1. Introduction

Meta-materials are defined as those composites having properties not found in nature. In electromagnetics research, there is great interest in developing composites which display precise values of electric permittivity (ε) and magnetic permeability (μ), for these dictate the movement of electromagnetic radiation in and around the material. Even brief surveys of the literature uncover designs for extraordinary devices made possible by the use of metamaterials, such as the invisibility cloak [1] and artificial black hole [2].

Particular attention has been directed toward achieving negative values for ε and μ, such that the refractive index ( $n=n^{\prime }+{\mathit{in}}^{″}=\pm \sqrt{\epsilon \mu }|n^{″}>0$) also becomes negative, and it is to this end that we devote our efforts. In addition, we will propose designs operating at frequencies lower than that of red light; it is at these frequencies where metals retain a good measure of conductivity, which will prove salient in our designs.

At these frequencies, past attempts to induce negative refractive index behaviour can be categorized into two main design routes. The first generates electric current flow in periodically-arranged metallic patterns (unit cells), which are printed onto dielectric backing material in the style of the famous split-ring resonator [3–5]. The second utilises the Mie resonance in small (usually spherical) particles dispersed in a host material [6–10]. Both share the issue of scalability, since the unit cell (metallic pattern and Mie particle, respectively) must be smaller than the incident wavelength. In addition, the performance of each composite is generally quite sensitive to small changes in the unit cell geometry.

These issues have motivated one particularly novel extension: to design metamaterials which can be manufactured using technology borrowed from the fiber optics industry. A metal-dielectric composite preform can be designed, which is then drawn into a long fiber. The fiber is cut and stacked, becoming the unit cell in the metamaterial (see Fig. 1). In this manner, researchers have realized negative permeability (μ′ < 0) by drawing fibers with cross-sections resembling the split-ring resonator [11–13]. To our knowledge, though, a design capable of simultaneously achieving ε′ < 0, and thus negative refractive index, is yet to be proposed.

 

Fig. 1 The proposed metamaterial. Fibers comprised of aligned metal wires and a solid metal core provide a route to strong electromagnetic resonances which can lead to negative refractive index behaviour.

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Such is the novelty of this article. We propose a design methodology capable of leveraging the advantages of fiber-based manufacturing and which also achieves a negative refractive index. Furthering disparity to the previously-mentioned fiber-based designs, ours is one utilising Mie resonance. We show that a very high refractive index fiber is required to induce negative material properties, and that this can be achieved by introducing aligned metallic wires into a dielectric or semiconductor fiber. We continue by showing that the addition of a metallic core to the fiber can effect a synchronization in electric and magnetic dipole resonance, resulting in a negative refractive index. We conclude by directly simulating a backward wave propagating through the proposed composites.

2. Theory of fiber-drawn metamaterials

To illustrate the emergence of metamaterial behaviour in a fiber-based composite, we simulated an isolated silicon fiber (material properties from [14]), exposed to a transverse magnetic (TM) plane wave as shown in Fig. 2(a). As predicted by the Mie theory [15], at certain (geometry-dependent) frequencies the fiber develops strong internal fields, known as magnetic (Fig. 2(b)) and electric (Fig. 2(c)) dipole resonance, respectively.

 

Fig. 2 Metamaterial behaviour from fibers. In the above, the arrows and colour scale represent the E⃗ and H⃗ fields, respectively.

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What of a composite material comprising such fibers? It can be shown that the resonant fields developed in the fibers can impart a marked effect on the electromagnetic properties (μ, ε) of the composite as a whole, causing them to adopt a resonant nature as well. The relevant expressions, known as the extended Maxwell-Garnett theory, are given as [16, 17]:

In the above, D₁(z) = J′₁(z)/J₁(z), and J(z) and H⁽¹⁾(z) are Bessel and Hankel fuctions, respectively. The parameters $x=\sqrt{{\epsilon }_b}{k}_{0}r$ (where k₀ and r are the incident wavenumber and fiber radius, respectively) is the fiber size parameter; m is the fiber refractive index, calculated relative to the background/host medium which has permittivity ε_b. Such formulae have proven to remain quite accurate even at very high fiber volume fractions, f [18]. It should be understood that μ and ε pertain to the material property values perpendicular to the fiber axis; different values exist parallel to the fibers, though we will not discuss these here.

Although not immediately obvious from Eq. (1), the higher the fiber refractive index m, the stronger the electric and magnetic resonances will be (such information is contained in the Mie coefficients a₁, b₁). This in turn produces greater variations in the properties ε, μ.

Of the materials amenable to large-scale fiber-drawing techniques, Silicon [14, 19] has the highest index of which the authors are aware. Though we have found that even a very dense collection of homogeneous silicon fibers does not produce strong-enough resonance to emulate negative material properties: Fig. 3(a) demonstrates one example, where the fiber diameter was selected such that the resonances occured between 100 and 150 THz. Similar results can be produced throughout the IR spectrum. The challenge, then, is to increase the refractive index of the fibers.

