1. Introduction

Double-negative (DNG) materials, where the real part of the electric permittivity ε′ and magnetic permeability μ′ are simultaneously less than zero, are candidates for a number of exciting applications including the superlens [1], light trap [2] and invisibility cloak [3].

Though not found in nature, DNG materials have been realized via a number of approaches, including split-ring resonator arrays [4] and fishnet surfaces [5]. Isotropy, however, is achieved only in the designs based on Mie resonance. These composites comprise (traditionally spherical) particles dispersed in a host medium, designed such that simultaneous electric and magnetic resonance is induced in the composite at the target frequency. Authors have proposed using two sets of high-permittivity inclusions, one set providing ε′ < 0 via the electric dipole resonance of the particle; the other providing μ′ < 0 via magnetic dipole resonance [6]. One could also use a single set of coated particles, where the core provides μ′ < 0 via magnetic dipole resonance, while surface plasmon resonance on the coating induces ε′ < 0 [7], or vice versa [8]. We have also recently shown that a particle comprising a conducting core with a high-permittivity shell can produce DNG behaviour via the synchronization of electric and magnetic dipole resonances in the shell [9].

Both of the above designs tend to break down in the visible spectrum, as it becomes very difficult to produce μ′ < 0. Strong magnetic effects require the inclusions to have a high permittivity relative to the host medium, and high-permittivity materials are difficult to find at visible frequencies. The highest permittivity we have observed in the literature is that of Silicon [10, 11], and we will soon show that even this material, when dispersed in a host medium with permittivity higher than that of vacuum, will not undergo strong-enough magnetic activity to force μ′ < 0 for most visible frequencies.

This issue has inspired a number of innovative solutions, such as using pairs or clusters of plasmonic nanoparticles to provide the required optical magnetic activity [12–14]. A multiscale design approach has also been proposed, where tightly-packed clusters of plasmonic nanoparticles behave as homogeneous spheres with high-permittivity, which are in turn capable of inducing μ′ < 0 in the composite as a whole [15, 16].

In this article, we propose a different multiscale design approach: Rather than designing inclusions which possess strong magnetic activity, we provide a method of strengthening the magnetic activity of the existing inclusions. To demonstrate the idea, we will undertake analysis in two dimensions, as it more easily lends itself to numerical validation (towards the end of the letter). The method is fully extensible to three dimensions with only minimal changes to the mathematics.

2. The multiscale approach

Consider, then, a material comprised of two sets of circular inclusions embedded in a host medium as shown in Fig. 1:

 

Fig. 1 Composite comprised of two sets of inclusions with radii r₁, r₂ and permittivities ε₁, ε₂ dispersed in a host medium with permittivity ε_b. The volume fractions in both of the above are equivalent.

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Note that although we visualize the composites of this article as being periodic, this is done for convenience only. The discussion that follows remains valid for random particle distributions. Now in the literature (see, e.g. [6, 17, 18]), the effective permittivity ε = ε′ + iε″ of such a composite is commonly predicted using the generalized Clausius-Mosotti relation, which in 2D reads [19]:

We reason that at some sufficiently small value of r₁/r₂, the set of larger particles will become “unaware” of the set of smaller ones. That is, the topological features of the smaller particles will become insignificant on scales relevant to the larger ones. The larger will then cease to behave as though submerged in a background with ε_b, and begin to behave as though submerged in a new (composite) background, formed from the host material and the smaller particles. This is visualized in Fig. 2:

 

Fig. 2 For sufficiently small r₁/r₂, the set of smaller inclusions and host material can be replaced by a homogenized composite background with permittivity ε_c.

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By this notion, the following should be used in lieu of Eq. (1):

We note that for composites akin to Fig. 1(b) (and Fig. 2(a)), where r₁ ≪ r₂, there must be close proximity between neighbouring particles. Effects brought about by particle proximity are a focus in the literature, where it has been shown that inter-particle coupling can result in the broadening and/or shifting of the resonance frequency of neighbouring particles [22, 23]. As far as the authors are aware, the hierarchy effect proposed in this article differentiates from those mentioned due to the constraint that r₁ ≪ r₂, and has not previously found application, at least in the context of DNG metamaterials.

