## Abstract

Metamaterials with hyperbolic dispersion based on metallic nanorod arrays provide a flexible platform for the design of bio- and chemical sensors and nonlinear devices, allowing the incorporation of functional materials into and onto the plasmonic metamaterial. Here, we have investigated, both analytically and numerically, the dependence of the optical response of these metamaterials on refractive index variations in commonly used experimental sensing configurations, including transmission, reflection, and total internal reflection. The strategy for maximising refractive index sensitivity for different configurations has been considered, taking into account contributions from the superstrate, embedding matrix, and the metal itself. It is shown that the sensitivity to the refractive index variations of the host medium is at least 2 orders of magnitude higher than to the ones originating from the superstrate. It is also shown that the refractive index sensitivity increases for higher-order unbound and leaky modes of the metamaterial sensor. The impact of the transducer’s thickness was also analysed showing significant increase of the sensitivity for the thinner metamaterial layers (down to few 0.01 fraction of wavelength and, thus, requiring less analyte) as long as modes are supported by the structure. In certain configurations, both TE and TM-modes of the metamaterial transducer have comparable sensitivities. The results provide the basis for the design of new ultrasensitive chemical and biosensors outperforming both surface plasmon polaritons and localised surface plasmons based transducers.

© 2015 Optical Society of America

## 1. Introduction

The refractive index sensitivity of plasmonic and waveguided resonances forms the basis of commercial and newly emerging optical sensing techniques for label-free biosensing and chemical identification [1–4] as well as active nanophotonic components [5, 6]. In the former class of applications, the presence of an analyte substance modifies the eingenmodes of the nanostructure, a change that can be detected by a shift of the resonant wavelengths of the structure or by changes of transmitted or reflected light intensity. In the latter, the refractive index changes are induced by external stimuli, such as temperature, acoustic pressure, external static electric or magnetic field, or indeed optical field via nonlinear effects in the surrounding dielectric or metal [6]. The strong modification of the optical response in plasmonic nanostructures arises from the strong confinement of the electromagnetic field near the metal/dielectric interface. Both sensing and active nanophotonic devices can make use of macroscopic thin metal films or nanostructured surfaces where surface electromagnetic waves called surface plasmon polaritons (SPPs) propagate, or nanoparticles and their assemblies supporting localised surface plasmons (LSPs) [2, 3, 7].

Surface plasmon resonance (SPR) biosensors use SPP waves for the detection of binding events, lifetime measurements or molecule concentration, based on the attenuated total internal reflection (ATR) configuration [1]. Due to a strong field confinement of SPPs, sensing limits are greatly enhanced, exceeding 3,000 nm per refractive index unit (RIU) [8] and can be even further boosted via phase-sensitive interferometry [9]. Nonetheless, SPP-based techniques have restrictions in detecting small molecule analytes, typically smaller than 500 Da, where Da is the atomic mass unit, making it problematic for modern nanoscale chemical and biochemical tasks [10]. An alternative route is to use LSP modes on plasmonic nanoparticles that provide an even stronger field confinement and, thus, are more sensitive to smaller-size molecules [2, 11]. However, the overall sensitivity provided by LSPs is typically orders of magnitude smaller than for SPPs, not exceeding 100-300 nm/RIU [2, 12].

Recently, plasmonic metamaterials have been demonstrated to provide the record refractive index sensitivity for biosensing, ultrasound detection and high effective Kerr-type nonlinearities [13]. In particular, the class of anisotropic metamaterials based on arrays of strongly interacting, aligned plasmonic nanorods, exhibits hyperbolic dispersion [14, 15], with one negative (${\epsilon}_{z}$) and two positive effective permittivity tensor components (${\epsilon}_{\text{x,y}}$), leading to a metamaterial with hyperbola-shaped isofrequency contours expressed as ${k}_{\text{x}}^{2}/{\epsilon}_{\text{z}}+{k}_{\text{y}}^{2}/{\epsilon}_{\text{z}}+{k}_{\text{z}}^{2}/{\epsilon}_{\text{x,y}}={(\omega /{c}_{0})}^{2}$, where $\omega $ is the angular frequency, ${c}_{0}$ is the speed of light, and ${k}_{\text{x,y,z}}$ are the wavevector components along Cartesian directions. This unique isofrequency surface enables a plethora of applications, from guiding and imaging beyond the diffraction limit [16, 17], to enhanced nonlinearities [5, 18], and chemo- and biosensing [5, 6, 19]. Additionally, selected resonances of plasmonic nanorod metamaterials have been shown to exhibit a strong sensitivity to the thickness change of the dielectric load [20], ultrasensitive detection of ultrasound [21], and ultrafast sub-ps response times due to optical nonlocality [18].

