1. Introduction

A flurry of recent theoretical and experimental activity has confirmed the idea first proposed nearly four decades ago that the refractive index of a medium could be negative as well as positive [1]. The first demonstration of negative index behavior was at microwave frequencies [2]. More recently, material systems closer to visible wavelengths that exhibit negative index properties have been reported, which include periodic structures such as photonic crystals [3], nano-patterned noble metal particles [4] or their complements [5, 6], as well as bulk metamaterials containing a random distribution of nano-particulates [7, 8]. All of these designs targeted a fixed wavelength for achieving negative index behavior with the exception of [8], where reconfigurability of the metamaterial properties was realized using liquid crystal as the host medium containing randomly dispersed coated dielectric spheres as guests.

In this paper, we introduce a new and novel reconfigurable metamaterial that incorporates a superstrate and a substrate of nematic liquid crystal onto a conventional negative-index meta-material (NIM) which utilizes a combination of magnetic resonators and metal films [9]. We demonstrate that the effective refractive index of the structure can be tuned from negative — through zero — to positive values, as the dielectric constant of the liquid crystal is varied. We also demonstrate that the bandwidth over which the metamaterial exhibits a negative index behavior can be changed (i.e., increased or decreased) by properly tuning the liquid crystal layers.

2. Liquid crystal properties and analysis methodology

In an aligned nematic liquid crystal, linearly polarized light incident as an extraordinary wave will see a permittivity ε_LC given by [10]

where ε _∥ and ε _⊥ are the respective permittivities for light polarized parallel and perpendicular to the director axis n̂. It is important to note that ε_LC is independent of frequency and only depends on the director axis orientation angle θ with respect to the optical wave vector k ⁱ. To date, nematic liquid crystal with very large dielectric anisotropy (or birefringence) has been synthesized (ε _∥ ≈ 4, ε _⊥ ≈ 2, and the resulting tuning range for the refractive index Δn ≈ 0.6) over the visible-infrared spectrum. In this spectral region, micron-thick nematic liquid crystal layers are nearly lossless. Studies over the last two decades have also conclusively demonstrated their unusually large electro- and all-optical (i.e., nonlinear-optical) response associated with the field induced director axis reorientation. These unique optical properties, in addition to their compatibility with almost all technologically important optoelectronic materials and their fluid nature, make them prime candidates for incorporation into nanostructured (electrically or all-optically) tunable materials/devices.

The optical properties of the metamaterials in this study are analyzed using a rigorous full-wave electromagnetic scattering analysis technique known as the finite-element boundary-integral method [11]. Electric field values in a single unit cell of an infinitely periodic structure are determined by imposing periodic boundary conditions in the computational domain. Once the complex reflection and transmission coefficients are determined from the numerical analysis, the effective index of refraction n = n′ + in″ of a homogenized layer can be unambiguously determined from well-established inversion procedures [12, 13]. In the following numerical analysis, the liquid crystal is treated approximately as a homogeneous isotropic dielectric layer with a relative permittivity of ε_LC.

 

Fig. 1. A two-dimensional metamaterial with thick liquid crystal superstrate and substrate layers having a reconfigurable index of refraction.

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3. Reconfigurable near-IR metamaterial

Figure 1 shows a single unit cell of a two-dimensional periodic metamaterial structure with reconfigurable negative-zero-positive index of refraction. The structure is infinite in the ±ŷ directions and periodic with period p in the ±x̂ directions. This is a slightly modified version of a near-infrared (near-IR) metamaterial geometry previously reported in [9]. A magnetic resonator comprises two strips of silver of width w and thickness t separated by a thin layer of alumina of thickness d. The negative permittivity needed for negative-index behavior is provided by thin silver films of thickness t_f bounding the periodic array of magnetic resonators from the ±ẑ directions. Finally, the space between neighboring magnetic resonators is filled with silica.

Unlike in [9] where thin protective layers of silica were placed over silver films, here a layer of liquid crystal with thickness t_LC is used both as the superstrate and the substrate. Finally, the entire metamaterial is sandwiched between thick glass substrate and superstrate layers which, for modeling purposes, are treated as two half-spaces. A monochromatic light wave with the electric field polarized in the x̂ direction is assumed to illuminate the structure at normal incidence propagating in the +ẑ direction.

