## Abstract

We propose a systematic method of producing isotropic, double-negative metamaterials which operate in the visible spectrum. The material comprises two sets of inclusions dispersed in a host medium. We demonstrate that if the inclusions in one set are much smaller than those in the other, then the larger will behave as though they are submerged in a composite background material, rather than the true host material. This *hierarchy effect* is shown to enrich the designer’s capacity to induce strong, simultaneous electric and magnetic resonance at an arbitrary visible frequency, leading to double-negative behaviour. The predictions of Mie theory are verified using full-wave analysis and backward waves directly measured in the proposed designs.

© 2014 Optical Society of America

## 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:

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]:

*f*,

_{j}*j*= 1, 2 is the volume fraction of the

*j*th set of inclusions,

*ε*is the host medium permittivity, and for a homogeneous circular particle, the electric polarizability

_{b}*α*

^{(E)}is calculated via the Mie theory as [20, 21]:

*D*

_{1}(

*z*) =

*J′*

_{1}(

*z*)/

*J*

_{1}(

*z*), and

*J*(

*z*) and

*H*

^{(1)}(

*z*) are Bessel and Hankel fuctions, respectively. The parameters

*x*and

*m*are the particle size parameter and refractive index, which should be calculated relative to the

*background medium*in which the particle resides. In all instances of which the authors are aware, the background medium for both sets of particles is assumed to be the same, and equal to the physical host material with permittivity

*ε*. Full-wave analysis of structures akin to Fig. 1(a) have shown that when

_{b}*r*

_{1}≈

*r*

_{2}, this technique gives excellent predictions [17]. However a question arises: Do the equations hold in the situation shown in Fig. 1(b), when

*r*

_{1}≪

*r*

_{2}?

We reason that at some sufficiently small value of *r*_{1}/*r*_{2}, 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:

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

*ε*, though ${\alpha}_{2}^{(E)}$ should be calculated with respect to

_{b}*ε*, being the homogenized composite medium comprised of the host medium and the set of smaller inclusions. We term this the

_{c}*hierarchy effect*, since the host material forms the background for the first set of (smaller) particles, which when combined form the background for the second set of particles (and so on, if desired).

We note that for composites akin to Fig. 1(b) (and Fig. 2(a)), where *r*_{1} ≪ *r*_{2}, 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 *r*_{1} ≪ *r*_{2}, 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

*ε*

_{1}= −1.6 and

*ε*

_{2}= 10, respectively. Let the volume fraction of the first set of inclusions be fixed at

*f*

_{1}= 0.15, and let the size parameters of both particles be sufficiently small such that the Mie coefficient

*a*

_{1}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

*r*

_{1}≈

*r*

_{2}, the permittivity function closely follows that of Eq. (1), however we found that for

*r*

_{1}/

*r*

_{2}less than approximately 1/5, Eq. (4) should be used instead.

## 3. Composite background design

The conclusion of the previous section is that for sufficiently small *r*_{1}/*r*_{2}, 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]:

*α*

^{(M)}is calculated with respect to

*ε*. As stated earlier, Silicon seems the most likely candidate regarding magnetic dipole resonance at optical frequencies. So consider, for example, a set of circular Silicon inclusions, with permittivity given in [25] and radius 50nm, dispersed in a host medium with permittivity

_{c}*ε*. Using Eq. (5), we calculate that such an inclusion undergoes magnetic dipole resonance near the middle of the visible spectrum. As shown in Fig. 4, the strength of the resonance (gauged by the amplitude of the

_{c}*μ′*function) can be seen to depend strongly on the permittivity of the background medium. For values of

*ε*> 1, it is not possible to produce

_{c}*μ′*< 0 using these inclusions.

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′*< 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].

_{c}Some natural materials, namely noble metals, satisfy this criterion (e.g. Silver has permittivity ca. −13 + 0.4*i* 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′*which is negative, though reasonably-close to zero, at an arbitrary optical frequency is most desirable.

_{c}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.

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 *a*_{1} 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′*< 0 behaviour in the visible spectrum by varying the thickness of the Silver coating.

_{c}## 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*= 2. We nominate

_{b}*f*

_{1}= 0.15,

*f*

_{2}= 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 *r*_{1}/*r*_{2}, being the size ratio of the smaller and larger inclusions; the reflection and transmission coefficients, *S*_{11} and *S*_{22}, are presented in Fig. 7. We list the retrieved effective material properties *ε*, *μ*, and refractive index
$n={n}^{\prime}+{\mathit{in}}^{\u2033}=\left\{\pm \sqrt{\epsilon \mu}:{n}^{\u2033}\ge 0\right\}$ at 550 THz. As seen, for each case of *r*_{1}/*r*_{2}, these are in good agreement. The most noticeable trend is a lessening in the drop of |*S*_{11}|, being the magnitude of the reflection coefficient, as *r*_{1}/*r*_{2} increases. Note that rapid changes in *S*_{11} are attributed primarily to the magnetic resonance in the large Silicon particles, and those in *S*_{21} to plasmonic resonance in the smaller particles (we undertook simulations of the smaller particles only and confirmed that the behaviour of *S*_{21} was very similar to that seen when simulating both sets of particles).

We thus interpret the lessening in the drop of |*S*_{11}| 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*), hence a weaker magnetic resonance is present.

_{c}The changes in the behaviour of *S*_{21} are generally less marked than those of *S*_{11}. 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.

## 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).

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