In this work, we present a comprehensive analysis of the dependence of the optical response of an anisotropic nanorod-based metamaterial on refractive index changes of analyte above and between the nanorods, in order to develop strategies for optimising its sensing properties. We consider only the wavelength range of hyperbolic dispersion and study reflection, transmission, and total internal reflection modalities of operation, examining the role of perturbations of both the real and imaginary parts of the refractive index in the superstrate, the host medium where the nanorods are placed, as well as from the metal itself. All components of the permittivity tensor determine the formation of these modes and, thus, their refractive index sensitivities. It is shown that the refractive index variations of the host medium most strongly influence the optical response of the metamaterial, providing highest sensitivity, outperforming both SPP and LSP-based biosensors. The sensitivity is greatly enhanced for higher-order modes of the metamaterial slab due to both the spectral positions of these modes, closer to the resonance of some permittivity tensor components at short wavelengths, and the increase of the field gradients with increasing mode order. This resonant contribution is significant for large nanorod filling factors, leading to comparable refractive index sensitivities of TE and TM-modes of the metamaterial slab. For lower metal filling factors, the sensitivity of TM-modes prevails. The role of the transducer’s thickness is also investigated showing an increased sensitivity for thinner metamaterial layers.

## 2. Numerical model

We have studied the refractive index sensing capabilities of a metamaterial in several configurations where the anisotropic metamaterial slab is illuminated in both conventional and total internal reflection conditions. Changes in the intensity transmitted and/or reflected upon modifications of the refractive index and absorption of the analyte are monitored as a function of wavelength [Fig. 1(a)]. The metamaterial is considered to be composed of an array of gold nanorods arranged periodically [Fig. 1(a)]. The thickness of the metamaterial slab (height of the nanorods) is *l,* and it is sandwiched between a substrate and superstrate with refractive indices of *n*_{sub} and *n*_{sup}, respectively.

The Maxwell-Garnett (MG) approximation was followed to derive the tensor of the effective permittivity of the anisotropic metamaterial [22]. Considering an array of rods in the x-y plane, the effective permittivities for ordinary and extraordinary axes take the form

*d*being the period of the array and

*r*is the nanorod radius, ${\epsilon}_{\text{Au}}$and ${\epsilon}_{\text{h}}$ are the permittivities of gold [23] and host medium, respectively, and $\tilde{\epsilon}=({\epsilon}_{\text{Au}}+{\epsilon}_{\text{h}})/2$. Note that the same period is considered in both

*x*and

*y*directions, thus, ${\epsilon}_{\text{x}}^{\text{eff}}={\epsilon}_{\text{y}}^{\text{eff}}$. The MG approximation breaks down for wavevectors close to the Brillouin zone boundary, however for a typical period of 100 nm considered here, the 1st Brillouin zone boundary is close to ${k}_{\text{x}}$ ≈31 μm

^{−1}, which is far from the investigated regime (${k}_{\text{x}}$ < 17 μm

^{−1}). Additionally, the wavelength range where $\mathrm{Re}({\epsilon}_{\text{z}}^{\text{eff}})$ vanishes (ENZ regime) requires special considerations; the ENZ and elliptic dispersion regime will not be considered in this work [24, 25].

