. 12, 063037 (2010). [CrossRef]

12.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

13.

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, 195104 (2002). [CrossRef]

14.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77, 195328 (2008).

15.

J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B. 80, 035109 (2009). [CrossRef]

16.

A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Yu. S. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B 81, 115109 (2010). [CrossRef]

17.

C. Garcia-Meca, J. Hurtado, J. Marti, A. Martinez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett. 106, 083104 (2011). [CrossRef]

18.

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef] [PubMed]

19.

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). [CrossRef] [PubMed]

20.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959).

21.

A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Yu. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011). [CrossRef]

22.

A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Yu. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]

OCIS Codes
(260.2065) Physical optics : Effective medium theory
(350.3618) Other areas of optics : Left-handed materials
(160.3918) Materials : Metamaterials
(350.4238) Other areas of optics : Nanophotonics and photonic crystals

ToC Category:
Metamaterials

History
Original Manuscript: May 11, 2012
Revised Manuscript: June 5, 2012
Manuscript Accepted: June 8, 2012
Published: June 20, 2012

Citation
Sergey S. Kruk, David A. Powell, Alexander Minovich, Dragomir N. Neshev, and Yuri S. Kivshar, "Spatial dispersion of multilayer fishnet metamaterials," Opt. Express 20, 15100-15105 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15100


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References

  1. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
  2. 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. B73, 075103 (2006). [CrossRef]
  3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B74, 075103 (2006). [CrossRef]
  4. M. A. Noginov, Yu. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett.94, 151105 (2009). [CrossRef]
  5. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79, 245127 (2009). [CrossRef]
  6. L. M. Custodio, C. T. Sousa, J. Ventura, J. M. Teixeira, P. V. S. Marques, and J. P. Araujo, “Birefringence swap at the transition to hyperbolic dispersion in metamaterilas,” Phys. Rev. B85, 165408 (2012). [CrossRef]
  7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science336, 205–209 (2012). [CrossRef] [PubMed]
  8. E. Narimanov and I. Smolyaninov, “Beyond Stefan-Boltzmann law: thermal hyper-conductivity,” arXiv:1109.5444v1.
  9. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature455, 376–379 (2008). [CrossRef] [PubMed]
  10. M. Beruete, M. Navarro-Cia, and M. Sorolla, “High numerical aperture and low-loss negative refraction based on the fishnet rich anisotropy,” Photonics Nanostruct. Fundam. Appl.10(3), 263–270 (2012). [CrossRef]
  11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys.12, 063037 (2010). [CrossRef]
  12. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
  13. 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. B65, 195104 (2002). [CrossRef]
  14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B77, 195328 (2008).
  15. J. Zhou, T. Koschny, M. Kafesaki, and C. Soukoulis, “Negative refractive index response of weakly and strongly coupled optical metamaterials,” Phys. Rev. B.80, 035109 (2009). [CrossRef]
  16. A. Minovich, D. N. Neshev, D. A. Powell, I. V. Shadrivov, M. Lapine, I. McKerracher, H. T. Hattori, H. H. Tan, C. Jagadish, and Yu. S. Kivshar, “Tilted response of fishnet metamaterials at near-infrared optical wavelengths,” Phys. Rev. B81, 115109 (2010). [CrossRef]
  17. C. Garcia-Meca, J. Hurtado, J. Marti, A. Martinez, W. Dickson, and A. V. Zayats, “Low-loss multilayered metamaterial exhibiting a negative index of refraction at visible wavelengths,” Phys. Rev. Lett.106, 083104 (2011). [CrossRef]
  18. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14, 8247–8256 (2006). [CrossRef] [PubMed]
  19. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using epsilon-near-zero materials,” Phys. Rev. Lett.97, 157403 (2006). [CrossRef] [PubMed]
  20. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1959).
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