Effective constitutive parameters of plasmonic metamaterials: Homogenization by dual field interpolation

We introduce a general implementation of the recently proposed homogenization theory [Tsukerman, J. Opt. Soc. Am. B 28, 577 (2011)] allowing one to retrieve all 36 linear constitutive parameters of any 3D metamaterial with parallelepipedal unit cells. The effective parameters are defined directly as linear relations between pairs of coarse-grained fields, in contrast with methods where these parameters are obtained from reflection and transmission data or other indirect considerations. The method is applied to plasmonic metamaterials with spherical gold particles and split-ring resonators (SRR), respectively. In both cases, the expected physical behavior is reproduced almost perfectly, with no unphysical artifacts.

Figure 1

(a) Sketch of the unit cell of size $2a\times 2b\times 2c$ containing an arbitrary metallic inclusion. (b) Cross section of a split-ring resonator (SRR) lying in the $xy$-plane with a bending radius and gap of 35 and 9 nm, respectively. The width and thickness of the SRR are also 35 nm.

Figure 2

(Reprinted by permission from Ref. [20].) A 2D analog of the vectorial interpolation function ${\mathbf{w}}_{\alpha }$ (in this case, associated with the central vertical edge shared by two adjacent cells). Tangential continuity of this function is evident from the arrow plot; its circulation is equal to one over the central edge and to zero over all other edges.

Figure 3

(Reprinted by permission from Ref. [20].) A 2D analog of the vectorial interpolation function ${\mathbf{v}}_{\beta }$ (in this case, associated with the central vertical edge). Normal continuity of this function is evident from the arrow plot; its flux is equal to one over the central edge and zero over all other edges.

Figure 4

The diagonal elements of (a) the effective permittivity matrix and (b) permeability matrix for a metamaterial consisting of cubic-ordered gold spheres with diameter of 40 nm embedded in a vacuum. The unit cell size is (80 nm)${}^3$. The ovals and arrows identify which curves belong to which axes.

Figure 5

The dominant elements of (a) the effective permittivity matrix, (b) effective permeability matrix, and (c) the magnetoelectric coupling matrices for a metamaterial consisting of cubic-ordered gold SRRs of Fig. 1. The unit cell size is (200 nm)${}^3$. The ovals and arrows identify which curves belong to which axes.

Figure 6

Comparison of the parameters ${\varepsilon }_{xx}$, ${\mu }_{zz}$ and ${\chi }_0$ for the SRR-metamaterial obtained by the direct method (‘D’) presented in this paper and the S-parameter method (‘S’) in Ref. [23].

Figure 7

Comparison of the complex reflection and transmission coefficients from a 1-$\mu$m-thick SRR-metamaterial slab for the two different homogenization methods. The incident wave is $x$-polarized and propagating along the $y$-direction. $r_{+}$ and $r_{-}$ correspond to the reflection coefficients for propagation along +$y$ and $-$$y$, respectively. The legend “Sim” denotes FEM full-wave simulation of a five-layer SRR-metamaterial.