Negative refraction, gain and nonlinear effects in hyperbolic metamaterials

: The negative refraction and evanescent-wave canalization effects supported by a layered metamaterial structure obtained by alternating dielectric and plasmonic layers is theoretically analyzed. By using a transmission-line analysis, we formulate a way to rapidly analyze the negative refraction operation for given available materials over a broad range of frequencies and design parameters, and we apply it to broaden the bandwidth of negative refraction. Our analytical model is also applied to explore the possibility of employing active layers for loss compensation. Nonlinear dielectrics can also be considered within this approach, and they are explored in order to add tunability to the optical response, realizing positive-to-zero-to-negative refraction at the same frequency, as a function of the input intensity. Our findings may lead to a better physical understanding and improvement of the performance of negative refraction and subwavelength imaging in layered metamaterials, paving the way towards the design of gain-assisted hyperlenses and tunable nonlinear imaging devices.


Introduction
Hyperbolic metamaterials [1][2][3][4][5][6][7] hold great promise to enable a wide range of novel, metamaterial-inspired electromagnetic devices that may become essential components of future microwave, infrared (IR) and optical circuits. These devices may realize lenses overcoming the diffraction limit [8][9][10], exhibit negative refraction [11,12], achieve perfect absorption [13,14] and broadband super-Planckian thermal emission [15], and increase the spontaneous radiation of emitters [16,17] in different frequency ranges. Negative refraction may be achieved in these metamaterials thanks to the hyperbolic dispersion of these structures [1,2], which may also enable the conversion of a portion of the evanescent spectrum into propagating modes. As a result, subwavelength imaging can also be achieved in these structures, even at optical frequencies. It should be stressed that these media support a moderately broadband negative refraction phenomenon without a negative index of refraction [18], but instead based on the anomalous frequency and spatial dispersion of the metamaterial eigenmodes. This leads to more broadband performance compared to conventional negative index materials, since many of the bandwidth limitations on the effective parameter dispersion are relaxed. This is also combined with lower sensitivity to material losses [19].
In this paper, we theoretically and numerically analyze a metamaterial composed of alternating layers of dielectric and metal. This geometry has been shown to support hyperbolic and flat dispersion, which leads to subwavelength resolution imaging [10]. In the following, we apply a closed-form analytical technique to analyze, demonstrate and assess the bandwidth performance of the device's negative refraction operation based on the transmission-line method [20], analogous to the transfer matrix technique [21]. This approach is based on the fact that the negative refraction operation can be directly connected to the condition for which the slope of the phase of the transmission coefficient is negative with respect to the angle of incidence [11]. This allows us to efficiently analyze a broad range of the design parameter space to explore and possibly optimize the negative refraction operation. We show that several bands of high transmission can exist in different frequency ranges, for which negative refraction can occur for a broad range of incident angles. In addition, our theory allows to easily introduce into the picture other interesting anomalous effects arising when optically active materials [22][23][24][25] and Kerr third-order nonlinear media [26] are included in the alternating layers of the hyperbolic metamaterial. Loss compensation with realistic gain materials is presented, opening new routes towards the design of broadband low loss hyperbolic metamaterials. Finally, all-optical switching between positive refraction, epsilon-near-zero behavior [27,28] and negative refraction with moderate input intensities is also reported. We envision that the proposed metamaterial structures may lead to the design of novel nonlinear electromagnetic devices, with exciting applications, such as all-optical switches and tunable subwavelength imaging systems.

