Homogenization of resonant chiral metamaterials

Homogenization of metamaterials is a crucial issue as it allows to describe their optical response in terms of effective wave parameters as, e.g., propagation constants. In this paper we consider the possible homogenization of chiral metamaterials. We show that for meta-atoms of a certain size a critical density exists above which increasing coupling between neighboring meta-atoms prevails a reasonable homogenization. On the contrary, a dilution in excess will induce features reminiscent to photonic crystals likewise prevailing a homogenization. Based on Bloch mode dispersion we introduce an analytical criterion for performing the homogenization and a tool to predict the homogenization limit. We show that strong coupling between meta-atoms of chiral metamaterials may prevent their homogenization at all.

Figure 1

(Color online) Metamaterial designs under consideration: (a) TC and (b) TSRR. Wave propagation and metamaterial stacking direction is along $z$ axis.

Figure 2

(Color online) Effective refractive index of left circular polarized light of a twisted-cross metamaterial as a function of frequency and for different numbers of layers that form the MM. Dense stacking (period $2a=250\text{nm}$) (a) $\mathfrak{R}\left(n_{\text{LCP}}\right)$, (b) $\mathfrak{I}\left(n_{\text{LCP}}\right)$; sparse stacking (period $2a=500\text{nm}$) (c) $\mathfrak{R}\left(n_{\text{LCP}}\right)$, (d) $\mathfrak{I}\left(n_{\text{LCP}}\right)$. Effective parameters are retrieved with the standard method (Ref. 15) (SM) for one (black line), two (green/gray line), and three (cyan/light gray line) monolayers and with the WPRM (Ref. 37) for 48 layers (red/dark gray dots).

Figure 3

(Color online) Bloch modes damping $\mathfrak{I}\left(k\right)$ dispersion spectra for twisted-cross MM for different separations: (a) 125 nm, (b) 250 nm, (c) 400 nm, and (d) 500 nm. Fundamental modes are shown as black line with circles and green/gray line with triangles, all the higher-order Bloch modes are shown in red/dark gray dots. The frequency region above the first Bragg resonance is shadowed.

Figure 4

(Color online) (a) The difference of the propagation constants $\Delta \mathfrak{I}\left(ka\right)={\text{min}}_{i\ge 3}\left[\mathfrak{I}\left(k_{i}a\right)\right]-\text{max}\left[\mathfrak{I}\left(k_{1}a,k_{2}a\right)\right]$ of the least damped higher-order Bloch mode and the most damped fundamental mode for various unit-cell sizes $a$: 125 nm (black line), 150 nm (red line with squares), 200 nm (green line with circles), 250 nm (blue line with triangles up), 300 nm (orange line with triangles down), 400 nm (magenta line with stars), and 500 nm (brown line with crosses). (b) The minimum of $\text{min}\Delta \mathfrak{I}\left(ka\right)$ in (a) as a function of the period (red circles). The straight line represents a linear fit of $\text{min}\Delta \mathfrak{I}\left(ka\right)$ on $a$. The dashed line corresponds to the homogenization limit $\Delta \mathfrak{I}\left(ka\right)=0.5$.

Figure 5

(Color online) The loss spectrum for TSRR MM for different periods: (a) 290 nm, (b) 450 nm, (c) 600 nm. Fundamental modes are shown as black lines with dots and green/gray lines with triangles, all higher-order Bloch modes are shown as red/dark gray dots. The frequency region above the first Bragg resonance is shadowed.

Figure 6

(Color online) The sketch of the split-ring resonators stacking (a) in the negative-permeability configuration and (b) in the chiral configuration. Arrows represent the direction of the wave propagation, electric and magnetic field vectors. Black lines in the lower pictures represent the cross-sectional view of the lines of the magnetic field created by SR A and penetrating SR B.

Figure 7

(Color online) (a) Mechanical analogy of the metamaterial in the photonic crystal regime, (b) homogeneous and (c) inhomogeneous medium. Small balls and springs represent host medium. Large balls represent resonant inclusions.