Metallodielectric broadband metamaterials

Materials often find an application after their synthesis, but sometimes, practical demands outrun production capabilities. As a result, many projects await materials whose physical properties overcome the current constraints. Materials science needs new ideas to develop these applications, beginning with optical nanocircuits and ending with intriguing cloaking devices. Most of the many publications devoted to metamaterials deal with negative-refractive-index and bianisotropic materials, both of which involve exotic magnetoelectric properties. Meanwhile, a lot of exciting opportunities exist if we could tailor only the electrical permittivity. Materials with near-zero permittivity¹ or a refractive index near or below unity² could start a revolution in photonics—acting, for example, as optical insulators or near-perfect absorbers. Such materials occur in nature, but they suffer from narrow bandwidths. The design of broadband metamaterials could considerably extend the range of potential applications.

How can we produce broadband low permittivity or low refractive index materials? We propose that metallodielectric photonic crystals (periodic composites) with a complex unit cell could be a natural solution. The combination of metal and dielectric constituents can generate local resonances (surface plasmons), and the many degrees of freedom in a complex unit cell allow more resonances to occur over a broad spectral range. Properly designed, such composites offer a wide scope for tailoring the tensor of permittivity and hence the optical properties. Our estimations show that for the band 415–675nm, almost completely covering the visible range, we can obtain a permittivity within ±0.008 of zero when the unit cell contains 21 layers.¹

Roughly speaking, the problem of designing broadband materials reduces to the following choices: the constituent materials, the basic geometry, and the specific parameters of the unit cell. In principle, the first two choices are not independent because a chosen geometry imposes a limitation on the possible constituents.

To be more specific, we explored a basic example of a broadband metamaterial design. Consider a one-dimensional composite made of periodically repeated unit cells. Each cell is made of N parallel layers (see Figure 1), and each layer has its own permittivity ε_i. The applied electric field is assumed to be normal to the phase interfaces. Because the normal component of the displacement is continuous at the boundaries, the effective permittivity of the composite, ε_e, may be written as the harmonic average of the phase permittivities. In the simplest case, the layers are alloys of two metals. A crude but widely used approximation for their permittivity is an average, weighted by the volume fraction of the metal phase in each layer.³

Figure 1. Schematic diagram of a unit cell consisting of 7 parallel layers, with the directions of the electric field, magnetic field, and wave vector as E, H, and k, respectively. ε_i: Permittivity of each layer.

To get a material with low and constant effective refractive index Re(n_e)=Re(ε_e)^1/2=n_d within a given frequency band, we should minimize the objective function. This gives the difference between the designed crystal's refractive index and the desired refractive index over that range, and it allows us to determine the sought volume fractions. Of special interest is the regime when the permittivities of constituents, ε₁ and ε₂, are of different sign, as this condition allows the effective permittivity to oscillate around the chosen value. (Otherwise the effective permittivity is expected to be a monotonic function of frequency within the band.) If we are dealing with metal alloys, the target frequency band should lie between the plasma frequencies of the metal phases (the frequencies at which the metals have permittivities of zero). In other words, one of the metals should have a positive permittivity—it should behave like a dielectric.

This example is simple, but dealing with metal alloys can involve technological difficulties. Instead, we could consider more complicated unit cell geometries. Among them, we distinguish porous materials and those made of wires with varied widths because they may be manufactured with existing technology. For example, Figure 2 shows the effective refractive index of a metamaterial made of parallel metal wires (with the metal permittivity of the Drude form) embedded in a dielectric host with ε₂=2:5. It oscillates around (and very near) 0.6 over a broad range of frequencies.

Figure 2. The effective refractive index (Re n_e) of metallodielectric photonic crystal fitted to n_d=0:6within the frequency range [0.32–0.42]ω_p, where ω_p is the plasma frequency. The unit cell consists of 16 layers.

Of course, that scenario is strongly simplified. A great deal needs to be done before practical implementation of broadband optical metamaterials, both in perfecting the theoretical models and in the technology to produce the photonic crystals. Now, we are developing the effective medium theory that more adequately describes interactions between crystal constituents and can optimize the design of two- and three-dimensional photonic crystals.

The practical feasibility of broadband metamaterials seems to be a challenging but realistic problem. At present, a number of well-developed methods can produce high-quality metal wires, as well as dielectric pores in metal films. Generally, both the wire width and length can be controlled. We believe that our results will encourage experimentalists to fabricate broadband metamaterials, providing new ways to manipulate light for versatile applications.