Theory of hyperbolic metamaterials
We start by studying the wave propagation in an inhomogeneous anisotropic slab composed of alternating layers of dielectric ( 2 d ε = ) and plasmonic (silver, Ag ε ) materials with thicknesses 1 d and 2 d respectively, as shown in Fig. 1(a). In the subwavelength limit , .
The dispersion relation of this anisotropic material for transverse-magnetic (TM) propagation with magnetic field polarized along ŷ is then governed by the second-order dispersion relation 2 where 0 k is the free space wavenumber and , x z k k are the transverse and longitudinal components of the wave vector, respectively. The ratio between the power densities carried in the transverse and normal directions is then found equal to  Assuming for the moment real-valued permittivities, i.e., neglecting losses, Eq. (2) implies that, when the product t z ε ε is negative (either wave is required to refract negatively at an interface with a conventional isotropic material with normal along ẑ , since z k and z S necessarily have opposite signs. This is consistent with the fact that the dispersion relation of the metamaterial becomes hyperbolic [6]. Inspecting Eq. (1), it is evident that for this to happen the condition The "canalization" regime [10] is a special situation at the boundary between the two cases, and it is realized in the limit of z ε → ∞ . In this case, a flat dispersion curve is obtained, opening the possibility for efficient subwavelength imaging. Ideally, an infinite number of spatial harmonics can be converted into propagating modes in this situation. To work in this regime, it is necessary for the parameters of the layered metamaterial to satisfy the relation . As a result, the canalization regime can ideally be achieved only for a single frequency. In the present work, we focus most of our attention on the negative refraction condition for propagating waves in the layered metamaterial structure, which can be accessible over a continuous bandwidth, and of which the canalization regime is one of the boundaries. Therefore, in the following, unless differently specified, we consider TM propagating waves with transverse wavenumber impinging on a metamaterial slab of length L as indicated in Fig. 1(a). Later in the discussion, we will also consider higher spatial harmonics that exhibit evanescent nature in free-space. For arbitrary spatial distribution, the magnetic field of the incoming beam traveling in air can be decomposed into a superposition of plane waves impinging upon the metamaterial slab at different oblique incident angles The interaction of each plane wave with the layered metamaterial may be efficiently, yet rigorously modeled using a transmission-line formalism [20], which allows calculating the overall ABCD matrix of the layered metamaterial through the multiplication of the ABCD matrices associated with each dielectric and plasmonic layer, as a function of the transverse momentum x k . All these expressions are known in closed analytical form. Next, the total transmittance through the structure may be analytically calculated, and the magnetic field at the output of the device is simply Assuming that the impinging beam is a weakly modulated plane wave propagating along the incident angle i θ , i.e., its spectrum is confined around the wavenumber 0 0 sin , where L is the total length of the layered structure. If the spatial shift Δ is negative, i.e., the slope of the phase of the transmission coefficient as a function of the angle of incidence is negative, then negative refraction is expected. Note that this formula is in general more accurate for thicker structures and its precision deteriorates as the illuminating beam approaches grazing incident angles [12]. By having ( ) x T k in closed analytical form allows calculating in an efficient, yet rigorous way the refraction properties of the metamaterial by simply looking at the derivative of the phase of the transmission coefficient with respect to the transverse wave number. This allows us to quickly analyze the effect of several design parameters of the layered metamaterial on its index of refraction, and potentially optimize its design to achieve broader bandwidths of negative refraction. In general, the proposed analytical method represents a powerful tool to analyze negative refraction phenomena, obtained from complex metamaterial structures.

Linear operation
The plasmonic layers of the structure in Fig. 1(a) are assumed to be made of Ag, for which the permittivity dispersion is based on experimental data retrieved in [30]. In this section we neglect absorption (Im[ ] 0) Ag ε = in order to demonstrate more clearly the bandwidth performance in terms of negative refraction. The effect of realistic losses will be discussed in the next section. The electric field distribution of the layered metamaterial structure is shown in Fig. 1(b)   1(c) and 1(d), respectively. The regions of special interest for our study are the ones in which the angle of transmission takes negative values, i.e., negative refraction, and at the same time a significant portion of the impinging energy is transmitted through the slab. The gray solid lines in the amplitude plot [ Fig. 1(c)] enclose the region in which 0 Δ < and negative refraction is dominant, in order to highlight our region of interest. With our analytical tools, and in particular with the derived condition to localize negative refraction, it is possible to efficiently and rapidly analyze arbitrary configurations and design parameters and localize the intersection between negative refraction and high transmittance, which is particularly interesting in order to improve the frequency and angular bandwidth of operation. Several bands of high transmission can be achieved, in which negative refraction occurs over a broad range of incident angles. Note that negative refraction and transmission can strongly depend on the angle of incident radiation, as it can be clearly seen in Figs. 1(e) and 1(f), in which we calculate and plot the transmission and negative refraction based on the effective anisotropic permittivity tensor given by the approximate model of Eq. (1), where the slab has the same thickness as the layered metamaterial of Fig. 1(a). It is obvious that the homogenized model, based on the effective medium approximation, cannot take into account the inherent nonlocal effects and spatial dispersion of the layered metamaterial, as originally noticed in [32]. On the contrary, these interesting effects are fully taken into account when considering the layered structure in our transmission-line based calculations shown in Figs. 1(e) and 1(f). Using the proposed transmission-line method we can efficiently and rapidly evaluate the negative refraction angle, taking into full account the nonlocal effects of the metamaterial. It is possible to apply the proposed theoretical analysis also to the evanescent portion of the impinging spectrum, in order to analyze the canalization of the evanescent spectrum into propagating waves and the potential for imaging with subwavelength resolution supported by the proposed structure. With this goal in mind, we extend the plot of transmittance of Fig.  1(c) to evanescent incident waves with wavenumbers 0 x k k > in Fig. 2. It is seen that evanescent waves get fully transmitted over a broad range of frequencies and wave numbers, up to the cut-off around , fundamentally determined by the granularity of the layered slab. The proposed analytical method is proven to be a useful tool not only for the prediction of negative refraction operation, but also to analyze the subwavelength resolution potentials of the proposed layered metamaterial hyperlens. It may be used to study and optimize different designs within the desired parameter space range, leading to a better understanding of the mechanisms of negative refraction and canalization of evanescent waves. Our analysis and exploration of the available parameter space, made possible by the efficiency of the proposed analytical method, shows that the bandwidth of negative refraction of the presented metamaterial (region enclosed by the solid gray lines plotted at the transmittance figures above) can be broadened in two different ways: first, dielectric layers with higher permittivity should be used, if possible, relaxing one of the boundaries to obtain negative refraction, i.e.,  Fig. 3(a) are lower compared to the previous case of Fig. 1(c), because of a larger impedance mismatch. Alternatively, the thicknesses of the plasmonic layers may be increased , which leads to an even broader area of negative refraction, as it can be seen in Figs. 3(c) and 3(d). The transmittance, however, is even more affected in this scenario compared to both previous cases shown in Figs. 1(c) and 1(d), due to the occurrence of larger resonance bands and a stronger dependence on the incident angle. A general trade-off exists between bandwidth and transmittance performance of the negative refracted waves.

