Electrodynamic Theory of Three-Dimensional Metamaterials of Hierarchically Organized Nanoparticles

Abstract

To date, the most promising candidate structures for exhibiting photo-induced magnetism and negative refractive index in the optical regime are the so called Mie resonance-based metamaterials which consist of scatterers of simple geometrical shape, e.g., spherical or cylindrical, and are made of a high-index material. When such a structure is illuminated by an electromagnetic wave of frequency around the Mie resonance of a single scatterer, strong polarization currents are generated within the surface of the scatterers resulting in a macroscopic magnetization of the metamaterial. Due to the lack of naturally occurring materials with high refractive index in the optical regime, one can envisage a metamaterial which consists of meta-atoms that are clusters of metallic nanoparticles wherein strong polarization currents can also be induced under illumination. These type of metamaterials are hierarchically organized as they possess two length scales: the inter-particle distance within the cluster and the inter-cluster separation within the metamaterial. The nanoparticle clusters can be formed by direct or template-assisted self-organization and are generally amorphous due to the random positioning of the nanoparticles in air or within a cavity. The amorphous arrangement of such strongly scattering objects constitutes a major challenge for the field of theoretical and computational nanophotonics. In order to tackle this computational problem in the framework of metamaterials, we adopt a hierarchical theoretical strategy in proportion to the hierarchical organization of such structures. To this end, we develop a layer-multiple-scattering formalism for electromagnetic waves in order to model the optical response of metamaterials formed as collections of cavities filled by amorphous clusters of hierarchically organized spherical nanoparticles. It is based on a three-stage process where we take fully into account all the multiple-scattering processes experienced by photons: (a) among the particles of the cluster inside the cavity, (b) between the cluster and the cavity and (c) among the cavities (containing the clusters) within the metamaterial. We demonstrate the applicability of the method to the case of a silica-inverted opal whose voids contain clusters of gold nanoparticles. We find, in particular, such a metamaterial acts as a super absorber over a wide frequency range, from 2–4 eV.

Appendix

The matrix Ω for a vector field is given by [52, 79]

(5.65)

(5.66)

where

$$ \psi_l=\sqrt{l(l+1)} $$

(5.67)

$$ \alpha_l^m=\frac{1}{2} \bigl[(l-m) (l+m+1) \bigr]^{1/2} $$

(5.68)

$$ \gamma_l^m=\frac{1}{2}\bigl[(l+m) (l+m-1) \bigr]^{1/2}\big/\bigl[(2l-1) (2l+1)\bigr]^{1/2} $$

(5.69)

$$ \zeta_l^m=\bigl[(l+m) (l-m)\bigr]^{1/2}\big/ \bigl[(2l-1) (2l+1)\bigr]^{1/2}. $$

(5.70)

$G_{LL'} ({\bf R}_{nn'};q_{h})$ transforms an outgoing scalar spherical wave about ${\bf R}_{n'}$ to a series of incoming scalar spherical waves around ${\bf R}_{n}$. It is given by

(5.71)

with

$$ B_{lm}\bigl(l''m'';l'm' \bigr)=\int d\varOmega Y_{lm}(\hat{{\bf r}}) Y_{l' -m'}(\hat{{\bf r}}) Y_{l''m''}(\hat{{\bf r}}). $$

(5.72)

$Y_{lm}(\hat{{\bf r}})$ are the usual scalar spherical harmonics [50].

The matrix Ξ for a vector field is given by [52, 79]

(5.73)

(5.74)

$\xi_{LL'} ({\bf R}_{nn'};q_{h})$ transforms an outgoing (incoming) scalar spherical wave about ${\bf R}_{n'}$ to a series of outgoing (incoming) scalar spherical waves around ${\bf R}_{n}$ [see (5.25) and (5.26)]. It is given by

(5.75)