To examine the refractive index sensitivity of the optical properties of the metamaterial, the effective permittivity has been designed to achieve hyperbolic dispersion throughout the visible spectral range. Figure 1(b) shows the real and imaginary parts of the components of the effective permittivity tensor for a typical nanorod array parameters and a water-like permittivity for the host medium. The validity of the EMT approach used here was checked with full vectorial 3D numerical simulations. For this geometry, ${\epsilon}_{\text{z}}^{\text{eff}}$, the permittivity component for light polarized along the nanorod axis, is negative for wavelengths longer than approximately 475 nm [Fig. 1(b)], while the transverse permittivity${\epsilon}_{\text{x,y}}^{\text{eff}}$, for light polarized perpendicular to the nanorod axes, is always positive, exhibiting a resonant behaviour near a wavelength of 540 nm, where the imaginary part of the permittivity reaches a maximum. For wavelengths above 540 nm the transverse component of permittivity’s imaginary part is small relative to its real counterpart and does not exhibit any resonances.

The reflectance and transmittance spectra of the metamaterial transducer have been calculated for angles of incidence (AOI) 0° ─ 90° using the transfer matrix method [26] and taking into account a glass substrate (*n*_{sub} = 1.5), a water superstrate (*n*_{sup} = 1.33), and Au nanorods of height *l* = 400 nm. Figures 1(c) and 1(e) along with Figs. 1(d) and 1(f) show the transmittance and reflectance dispersions for both TM and TE-polarizations, respectively. Unbound modes that leak in both substrate and superstrate exist below the critical angle *θ*_{c} appear as maxima (minima) in the transmittance (reflectance), while “leaky” modes, homogeneous in the substrate, are only present above *θ*_{c} and their effect is only visible in the reflectance of the structure. These modes are quantized solutions of the wavevector (${k}_{z}$) due to the 1D confinement of the metamaterial slab in the *z*-direction. For the glass/water (substrate/superstrate) interface, total internal reflection (TIR) occurs at *θ*_{c} = 62.46°, above which there is no transmission, and thus transmission-detection sensing is possible only below TIR angles.

In order to compare the sensing capabilities of different transducers and geometries, it is convenient to introduce two figures of merit (FoMs): FoM_{λ} is to characterize the spectral shift induced by the refractive index changes of an analyte, while FoM_{I} is to characterize induced intensity variations of the transmitted or reflected light. We define FoM_{λ} = (Δ*λ*/δλ)/Δ*n*, where Δ*λ* is the resonance shift of a metamaterial resonance to a refractive index change Δ*n*, and δλ is the reference full-width at half maximum of the resonance [27]. This definition accounts simultaneously for both the wavelength shift of a given mode per refractive index change, and the sharpness of the resonance. For intensity measurements, FoM_{I} = (Δ*I*/*I*_{0})/Δ*n*, where Δ*I* is the change of the transmitted or reflected intensity corresponding to a refractive index change *Δn*, and *I*_{0} is the initial intensity [27]. While intensity measurements are simpler to implement than spectral shift measurements, in some cases the latter may provide better sensitivity [19]. Both FoMs represent the sensitivity to an average analyte refractive index changes rather than to local binding events of sensed molecules to nanorods. We will use these FoMs to evaluate the sensing capabilities of the metamaterial in the numerical results obtained below.

## 3. Effective permittivity sensitivity to refractive index variations of constituents

We now examine the sensitivity of the effective permittivities ${\epsilon}_{\text{x,y}}^{\text{eff}}$ and${\epsilon}_{\text{z}}^{\text{eff}}$, which determine the optical response of the metamaterial, on the refractive index changes of the metamaterial’s constituents ${n}_{\text{h}}=\sqrt{{{\epsilon}^{\prime}}_{\text{h}}+i{{\epsilon}^{\u2033}}_{\text{h}}}$and${n}_{\text{Au}}=\sqrt{{{\epsilon}^{\prime}}_{\text{Au}}+i{{\epsilon}^{\u2033}}_{\text{Au}}}$, for the host medium and Au, respectively. The modification of the refractive index of the metal can be achieved for example in nonlinear optical experiments, when the metamaterial is under femtosecond optical excitation due to the Kerr-type third-order nonlinearity of Au, but can also be induced by environmental change in temperature or pressure [19, 25]. The refractive index of the host medium can be changed in sensing experiments when analyte is incorporated between the nanorods or if nonlinear or temperature effects are important in the host dielectric. To evaluate this sensitivity, we study the partial derivative of Eqs. (1) and (2) with regards to the permittivity components of each constituent, obtaining:

