Negative refractive index induced by percolation in disordered metamaterials

Abstract

An effective medium model is developed for disordered metamaterials containing a spatially random distribution of dielectric spheres. Similar to effective medium models for ordered metamaterials, this model predicts resonances in the effective permeability and permittivity arising from electricand magnetic-dipole Mie resonances in the spheres. In addition, the model predicts a redshift of the electric resonance with increasing particle loading. Interestingly, when the particle loading exceeds the percolation threshold of 33%, the model predicts that the electric resonance overlaps with the magnetic resonance, resulting in a negative refractive index.Metamaterials, inhomogeneous composite materials that behave macroscopically as homogeneous materials, are of great fundamental and technological interest. The ability to tailor the macroscopic electromagnetic parameters, namely the permittivity and permeability, by changing the geometry and arrangement of the included elements has led to a number of important material designs and applications [1]. Most notable among these are materials with negative refractive index [2,3], which have both negative permittivity and permeability. Negative index materials exhibit unusual properties such as negative refraction and exponential growth of evanescent near fields, which have been utilized for applications such as cloaking [4] and superresolution imaging [5].To date, the metamaterial literature has focused primarily on designs where the included elements are arranged in a periodic lattice [6]. This is largely due to the ease of computation and suppression of scattering, as disorder of the lattice leads to spatial inhomogeneities. However, fabrication of periodic metamaterials on a large scale can be cost prohibitive, particularly for metamaterials operating in the optical band, where the lattice dimensions are submicron and require the use of electronbeam lithography [7,8]. Self-assembly based nanosphere lithography has been used to fabricate centimeter-scale optical metasurfaces [9], but this approach is not scalable to much larger areas.Disordered metamaterials provide a scalable alternative to periodic structures, provided that the scattering can be managed and their performance can be predicted. However, relatively few works have focused on disordered metamaterials. These limited studies have shown that size [10-13] and positional [14-23] disorder of a metamaterial's resonator elements lead to resonance broadening and effective scattering loss. While these works provide important insights, they overlook one of the most important aspects of disorder: percolation. Percolation occurs in random composites at high loading when the particles form a continuous path through the composite, and it can have dramatic consequences on the effective electromagnetic parameters [24,25] In this Letter, it is shown that percolation induces a negative refractive index in disordered metamaterials containing Mie-resonant particles. To this end, the classic Bruggeman percolation theory [24,26] ...

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“…The number of particles in the sample with the size of µm was fixed to 725, the radius of the spheres was varied. For comparison, the real parts of the effective index of refraction resulting from the effective medium formulas [ 18,19 ] are shown. It is worth mentioning that Lewin's theory [ 18 ] describes the cubic lattice of the spheres taking into consideration the first electric and magnetic Mie resonances.…”

“…The similar effective medium approximation for the random distribution of dielectric spheres is developed in ref. [19]. The dips of the curve correspond to the Mie resonances.…”

“…As can be seen from Figure 3, the dependence obtained using FDTD simulations correlates with Lewin's theory while the first electric resonance is shifted by the model of ref. [19]. The discordance between these calculations and theory [ 19 ] may arise from an assumption of zero scattering from the nanocomposite, whereas it is doubtful for the resonant disordered metamaterial.…”

“…The discordance between these calculations and theory [ 19 ] may arise from an assumption of zero scattering from the nanocomposite, whereas it is doubtful for the resonant disordered metamaterial. Theory [ 19 ] was tested in the limiting case of small inclusions where it reduces to the classic Bruggeman formula [ 20 ] and for the diluted composite of the spheres with sizes far from the resonances transiting to Lewin's expressions. Thus, the present study showed that, at least, the first magnetic and electric resonances are not changed appreciably for moderate volume fractions of the randomly positioned spheres.…”

“…The effective refractive index is calculated using FDTD modeling, and are results of the effective medium theories. [ 18,19 ]…”