## 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] 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 [2–6]. 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.

## 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 (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 *E⃗* 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]:

*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

*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] 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 *E _{x}* component at wavelength 1.25

*μ*m through the

*y*−

*z*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

*E*and magnetic

_{x}*H*transverse components, respectively. The lower row corresponds to the longitudinal component of electric field

_{y}*E*, which represents the displacement current through the dielectric layers induced by the magnetic field. Importantly, the

_{z}*E*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].

_{z}## 3. Response at oblique incidence

Next we consider the case of a wave impinging onto the structure at oblique incidence, fixing the electric field of the incident wave in the *x* − *z* plane. The two cases of TE and TM polarizations are shown in Figs. 3(a,e). A snapshot of the *E _{x}* component is shown in Fig. 2(b) for 1.20 μm wavelength.
Media 2 demonstrates the propagation of the transverse component of the electric field through the structure for the cases of: (a,b) normal incidence at wavelength 1.25 μm; (c,d) TE polarization, 45° angle of incidence at wavelength 1.23 μm; and (e,f) TM polarization, 45° angle of incidence at wavelength 1.42 μm. 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 *k⃗* inside the metamaterial. For the *k _{z}* component we use Eq. (1) while the boundary conditions ensure that

*k*and

_{x}*k*are continuous at the interface. Figure 3(c) shows dispersion

_{y}*Re*[

*k*] versus

_{z}*k*above the light cone for three different wavelengths for the TE-polarization. At wavelength 1.03 μm (red curve)

_{y}*Re*[

*k*] > 0 and the structure exhibits ordinary elliptic dispersion. At wavelength 1.17 μm (blue curve)

_{z}*Re*[

*k*] < 0 and the structure exhibits

_{z}*hyperbolic dispersion*[1, 18]. At wavelength 1.09 μm (green curve)

*Re*[

*k*] = 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].

_{z}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*increases it crosses through zero and becomes a forward-wave. This zero

_{x}*k*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

_{z}*k*. 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

_{z}*k*, indicating that this media should exhibit number of indefinite medium properties, such as high density of states and strong Purcell factor enhancement.

_{x}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]:

*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 *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*] 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].

## 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 *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*component. As we consider

_{z}*E⃗*to be in the

*x*−

*z*plane, the components

*ε*and

_{y}*μ*do not contribute to the response in our simulations.

_{x}Dispersion relations for the real parts of the *k⃗* can be derived from Maxwell’s equations:

*ε*= 0.45,

_{x}*μ*= 0.18 for the elliptical dispersion [Fig. 3(c), red curve] and

_{y}*ε*= −0.20,

_{x}*μ*= −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

_{y}*ε*to be dependent on

_{x}*k*. Therefore, to describe our numerical data the permittivity and permeability must take the form:

_{x}*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].

## 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 *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.

## Acknowledgments

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

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