 

Fig. 3 Effective properties of fiber-based composites. In each of the above, the fiber filling fraction is f = 0.7 and the background medium has ε_b = 1.0.

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We propose a method thus: Introduce, into the fiber pre-form, a collection of metallic wires, which should remain in position when the (now composite) fiber is drawn. Such a technique is well-established for range of metals, glasses and semiconductors [20, 21], and has been proposed for use in metamaterials for different applications than are considered here [22,23]. Here we will continue to assume silicon as the fiber material, though the analysis is extensible to other materials in principle.

To see why metallic wires should increase the refractive index of the fiber, we can apply Eq. (1) to the fiber material itsself. Asymptotically expanding Eq. (1d) for small inclusion size:

Now note that for conductors, |m| ≫ 1 and we can use Eq. 2 and 1b to solve explicitly the effective permittivity of a silicon-metallic fiber:

The above (and accordingly, the fiber refractive index) becomes very large for high wire filling fractions (f_wire), and Fig. 3(b) exemplifies negative properties being achieved using this technique, whereby copper wires (material properties from [24]) were arranged in a circular array such that the volume fraction of the wires (f_wire) was set to 0.5, a value high enough to demonstrate the effect, though small enough to simulate (and presumably manufacture) efficiently. Note here that we have effectively designed a composite on two length scales simultaneously – the smaller being the silicon-metallic (fiber) material, the larger being a collection of such fibers in a host material (Fig. 1).

The limitation on this effect occurs when the small inclusion limit breaks down, which typically occurs when the conductor cross-section exceeds its skin depth [25]. For this reason, the proposed technique will be effective only in regions where the chosen conducting wire material has moderately-high conductivity, as the skin depth will be non-zero. For most metals, this is satisfied in the infrared region, and the copper from the above example could reasonably be substituted for most metals. The higher the conductivity, the smaller the wire diameter must be in order to be effective.

3. Negative refractive index behaviour

We now turn to the requirement of simultaneously-negative ε and μ. Since the corresponding resonance frequencies can be tuned via the diameter of the fiber, one strategy would be to use two sub-lattices of fibers – one providing ε′ < 0, the other μ′ < 0. In this particular case, though, our simulations show this strategy to be futile, the reason being that the maximum volume fraction of a given fiber in a two lattice system is too low to produce negative effective properties.

Fortunately, we have previously shown that the addition of a conducting core to a fiber (or sphere) can effect a synchronization in the electric and magnetic dipole resonances [26]. Since now only a single lattice of fibers is required, the volume fraction (and strength of the resulting resonances) can be much higher. It then becomes a task of optimizing the size and volume fractions of the fiber, wires and core in order to achieve a negative refractive index. As an illustrative example, we choose 100 THz as a target frequency and optimized the parameters using a custom particle swarm algorithm [27]. As above, copper was nominated for the core material, though as above, most other metals could equally-well be used. Figure 4 demonstrates the geometry and overlapping resonances of such a composite. Although we chose 100 THz as a target here, this strategy can quite generally be applied to the entire infrared region.

 

Fig. 4 Negative refractive index behaviour. Effective properties were calculated for a composite with ε_b = 1.0, silicon fiber (∅0.47 μm, f = 0.7), metallic wires (∅0.25 μm)

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It should be noted that the sharp increase in n″, the imaginary component of the refractive index near the resonance frequency, does not imply that the composite material is lossy, as would be the case for a homogeneous material. Rather, as a result of the extended Maxwell-Garnett formalism assumed in this article, the peak in n″ encapsulates information regarding both the absorption and scattering from the composite inclusions [28,29]. This being an article pertaining predominantly to conceptual design, we will not further explore techniques to estimate the true loss in the material; we refer interested readers to the methodology described in Refs. [30, 31] for such an endeavour.

As further verification of the predicted material behaviour, we utilised COMSOL Multi-physics to simulate a plane wave travelling through a (5×5) slab of the proposed composite material as shown in Fig. 5. A backward wave, confirming that a negative refractive index has been achieved, is quite clearly visible in the region of the composite, as evidenced by the reversal of the direction of phase change at the boundaries of the composite.

 

Fig. 5 Backward wave propagation in the proposed composite. Inside the composite, the field was measured between the fibers and interpolated elsewhere. The color scale shows H′_z, and was limited to ±1 for clarity; higher fields exist inside the fibers.

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4. Conclusion

We have thus successfully demonstrated a method for designing a new class of fiber metamaterials. By leveraging effective medium phenomena at two length scales simultaneously, we are able to enhance and synchronize the Mie resonance modes of the fiber. By fine tuning the design parameters, we have proven capable of achieving a composite with a negative refractive index, a strategy which can quite generally be applied in the IR spectrum using the materials suggested herein.