Effect of losses and their compensation with gain materials
The results presented so far have been limited to lossless plasmonic layers, i.e., absorption has been neglected. We may expect significant changes in the response of the metamaterial structure when realistic losses are included. In the following, we include realistic Ag losses in the plasmonic layers and observe the effect on the negative refraction functionality. Now, the dielectric layers have permittivity 2.5 and 15 alternating layers. The metamaterial is composed of fewer and thinner layers to make sure that the impinging radiation is less dissipated. Moreover, we chose a different ratio of dimensions and permittivity values to achieve improved negative refraction bandwidth performance, applying the method described in the previous section. Realistic absorption is considered by introducing a non-zero imaginary part in the permittivity dispersion of silver (Im[ ] 0) respectively. Very poor transmission is obtained in Fig. 4(a) compared to the lossless case of Fig. 1(c). However, it is interesting that a relatively wide region of negative refraction is still present in Fig. 4(b), which is slightly affected by the introduction of losses, following a similar trend with the lossless response presented in Fig. 1(d). Note that the imaginary part of silver in the frequency region of interest is almost constant and equal to Im[ ] 0.6 This is consistent with recent experiments on structures realizing this effect based on semiconductor layers, which indeed supported negative refraction despite the very low transmission levels due to Ohmic losses [5]. We can conclude that negative refraction still exists for lossy structures and our method can be applied to efficiently analyze them and improve their performance, but the transmission levels are in general strongly affected, making the design of a realistic hyperlens with sufficient thickness quite impractical.
In order to alleviate this severe limitation, it has been proposed to use different semiconductor materials, such as aluminum-doped zinc oxide (AZO), which exhibit plasmonic response at IR frequencies combined with lower losses [33]. As an alternative approach, gain media can be introduced instead of ordinary dielectrics in the layers of the metamaterial structure. This is expected to compensate the inherent losses of the plasmonic material and the transmission will be restored to moderately higher values [24,25]. Here, we assume a gain material with inverse Lorentzian dispersion in the dielectric layers of the structure of Fig. 1(a) Fig. 4(a)]. Obviously this small gain cannot fully compensate the losses in silver, but our optimized design ensures strongly enhanced fields in the gain material, maximizing its effect. Higher transmission through the gain-assisted hyperlens is obtained for normal incidence. Hence, our analysis demonstrates that realistic gain media can indeed partly compensate the losses of plasmonic layers within a moderately wide range of frequencies and incident angles, improving the metamaterial negative refraction performance.

Nonlinear operation
To further extend the reach of these concepts and highlight the efficiency and generality of the proposed analytical method, in this section we consider the introduction of third-order (3) χ nonlinear effects in the dielectric layers of the lossless structure shown in Fig. 1(a) induced by the incident radiation intensity. The local fields inside each layer are almost uniform, due to the subwavelength thickness and, therefore, with very good approximation we can consider their average value to compute the nonlinear permittivity of each layer as a function of input intensity, as commonly done in similar configurations [28,37]. For simplicity, we neglect in this analysis the metal losses, but similar nonlinear response, yet with smaller excursions, is expected also when losses are considered. , where different behavior is obtained: positive refraction, epsilon-near-zero behavior, with an angle of refraction along the normal, like in a hyperlens, and negative refraction, respectively. Such high input intensities may be achieved with ultrafast pulsed laser sources.
We also computed the transmittance and angle of transmission for the same layered metamaterial at 311nm λ = , for which now the plasmonic layers have permittivity 0.1