*p*[Eqs. (5) and (6)]. On the other hand, ${\epsilon}_{\text{x,y}}^{\text{eff}}$ exhibits the strongest changes close to its resonance at 540 nm [Fig. 1(b)] and strongly decreases for lower nanorod filling factors. Beyond the resonance at longer wavelengths, inspecting the dependence of ${\epsilon}_{\text{x,y}}^{\text{eff}}$ on the host medium refractive index, one notices that it is always exceeds the sensitivity of ${\epsilon}_{\text{z}}^{\text{eff}}$[Figs. 2(a) and 2(b)], while both are similar at low nanorod filling factors. The ${\epsilon}_{\text{z}}^{\text{eff}}$ sensitivity to the refractive index of the plasmonic material dominates at long wavelengths, while the resonant behaviour of ${\epsilon}_{\text{x,y}}^{\text{eff}}$ proves in general more strongly sensitive to the host medium refractive index. To examine the impact of filling factor

*p*more extensively, the dispersion diagrams of the derivative of ${\epsilon}_{\text{x,y}}^{\text{eff}}$ with respect to ${{\epsilon}^{\prime}}_{\text{h}}$ are presented in Figs. 2(e) and 2(f); the derivative of ${\epsilon}_{\text{x,y}}^{\text{eff}}$ with respect to ${{\epsilon}^{\u2033}}_{\text{h}}$ can then be trivially extracted from Eq. (3). All refractive index sensitivity dispersions are dominated by a dispersive-shape resonance, which red-shifts and increases in magnitude with increasing nanorod filling factor. Both the dispersive behaviour and spectral shift observed in Figs. 2(e) and 2(f) can be linked to the spectral sensitivity of the resonance in ${\epsilon}_{\text{x,y}}^{\text{eff}}$ shown in Fig. 1(b), which red-shifts with both an increase in ${\epsilon}_{\text{h}}$ and nanorod filling factor. The same calculations were repeated for ${{\epsilon}^{\prime}}_{\text{Au}}$ showing a trend similar to Figs. 2(e) and 2(f), but with the resonant sensitivity approximately one order of magnitude smaller (data not shown).

## 4. Mode frequency dependence on the refractive index of analyte

All the components of the effective permittivity tensor are important for defining the behaviour of TM-modes of the metamaterial and, thus, their sensing capabilities. At the same time, only ${\epsilon}_{\text{x,y}}^{\text{eff}}$ components define TE modes. To get better insight into the sensing capabilities of the unbounded, leaky, and waveguided modes of the metamaterial transducer, let us consider an approximate analytic model for extraordinary modes in the case of the anisotropic metamaterial layer [17]. We restrict ourselves primarily to TM-modes, while TE-polarization can be treated similarly. Neglecting phase shifts at the metamaterial’s boundaries, we obtain a simple analytical approximation for the wavevector component ${k}_{x}$ of TM-modes supported by the metamaterial slab as [17]

*qπ*/

*l*emerges from the transversal confinement of the wavevector component perpendicular to the metamaterial’s interfaces with the integer

*q*> 0 referring to the mode number. Unbounded, leaky, and waveguided modes then satisfy the condition${k}_{\text{x}}^{}<{n}_{\text{sub}}{k}_{0}^{}$, ${n}_{\text{sub}}{k}_{0}^{}\le {k}_{\text{x}}^{}<{n}_{\text{sup}}{k}_{0}^{}$, and${k}_{\text{x}}^{}\ge {n}_{\mathrm{sup}}{k}_{0}^{}$, respectively, with the mode frequency

*q*and ${k}_{x}$, immediately follows as:

