Spatial dispersion of multilayer fishnet metamaterials

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 Optical Society of America OCIS codes:(160.3918) Metamaterials; (260.2065) Effective medium theory; (350.3618) Lefthanded materials; (350.4238) Nanophotonics and photonic crystals. 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). 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). 3. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B 74,075103 (2006). 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). 5. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B79,245127 (2009). 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). 7. H. N. S. Krishnamoorthy, Z. Jacob, E. Narimanov, I. Kretzschmar, and V. M. Menon, “Topological Transitions in Metamaterials,” Science 336,205–209 (2012). 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). 10. M. Beruete, M. Navarro-Cia, 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). 11. M. Beruete, M. Navarro-Cia, and M. Sorolla, “Strong lateral displacement in polarization anisotropic extraordinary transmission metamaterial,” New J. Phys. 12,063037 (2010). 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). 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). #168289 $15.00 USD Received 11 May 2012; revised 5 Jun 2012; accepted 8 Jun 2012; published 20 Jun 2012 (C) 2012 OSA 2 July 2012 / Vol. 20, No. 14 / OPTICS EXPRESS 15100 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). 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). 18. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express14,8247–8256 (2006). 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). 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). 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).


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 isofrequency 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] 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][3][4][5][6].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. [2,3,7,8].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].Such structures have previously been shown to exhibit complex anisotropic behavior in the mesoscopic regime [10,11].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.

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 (MgF 2 ) layers.Figs.1(b) and 1(c) show the single-layer fishnet consisting of two Ag layers, separated by a MgF 2 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], while for MgF 2 we use fixed permittivity of 1.90.Without lack of generality we assume the electric vector E to be in the x − z coordinate plane.Figure 1(d) shows the transmission spectra for the single-layer and 22-layers fishnets consisting of 23 Ag layers separated by 22 MgF 2 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].
Next, we extract the effective wavevector k by using the inverted Fresnel formula [13,14]: 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.[13,14].We define the effective refractive index as n = k/k 0 , where k 0 is the wavenumber in free space; and the figure of merit Similar to Ref. [15] we see that after approximately ten functional layers the optical properties of the fishnet metamaterial are essentially independent of the number of layers.Here we can directly observe the backward direction of phase velocity and negative refraction.Figures 3(b,f) show transmission of the ten functional layer fishnet versus wavelength and angle of incidence for the TE and TM cases, respectively.As one can see, for the TE case the transmission remains practically constant with changing angle of incidence.In contrast, for the TM case the transmission changes dramatically, showing the splitting of the transmission peak into two resonant modes.A similar effect was previously observed in the case of a single functional layer fishnet [16,17] and is closely related to the change of the incident electric field across the gap between the metal layers.
Next, we extract the wavevector k inside the metamaterial.For the k z component we use Eq. ( 1) while the boundary conditions ensure that k x and k y are continuous at the interface.Figure 3(c) shows dispersion Re[k z ] versus k y above the light cone for three different wavelengths for the TE-polarization.At wavelength 1.03 µm (red curve) Re[k z ] > 0 and the structure exhibits ordinary elliptic dispersion.At wavelength 1.17 µm (blue curve) Re[k z ] < 0 and the structure exhibits hyperbolic dispersion [1,18].At wavelength 1.09 µm (green curve) Re[k z ] = 0 and the structure is in a regime of topological transition between elliptic and hyperbolic dispersions [7], corresponding to a uniaxial version of epsilon-near-zero metamaterials [19].
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].The blue curve, corresponding to the backward-wave case, is elliptical for small k x , but as k x increases it crosses through zero and becomes a forward-wave.This zero k z feature corresponds to excitation of the gap plasmon,  (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 n eff 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 n eff , dashed curves correspond to the n eff with sign chosen as proposed in [13,14].
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 k z .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 k x , 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 n eff cos(θ ) = k z k 0 , where θ is the angle of refraction, and then define the real and the imaginary parts as [20]: where Here φ is the angle of incidence.Figures 3(d 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].

Effective parameters
Having demonstrated the different dispersion regimes, and shown the inadequacy of an angledependent 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 ε and permeability μ 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 µ z = 1 as the structure does not exhibit artificial magnetism for the H z component.As we consider E to be in the x − z plane, the components ε y and µ x do not contribute to the response in our simulations.
Dispersion relations for the real parts of the k can be derived from Maxwell's equations: For the TE case, the dispersion curves obtained with Eq. ( 4) fit numerical data if ε x = 0.45, µ y = 0.18 for the elliptical dispersion [Fig.4) only if we assume ε x to be dependent on k x .Therefore, to describe our numerical data the permittivity and permeability must take the form: 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 μ 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] 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].

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 ε and μ 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.

Fig. 1 .
Figure of Merit

Figures 1 (
e,f) show the calculated spectra for n and FOM for the singlelayer 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.

Fig. 3 .
Fig. 3. (a,e) TE and TM polarization of incident wave.(b,f) Transmission of the tenlayer 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 n eff 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 n eff , dashed curves correspond to the n eff with sign chosen as proposed in[13,14].
) and3(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 k z 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] 3(c), red curve] and ε x = −0.20,µ y = −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. (