PHYSICAL REVIEW B 83, 245119 (2011) Scattering properties of meta-atoms C. Rockstuhl, 1 C. Menzel, 1 S. M¨ uhlig, 1 J. Petschulat, 2 C. Helgert, 2 C. Etrich, 2 A. Chipouline, 2 T. Pertsch, 2 and F. Lederer 1 1 Institute of Condensed Matter Theory and Solid State Optics 2 Institute of Applied Physics, Abbe Center of Photonics, Friedrich-Schiller-Universit¨ at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany (Received 28 June 2010; revised manuscript received 12 April 2011; published 23 June 2011) Metamaterials consist of a periodic or aperiodic arrangement of so-called meta-atoms. Usually their optical properties are derived from the collective response of this ensemble. However, it is highly desirable to deduce them from the scattering properties of individual meta-atoms because frequently the periodic arrangement has a spurious effect on the desired functionality. Moreover, understanding the scattering properties of individual meta-atoms permits introducing guidelines for their design and predicting effective properties of amorphous metamaterials. To achieve this we introduce a genuine approach to quantify the properties of individual meta-atoms. To this end we evaluate spectrally resolved the composition of the rigorously calculated scattered field in terms of contributions of electromagnetic multipoles, such as electric and magnetic dipoles, quadrupoles and, in principle, arbitrary higher order moments. Beyond its direct application to metamaterial’s design and characterization, the approach will be significant in the entire field of nanooptics as, for example, for optical nanoantennas. DOI: 10.1103/PhysRevB.83.245119 PACS number(s): 42.70.Mp, 73.20.Mf, 78.20.Ek, 78.67.Bf Metamaterials (MMs) composed of micro- or nanostruc- tured materials emerged as a novel kind of artificial matter that allows affecting the light propagation in a manner inaccessible with natural media. 1 With the aim to observe dispersive regimes beyond those achievable by a mere averaging of its constituents, resonances are the key ingredient one aims to exploit. Two distinctive kinds of resonances are usually at the focus of interest. The first one requires a strict spatially periodic arrangement of the unit cells, for convenience called the meta-atoms. A resonance may be encountered if light scattered at consecutive periods interferes constructively. The structure acts then as a photonic crystal and it cannot be homogenized. Such a homogenization can be understood as one of the leitmotifs in the research on MMs. 2 Consequently, it requires that the resonance is associated with the meta-atoms itself and not with their arrangement. This is the second type of resonances to be exploited. Besides Mie resonances in dielectric particles, 3 localized plasmon polaritons allow observing such a resonant response. Accordingly, most of the current optical meta-atoms are metallic nanostructures. 4 Nevertheless, despite such understanding, in most cases the properties of MMs are evaluated for periodically ar- ranged meta-atoms. This simplifies both their fabrication and foremost their theoretical analysis. Concepts such as normal modes (Bloch modes) with the respective dispersion relation can be applied and dedicated numerical schemes may be exploited. 5 However, such approaches will not allow deriving the scattering properties of individual meta-atoms. But just this information is required to distinguish between properties induced by the lattice or by the meta-atoms themselves. In most cases this distinction is even pointless since it will be rather a combination of both. Dropping the periodicity and analyzing the properties of individual meta-atoms is presently urgently required for two reasons. 6 At first, it would allow revealing whether the involved optical response that prohibits the assignment of effective material properties to MMs is either due to the periodic arrangement of meta-atoms or to the excitation of higher order electromagnetic multipoles in the meta-atoms. 7 The use of simple constitutive relations requires that the scattering properties of the individual meta-atom are evoked by electric and magnetic dipoles, induced along the polarization of the incident field, and an electric quadrupole. 8 The latter may introduce certain ambiguities since it may be accounted for in the wave equation as either a spatially dispersive permittivity or an effective permeability. 9,10 Second, self-organized MMs that are fabricated by bottom- up approaches are emerging as a promising option to get rid of cost intensive top-down techniques which are largely restricted to a two-dimensional patterning of thin films, making it practically impossible to fabricate bulk MMs. 11,12 Although stacking of such functional layers may extend the options of top-down technologies, 13 it cannot be regarded as an ultimate route toward bulk MMs. Metamaterials fabricated by bottom-up approaches might be potential candidates for future bulk MMs. 14 However, the fabrication of perfectly periodically arranged meta-atoms is unlikely to be achieved with such methods which provide only a short-range order of the meta-atoms. The assignment of effective properties to such fully disordered MMs constitutes a challenging task. It can potentially be achieved by analyzing the scattering prop- erties of individual meta-atoms in terms of electromagnetic multipole contributions and a succeeding incorporation of the actual dipole moments into averaging procedures that do not require a certain spatial arrangement. Here we aim at providing the theoretical and numerical means for the spectrally resolved expansion of the scattered field of meta-atoms into multipole contributions. We further- more demonstrate that the strength of the electric and magnetic dipole moments can be exploited as an explicit criterion for the design of meta-atoms where undesired electromagnetic multipole moments may be fully suppressed. We finally show that effective properties of amorphous MMs can be retrieved with sufficient precision using such a multipole expansion. To outline the approach we start the analysis with a referential meta-atom, the split-ring resonator (SRR). At first we calculate spectrally resolved the scattered field from the SRR upon plane wave illumination [Fig. 1(a)]. Illumination 245119-1 1098-0121/2011/83(24)/245119(5) ©2011 American Physical Society