The mode position dependence with respect to the real part of the constituents’ permittivities is plotted in Fig. 3 within the 1st Brillouin zone to ensure EMT validity. As the mode number increases from *q* = 1 to *q* = 5 there is an order of magnitude increase in the sensitivity of the mode frequency to the host medium refractive index variations [Fig. 3(a)] and a two orders of magnitude increase to that of Au [Fig. 3(c)]. The superior refractive index sensitivity of high-order modes is a consequence of their spectral position close to the resonance in ${\epsilon}_{\text{x,y}}^{\text{eff}}$ [Fig. 2] and the increased field gradients inside the metamaterial for higher-order modes. These gradients are determined by the mode’s spatial frequency *qπ*/*l*, increasing with increasing *q* value or decreased sensor thickness *l*. Interestingly, this leads to an increase in the mode sensitivity as the mode shifts to shorter wavelengths, a trend opposite to that observed with conventional SPR or LSPR transducers. In particular, shifting the resonance frequency of the fundamental mode (*q* = 1) from 0.5 eV to 1.5 eV, by decreasing the metamaterial layer thickness from 500 nm to 130 nm, results in a ~500% increase in sensitivity as monitored in both transmittance and reflectance at normal incidence. An increase in the nanorod filling factor *p*, all other parameters being kept constant, leads to a decrease in the sensitivity of the mode position. Again, this is reminiscent of the increased mode delocalization within the metamaterial with decreasing frequency. Comparing Figs. 3(a) and 3(c) we can also observe that, for a given mode *q*, the rate of change of the resonance position is stronger with respect to changes in the real part of the host permittivity compared to changes in the real part of Au permittivity, $\partial {\omega}_{\text{q}}/\partial {{\epsilon}^{\prime}}_{\text{h}}\gg \partial {\omega}_{\text{q}}/\partial {{\epsilon}^{\prime}}_{\text{Au}}$. This is the result of the modal field distribution being mainly present in the host medium.

The mode frequency sensitivity with respect to${{\epsilon}^{\u2033}}_{\text{h}}$ and${{\epsilon}^{\u2033}}_{\text{Au}}$is not shown here but was also examined, giving a similar trend to Figs. 3(a) and 3(c), but with a one-to-two orders of magnitude smaller sensitivity to the variations in either or${{\epsilon}^{\prime}}_{\text{Au}}$. This indicates that the real parts of both the host medium and Au will affect sensing more than the imaginary counterparts. Monitoring the analyte’s refractive index changes at normal incidence provides the lowest sensitivity with regards to the modes’ spectral shift [Fig. 3].

Next, the impact of the thickness of the metamaterial transducer, *l*, was examined. Figures 3(b) and 3(d) show the *q* = 1 mode frequency sensitivity with respect to${{\epsilon}^{\prime}}_{\text{h}}$ and${{\epsilon}^{\prime}}_{\text{Au}}$as *l* varies from 300 nm to 600 nm. In both cases, as the waveguide thickness increases, the sensitivity drops, in agreement with the mode position shift to lower frequencies. A similar behaviour is observed for higher-order modes. This behaviour can be clarified more explicitly by examining the normal incidence behaviour of the mode resonance frequency, simplifying Eq. (9) to$\partial {\omega}_{\text{q}}/\partial ({{\epsilon}^{\prime}}_{\text{h}},{{\epsilon}^{\prime}}_{\text{Au}})=-[{c}_{0}\pi {({\epsilon}_{\text{x,y}}^{\text{eff}})}^{-3/2}/2l]\partial {\epsilon}_{\text{x,y}}^{\text{eff}}/\partial ({{\epsilon}^{\prime}}_{\text{h}},{{\epsilon}^{\prime}}_{\text{Au}})$, clearly showing the inverse proportionality of the mode resonance sensitivity to the metamaterial thickness *l* for unbound modes. Thus, for a given mode number, a thinner transducer will be more sensitive. This result has been verified using numerical simulations as long as the mode considered had a frequency not exceeding that corresponding to a free-space wavelength of 650 nm, or beyond the range of high losses due to $\mathrm{Im}({\epsilon}_{\text{x,y}}^{\text{eff}})$ [Fig. 1(b)].