In order to demonstrate the validity of Eq. (4), we use the S-Parameter extraction method detailed in [24]. Briefly, the method ascertains the effective electromagnetic parameters of a composite by comparing the scattering characteristics of a unit cell (or otherwise representative section) of the composite to an equivalent homogeneous slab. The method relies on full-wave analysis (FEA or equivalent) of the composite geometry, and is thus a rigorous form of validation for the predictions of Mie theory.

As an example, consider a composite comprised of a host medium with permittivity ε_b = 2, and two sets of inclusions with permittivities ε₁ = −1.6 and ε₂ = 10, respectively. Let the volume fraction of the first set of inclusions be fixed at f₁ = 0.15, and let the size parameters of both particles be sufficiently small such that the Mie coefficient a₁ converges (to its electrostatic value). In this case, only the relative sizes of the particles are important. Figure 3 depicts the results of such a calculation. The marks are the values calculated via the S-parameter extraction method, with indicative FEA models shown on the right. When r₁ ≈ r₂, the permittivity function closely follows that of Eq. (1), however we found that for r₁/r₂ less than approximately 1/5, Eq. (4) should be used instead.

 

Fig. 3 Effective permittivity calculation for different values of r₁/r₂, verified via the S-parameter extraction method. In the above, ε_b = 2, ε₁ = −1.6, ε₂ = 10 and f₁ = 0.15.

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3. Composite background design

The conclusion of the previous section is that for sufficiently small r₁/r₂, the hierarchy effect is indeed present, and the set of larger inclusions behaves as though submerged in a background medium which is itself a composite, formed from the host material and the set of smaller inclusions, as per Eq. (4). In other words, we now have a method of designing the composite background. Though in the context of DNG metamaterials, which background properties are advantageous?

To see this, we recall the magnetic analogue of the Clausius-Mosotti relation. Making the safe assumption that the smaller particles are too small to be undergoing magnetic activity, the effective permeability μ = μ′ + iμ″ of the composite is predicted in a similar manner to the electric case [20, 21]:

 

Fig. 4 Silicon particles of radius 50nm are dispersed in a background medium with permittivity ε_c, which is varied. The volume fraction in all cases was set to 0.5.

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For at least some optical frequencies, then, we require a method of producing low (i.e. ε′_c < 1) permittivity in our composite background medium to ensure μ′ < 0 while using Silicon inclusions. For DNG behaviour, we also must ensure that ε′ < 0. Since the electric dipole resonance always occurs at a different frequency to that of the magnetic dipole resonance in a homogeneous particle, we cannot rely on the same inclusions for this task. We conclude from this that the composite background medium must have a permittivity with real part less than zero, i.e. ε′_c < 0. We note that a composite with such properties has previously been studied and proven to have the desired behaviour of a DNG (left-handed) material in [26].

Some natural materials, namely noble metals, satisfy this criterion (e.g. Silver has permittivity ca. −13 + 0.4i at 550 THz [27]). Using noble metal as a background material is not desirable for obvious cost reasons. Also, as implied by Fig. 4, the more negative is ε′_c, the sharper the resonance in μ′ tends to be. This reduces the practicality whenever some bandwidth of operation is required. For these reasons, a method of producing ε′_c which is negative, though reasonably-close to zero, at an arbitrary optical frequency is most desirable.

One such method is to use for our smaller inclusions, dielectric particles which are coated in a thin layer of noble metal, as in Fig. 5. This method has been analysed in [28], though not in the context of DNG metamaterials.

 

Fig. 5 The smaller inclusions are in fact dielectric particles such as Silica, coated with a noble metal such as Silver. This method provides a route to tunable ε′_c < 0 behaviour at optical frequencies.

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In this case, the effective permittivity of the composite background is still predicted via Eq. (4a), though a more complicated form of the scattering coefficient a₁ is required to account for the particle coating. The details of such can be found in, for example [21]. As in [28], we nominate Silica for the core and Silver [27] for the coating, and assume the particles are dispersed in a host medium with ε_b = 2. Figure 6 demonstrates the versatility of this coated particle method in inducing ε′_c < 0 behaviour in the visible spectrum by varying the thickness of the Silver coating.