The magnetic resonators can cause the effective permeability μ = μ′ + iμ″ to deviate from unity and possibly reach negative values. In contrast, control over the effective permittivity is achieved mainly through an averaging effect, i.e., by averaging over the permittivities of constituent components of the metamaterial. Varying ε_LC will have a primary influence on the effective permittivity ε =ε′ + iε″ of the metamaterial. Hence, it follows that positive, zero, or negative values of ε′ properly combined with magnetic resonances can produce any desired positive, zero, or negative value of n′.

Figure 2 shows plots of the three effective parameters n, ε, and μ for a near-IR version of the metamaterial depicted in Fig. 1. The geometrical parameters of the base NIM design are the same as those used in [9], i.e., p = 600 nm, w = 300 nm, t = 30 nm, d = 40 nm, and t_f = 20 nm. The thickness of the liquid crystal layers is assumed to be t_LC = 200 nm. The silica, alumina, and glass are assumed to have refractive indices equal to 1.445, 1.62, and 1.50, respectively. For the silver, a dielectric function based on published experimental results [14] was employed.

 

Fig. 2. Effective parameters of the reconfigurable near-IR metamaterial for different values of ε_LC: (a) n′, (b) n″, (c) ε, and (d) μ with respect to wavelength.

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Figure 2(a) shows n′ plotted for five different values of ε_LC in the range 2 ≤ ε_LC ≤ 3. With ε_LC = 2, the metamaterial exhibits a negative index band between 1.37 μm and 1.47 μm. As ε_LC is increased, it is seen that the negative index band decreases from shorter wavelengths toward the longer wavelength side, which remains fixed as long as the negative index band is present. Therefore, as ε_LC is increased, the negative index band diminishes from the shorter wavelength side until it completely vanishes. No negative index characteristic is observed for values of ε_LC ≥ 2.7. Taking ε_LC = 2 as the minimum practical value for the permittivity of a liquid crystal, the proposed near-IR metamaterial possesses a reconfigurable index of refraction over a negative-zero-positive range from λ = 1.37 μm to λ = 1.47 μm, which represents a 7.1% fractional bandwidth.

The dependence of the effective permittivity on the parameter ε_LC is shown in Fig. 2(c). The averaging effect on the bulk metamaterial permittivity is evident by observing that ε′ monotonically increases as the value of ε_LC is increased. Moreover, the fact that the dielectric function of silver is a decreasing function of λ explains why the negative index band diminishes with increasing values of ε_LC. To this end, as ε_LC is increased, the starting wavelength for which ε′ < 0 moves toward longer wavelengths, causing the negative index band to decrease. For all cases exhibiting a negative index band (i.e., for ε_LC ≤ 2.6), magnetic resonances are observed in Fig. 2(d) to exist within the 1.39 μm to 1.46 μm range with varying strengths and bandwidths.

Comparing n′ and n′′ in Figs. 2(a) and 2(b), respectively, for different values of ε_LC over the region where n′ < 0, one can observe a trade-off relationship between bandwidth and loss. As the negative index bandwidth becomes narrower with increasing ε_LC, n″ monotonically decreases over the same bandwidth. This consistent relationship between n′ and n″ vanishes as soon as the negative index band disappears with ε_LC = 3. The trade-off between bandwidth and loss of the NIM can also be understood in terms of the Kramers-Kronig relations [15].

 

Fig. 3. The effective index of refraction n with respect to ε_LC at two different wavelengths λ = 1.4 and 1.45 μm.

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A different perspective is offered by Fig. 3, where the variations of n are shown plotted with respect to ε_LC for the metamaterial to demonstrate its ability to be reconfigured between a NIM, a zero-index metamaterial (ZIM), and a positive-index metamaterial (PIM) at a fixed wavelength. At this point two wavelengths (λ = 1.40 μm and 1.45 μm) within the negative index band corresponding to ε_LC = 2 are chosen for further consideration. The value of n′ is seen to increase from -1 to +1.8 or +1.2 when ε_LC is varied from 2 to 6, respectively, demonstrating reconfigurability for NIM-ZIM-PIM behavior. It is noted that n′ does not vary linearly with ε_LC. Instead, n′ increases abruptly from - 1 to about +0.2 over a relatively narrow range of ε_LC values for both of the targeted wavelengths. The rate of change of n′ over this interval is higher for the longer wavelengths. Within the range of ε_LC values considered and for both wavelengths, it is observed that n″ has a sharp dip and reaches the minimum value where n′ increases rapidly from negative values towards zero.