## 5. Numerical simulations

In order to show the complete pattern of the metamaterial behaviour with the refractive index modifications of its constituents, which are used to describe the sensing capabilities of the metamaterial and its active nanophotonic properties, we have also numerically studied the modification of the metamaterial resonances with changes of the complex refractive index of the superstrate, *n*_{sup}, and embedding medium, *n*_{h}. For all numerical calculations the refractive indices were the same before introduction of the analyte. These changes may originate from an unknown analyte or a material with nonlinear or electro-optical properties used for switching/modulation purposes. It may also be represented by Au itself as a source of nonlinearity for all-optical control or as temperature sensor. We examine how the latter two parameters translate into the sensitivity of the metamaterial transducer for the detection of the bulk refractive index changes of an analyte. We first consider a non-absorptive analyte (Section 5.1) for which only the real part of the refractive index changes. Then, the effect of changes in the absorption of analyte is studied (Section 5.2). Finally, the impact modifications of the refractive index of Au have on the optical properties of the metamaterial is discussed in Section 5.3. For all simulations, the sensing geometry was kept the same as in Fig. 1(b) with a nanorod height fixed at 400 nm.

#### 5.1 Sensing refractive index variations

We start with the nonabsorptive case, corresponding to the situation of an analyte consisting of nonresonant molecules. In this situation, changes in the refractive index may be induced through Kerr nonlinearities of the host medium or superstrate, electro-optical, thermo-optical or pressure effects. Figures 4(a)-4(c) show the reflectance sensitivity (FoM_{I}) for different wavelengths and angles of incidence for TM-polarization, when changes of the refractive index originate from either the superstrate (Δ*n*_{sup}), host medium (Δ*n*_{h}), or both simultaneously (Δ*n*_{b}). The FoM_{I} has a dispersive behaviour, changing sign in the vicinity of the mode resonances [Figs. 4(b) and 4(c)]. This behaviour is observed as a result of a simultaneous change in both the intensity (Δ*I*) and wavelength (Δ*λ*) of the metamaterial modes. However, when Δ*I* has a dominating role, the FoM_{I} retains its sign (for example, Fig. 4(a) for the angular range between 0° and 50°). From Fig. 4(a), it is clear that the strongest variations of the optical response are observed in the vicinity of the modes of the metamaterial transducer, but with a sensitivity maximised near the critical angle *θ*_{c}. The reason is that when the refractive index of the superstrate is changed by Δ*n*_{sup}, the critical angle itself will be affected leading to a strong sensitivity along the superstrate light line, the so-called near cut-off regime.

A different behaviour is observed for the variations of the refractive index of the host medium Δ*n*_{h} [Fig. 4(b)]. In this case, the sensitivity of leaky modes is higher than that of unbound modes. When both the refractive index of the host medium and the superstrate change, a combination of the above individual cases is observed, although not just a simple addition of individual sensitivities since modes from the metamaterial layer are penetrating the superstrate medium [Fig. 4(c)]. Not surprisingly, the results show that a small change in the refractive index of the superstrate affects the sensitivity only at angles very close to *θ*_{c}, while for any other incidence angle the host medium has a dominating role. Both TM [Figs. 4(a)-4(c)] and TE [Figs. 4(d)-4(f)] modes of the metamaterial transducer have comparable sensitivities if the analyte is incorporated between the rods. This is not surprising giving the nature of the anisotropic waveguided modes determined by all components of the effective permittivity tensor. For low filling factors, the role of TM-modes increases, when the ${\epsilon}_{\text{x,y}}^{\text{eff}}$ sensitivity becomes smaller. The changes in the refractive medium of the analyte between the nanorods results in the effective permittivity changes, affecting both TM and TE-modes.

Figures 4(g)-4(i) show the sensitivity for selected wavelengths and angles of incidence. The spectral dependence of FoM_{I} at normal incidence [Fig. 4(g)] reveals that the mode with the highest sensitivity (*q* = 3) reaches FoM_{I} exceeding 400, with the host medium being mainly responsible for this enhanced sensitivity. As mode number increases from *q* = 1 to *q* = 3, the FoM_{I} also increases in agreement with the analytical examination [Fig. 3(a)]. However, the *q* = 4 mode, located close to 2 eV, shows a decreased sensing capability contrary to the analytical scenario. This is due to the increased absorption of ${\epsilon}_{\text{x,y}}^{\text{eff}}$ [Fig. 1(b)] at wavelengths below 650 nm (> 1.9 eV), which are not tracked with the analytical model discussed in Section 4. Figure 4(h) examines the angular dependence of the sensitivity for the modes with the highest sensitivities: (*q* = 3) at 1.7 eV for TM-polarization and *q* = 2 at 1.4 eV for TE-polarization. These plots show a typical dispersive behaviour of the mode position, with high sensitivities observed on both sides of the resonance. As expected, the highest FoM_{I} is observed above *θ*_{c} for leaky modes. This behaviour is better observed in Fig. 4(i) which depicts the global maximum FoM_{I} for TM-polarization, for the *q* = 4 mode located at a wavelength of about 645 nm and for an angle of incidence of about 79°, above the critical angle. Similarly for the TE-case, the highest FoM_{I} is observed for the *q* = 2 mode for a wavelength close to 870 nm and at 84°. Only variations in the superstrate refractive index lead to high sensitivity lies just below *θ*_{c}, at 58.8° for reasons already explained above. In the following sections, we will concentrate only on the sensing with TM-modes, pointing out that in the case of absorption variation sensing, discussed below, TE-modes also have comparable but slightly lower FoM_{I}.