 

Fig. 6 Effective permittivity, ε_c = ε′_c + iε″_c, of a dispersion of Silver-coated Silica particles, as per Eq. (4a). In each of the above, r₀ = 5 nm, f₁ = 0.15, ε_b = 2.

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4. Double-negative behaviour

We now have the necessary tools to design a DNG metamaterial which operates in the visible spectrum. Rather arbitrarily, we nominate 550 THz to be the target frequency. From Fig. 4, we know that our larger (Silicon) particles should be in the order of 50 nm in radius, and we want our background permittivity ε′_c to be negative though close to zero at this frequency. From Fig. 6, we can achieve this background using ca. 5 nm radius Silica particles coated in Silver, such that the ratio of the core to shell radii is ca. 0.70, and dispersed in a host medium with ε_b = 2. We nominate f₁ = 0.15, f₂ = 0.3, primarily because it makes for convenient finite element model geometry.

We evaluated this design using the aforementioned S-Parameter extraction technique for several values of r₁/r₂, being the size ratio of the smaller and larger inclusions; the reflection and transmission coefficients, S₁₁ and S₂₂, are presented in Fig. 7. We list the retrieved effective material properties ε, μ, and refractive index $n=n^{\prime }+{\mathit{in}}^{″}=\left\{\pm \sqrt{\epsilon \mu }:n^{″}\ge 0\right\}$ at 550 THz. As seen, for each case of r₁/r₂, these are in good agreement. The most noticeable trend is a lessening in the drop of |S₁₁|, being the magnitude of the reflection coefficient, as r₁/r₂ increases. Note that rapid changes in S₁₁ are attributed primarily to the magnetic resonance in the large Silicon particles, and those in S₂₁ to plasmonic resonance in the smaller particles (we undertook simulations of the smaller particles only and confirmed that the behaviour of S₂₁ was very similar to that seen when simulating both sets of particles).

 

Fig. 7 Full-wave analysis of composite, and the resulting reflection S₁₁ and transmission S₂₁ coefficients. Note that |S_ij| and ∠S_ij represent the magnitude and phase of S_ij, respectively. In each, the larger (Si) inclusions have radius r₂ = 54 nm, and the coating on the smaller particles is given by r₀/r₁ = 0.73. The unit cell models are shown on the left, and effective property values are given at 550 THz.

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We thus interpret the lessening in the drop of |S₁₁| as a weakening of the magnetic activity in the larger Silicon particles, corresponding to a reduction in the hierarchy effect: The larger Silicon inclusions are beginning to behave as though they are submerged in the true host (with permittivity ε_b) as compared to the composite background (with permittivity ε_c), hence a weaker magnetic resonance is present.

The changes in the behaviour of S₂₁ are generally less marked than those of S₁₁. This can be attributed to the fact that the plasmonic resonance effect in the smaller particles is quite insensitive to the actual size of the smaller particles, provided they remain small in comparison to the incident light wavelength.

Full-wave analysis also makes possible the direct measurement of backward waves propagating in the proposed composites. Figure 8 demonstrates the magnitude and phase of an incident wave propagating through a slab 5 unit cells deep, with each cell having the geometry of Fig. 7(b). To the left and right of the slab, the phase is seen to decrease in the propagation direction (left-to-right), though inside the slab, the phase increases. Such is the behaviour of a backward wave [17], and excellent agreement is obtained when comparing the composite to an equivalent homogeneous slab.

 

Fig. 8 A backward wave is seen to propagate inside the proposed composite, evidenced by the change in sign of the phase gradient. In the above, the black dashed lines were measured in the composite shown; the red lines were measured in a homogeneous slab with ε = −0.97 + 0.04i, μ = −1.00 + 0.56i.

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5. Conclusion

We have successfully demonstrated a new method of obtaining double-negative (DNG) behaviour in the visible spectrum. For composites comprised of two sets of inclusions, the relative size of the particles becomes paramount in regard to the effective electromagnetic behaviour. Judicious design can bring about a hierarchy effect, leading to strong, simultaneous electric and magnetic resonance. Future work will be undertaken towards the physical realization of such designs. This work was funded by the Australian Research Council (DP110104698, FT120100947, DE120102906).