4. Scaling to mid-infrared wavelengths

Scaling the near-IR reconfigurable design to shorter wavelengths poses difficulty because thinner liquid crystal layers would be required. In contrast, scaling into longer mid-infrared (mid-IR) wavelengths is relatively straightforward. Considering the same base NIM structure, it is observed that the dielectric function of silver yields a larger negative permittivity together with higher losses at mid-IR wavelengths compared to near-IR wavelengths. To compensate for the larger negative permittivity of silver requires a correspondingly larger volume of positive permittivity material in order to achieve an average permittivity close to zero. This can be achieved by cladding thicker layers of liquid crystal on both sides of the base NIM. Then, as in the previous case, the permittivity of the liquid crystal can be appropriately adjusted to tune the metamaterial response throughout a desired negative-zero-positive index range.

Figure 4 shows the characteristics of a mid-IR reconfigurable metamaterial in the 4 to 6 μm range. The geometrical parameters used for this design example are given by p = 2.4 μm, w = 1.2 μm, t = 120 nm, d = 160 nm, and t_f = 20 nm. A liquid crystal superstrate and substrate are incorporated into the design, each having a thickness of t_LC = 600 nm. Finally, a Lorentz-Drude model [16] was used to represent the bulk silver properties in the mid-IR range.

Several performance aspects can be compared with those of the near-IR design. For the lowest value of ε_LC (i.e., ε_LC = 2), it is seen from Fig. 4(a) that the bandwidth over which a negative index of refraction exists is 4.32 μm ≤ λ ≤ 5.08 μm. This range of wavelengths also corresponds to where the index of refraction is reconfigurable between negative, zero, and positive values. The corresponding fractional bandwidth for negative index behavior is equal to 16.2%, which is much wider than in the near-IR design. However, this bandwidth enhancement is accompanied by a higher loss. Figure 4(a) shows that over the bandwidth of negative n′, the value of n″ ranges from 1.51 to 4.60 when ε_LC = 2, whereas n″ ranges from 0.51 to 2.74 (see Fig. 2(b)) even though the two designs have the same silver film thickness of t_f = 20 nm This phenomenon is due to the higher loss present in silver at longer wavelengths. As ε_LC is increased, the negative index bandwidth exhibits the same trend of diminishing from the lower wavelength side toward longer wavelengths.

 

Fig. 4. The effective index of refraction n for a mid-infrared reconfigurable metamaterial design: (a) n with respect to wavelength for different values of ε_LC, (b) n with respect to ε_LC at two different wavelengths λ = 4.5 and 4.8 μm.

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In Fig. 4(b), variations of n at two wavelengths, λ = 4.5 μm and 4.8 μm, are shown plotted with respect to ε_LC. The value of n′ for both wavelengths changes from - 1 to +1.7 over the tuning range 2 ≤ ε_LC ≤ 6. Similar to the near-IR design shown in Fig. 3, n′ for the mid-IR design changes abruptly over a relatively narrow range of ε_LC, with the rate of change greater at the longer wavelength λ = 4.8 μm. Furthermore, minimum loss is observed again over the same narrow bandwidth corresponding to the largest change in n′.

5. Conclusion

Near-IR metamaterials incorporating tunable liquid crystal layers as superstrates and substrates have been presented. Having the ability to vary the permittivity of the liquid crystal layers provides a means for controlling the overall effective permittivity of the metamaterial in an averaging fashion. Coupled with properly designed periodic magnetic resonators, the effective refractive index of the metamaterial structure can be readily reconfigured or tuned between negative, zero, and positive values at a given wavelength. Moreover, the bandwidth over which the metamaterial exhibits a negative index behavior can be controlled (i.e., increased or decreased) via the liquid crystal tuning. A 0.54 μm-thick near-IR metamaterial design was presented which exhibits tunability of n′ from -1 to +1.8 around λ = 1.4 μm by varying ε_LC from 2 to 6. It was also observed that the negative index band diminishes from the short wavelength side upward as ε_LC is increased, until eventually it completely disappears. The near-IR design has been extended to a mid-IR design that provides reconfigurability of the negative index behavior over a wider bandwidth at the expense of higher losses.