The *q* = 1 mode exhibits the highest resonant shift reaching almost 4000 nm/RIU, while the highest FoM_{λ} ≈167 is observed for the mode *q* = 3, outperforming both SPR and LSPR sensors which have typical FoM_{λ} around 23 [2] and 8 [2, 28], respectively. The origin of such high FoM_{λ} is in the influence of the analyte not only directly on the electomagnetic mode properties, as usual for conventional sensors, but also on the effective permittivity of the metamaterial which is connected to the host medium refractive index [Fig. 2], on the microscopic level, this is the result of plasmon-plasmon interactions within the nanorod array [20, 29, 30]. The intensity figure of merit (FoM_{I}) is proportional to (Δ*R*/Δ*n*)/*R _{0}* and can be directly observed in Fig. 4. Depending on the wavelength and AOI, FoM

_{I}can be as high as 25,000, significantly higher that any LSP- and SPP-based [27] sensors [Fig. 4(i)]. It should be noted that given the standard definition of the FoM

_{I}commonly used in the literature, which includes a normalization to the initial reflected or transmitted intensity

*I*

_{0}, the highest values of FoM

_{I}occur in spectral regions closed to the metamaterial’s resonances where

*I*

_{0}is small, the absolute variations of the intensity, however, have the same angular and spectral dependences as FoM

_{I}. Both FoM

_{I}and FoM

_{λ}can be further tailored and enhanced with geometrical parameters of the transducer, with FoM

_{λ}reaching up to 300 has been experimentally demonstrated [18].

#### 5.2 Sensing absorption variations

Turning to the impact of the absorption variations on the metamaterial transducer response, Figs. 5(a)-5(c) show the reflectance variations for different wavelengths and angles of incidence for TM-polarized incident light with changes in the imaginary part of the refractive index of either the superstrate (Δ*n*_{sup}), the host medium (Δ*n*_{h}), or both simultaneously (Δ*n*_{b}). Similar to changes in the refractive index [Figs. 4(a)-4(c)], the sensitivity of the transducer originates from the modal dispersion of the metamaterial slab. Interestingly, an increase in the absorption leads to an increased reflection for some angles, while others behave oppositely. Again, an increased sensitivity is observed for higher mode order moving from *q* = 1 to 3, with the decreased sensitivity for the *q* = 4 mode due to losses [Fig. 5(d)].

Similarly to the nonabsorbing case, the mode with the highest FoM_{I} (*q* = 3) depicted in Fig. 5(d), showed a resonant shift of more than 700 nm/RIU, while the *q* = 1 mode exhibits the highest resonant shift reaching values close to 4,000 nm/RIU. Tracking the *q* = 3 mode at various AOI [Fig. 5(e)], its sensitivity is enhanced around both 20° and 85°, where the mode is leaky. The FoM_{λ} of the *q* = 3 mode in Fig. 5(d) is close to 167, similar to the nonabsorbing analytes discussed in Section 5.1. However, its FoM_{I} reaches a value of almost 1,000, or at least twice higher than the corresponding one in Fig. 4(g) for the same mode (*q* = 3) since the influence of the imaginary part of the permittivity is on reflection is stronger than the real one. The highest FoM_{I} in Fig. 5(f) exceeds 24,000, which is a significant 25% higher than the TM-FoM_{I} for sensing nonabsorbing analytes [Fig. 4(i)].

#### 5.3 The effects of refractive index variations in Au

Lastly, the dispersive sensitivity of the metamaterial for refractive index variations of the gold nanorods (Δ*n*_{Au}) was investigated. These can be induced either optically, via the Kerr-type nonlinearity of the metal, or via thermo-optical effects. Figures 6(a) and 6(b) depict the reflectance variations when either the real or the imaginary part of the refractive index of Au varies. Note that the variations were set to be wavelength independent, which would correspond to different electronic/lattice temperatures at different wavelengths, to allow for comparison with the previous section but without altering the generality of the results. The sensitivity is smallest below the superstrate light line. More specifically, low-*q* modes exhibit lower sensitivities compared to high-*q* modes, located close to 2 eV. Again, material losses at higher frequency eventually limit the sensitivity of these modes as well. Interestingly, the sensitivity observed here is smaller than those resulting from changes in either host medium or superstrate discussed above. Examining again the mode with the highest sensitivity (*q* = 3) at normal incidence as shown in Figs. 6(c) and 6(d), one can clearly see that FoM_{I} is approximately one order of magnitude less than for the scenarios in Figs. 4(g) and 5(d). The FoM_{λ} of the aforementioned mode to Au permittivity variations is 168, very close to FoM_{λ} observed for sensing of the analytes above. At the same time, the intensity figure of merit for Au permittivity modification is much smaller than for the sensing of analytes giving a maximum value of only 1,200 [Fig. 6(e)]. It should be mentioned that the resonant shift per RIU is only a few nm regardless of the mode chosen; a result that applies for variations in both the real and imaginary part of the refractive index of Au.

## 6. Conclusion

We have analyzed, both analytically and numerically, the performance of a metamaterial transducer in sensing applications varying different geometrical and optical parameters such as the refractive indices of host medium, superstrate, and Au as well as the filling factor and height of nanorods. The analytical model shows the increased refractive index sensitivities for higher-order modes (*q*) and decreasing thickness (*l*) of the transducer. The numerical model captured the same trend as long as the excited modes are away from the resonant absorption associated with ordinary modes of the metamaterial. The sensitivity to changes in the refractive index of a host medium is much stronger than that of the superstrate refractive index, confirming the important role of the interaction between plasmonic resonances of the nanorod assembly in the sensing performance of the metamaterial. Furthermore, the sensitivity is increased when sensing the analyte’s absorption variations.

The variations of reflection and transmission of the metamaterial to changes in the refractive index of Au, which can be induced either via Kerr-nonlinearity of metal or thermal effects, have been shown to be more modest. In the studied purely hyperbolic regime, the metamaterial’s response was found to be smaller to Au permittivity modifications (the mode resonance shifts of a few nm/RIU) than that resulting from the similar changes of the refractive index of either the superstrate or the host medium (reaching many hundreds of nm/RIU). Furthermore, the reflection variations were an order of magnitude higher with refractive index changes of a dielectric (reaching several tens of percent) than for the same variations of Au refractive index.

Interestingly, in certain configurations, both TE and TM-modes of the metamaterial transducer have comparable sensitivities, opening up opportunities for polarization multiplexing in sensing experiments. The metamaterial transducer is shown to provide enhanced sensing performance compared to both SPP and LSP-based geometries presented in the literature to date, both in terms of FoM_{λ,I} and nm/RIU characteristics achievable. The above conclusions are valid in the hyperbolic dispersion regime and may be modified if the operating wavelength overlaps the ENZ regime of the metamaterial dispersion. The results can be used as a design strategy to enhance flexibility of ultrasensitive transducers for bio- or chemical sensors or nonlinear photonic devices based on plasmonic hyperbolic metamaterials.

## Acknowledgments

This work was supported, in part, by EPSRC (U.K.), the ERC iPLASMM project (321268), and NSF Materials World Network program (grant # DMR-1209761). A.Z. acknowledges support from the Royal Society and the Wolfson Foundation. G.W. acknowledges support from the EC FP7 project 304179 (Marie Curie Actions).

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