Aperiodic Order and Geometry

The development of exotic new materials, such as metamaterials, has created strong interest within the electromagnetics (EM) community for possible new phenomenologies and device applications, with particular attention to periodicity-induced phenomena, such as photonic bandgaps. Within this context, motivated by the fairly recent discovery in X-ray crystallography of "quasi-crystals", whose diffraction patterns display unusual characteristics that are associated with "aperiodic order", we have undertaken a systematic study of how these exotic effects manifest themselves in the radiation properties of aperiodically configured antenna arrays. The background for these studies, with promising example configurations, has been reported in a previous publication [V. Pierro et al., IEEE Trans. Antennas Propag., vol. 53, pp. 635-644, Feb. 2005]. In this paper, we pay attention to various configurations generated by Rudin-Shapiro (RS) sequences, which constitute one of the simplest conceivable examples of deterministic aperiodic geometries featuring random-like (dis)order. After presentation and review of relevant background material, the radiation properties of one-dimensional RS-based antenna arrays are analyzed, followed by illustrative numerical parametric studies to validate the theoretical models. Design parameters and potential practical applications are also given attention.

In this paper, we propose a general and efficient method to analyze the dipolar modes of aperiodic arrays of metal nanoparticles with ellipsoidal shapes and their electromagnetic coupling with external fields. We reduce the study of the spectral and localization properties of dipolar modes to the understanding of the spectral properties of an operator L expressing the electric field along the chain in terms of the electric-dipole moments within the electric quasistatic approximation. We show that, in general, the spectral properties of the L operator are at the origin of the formation of pseudoband gaps and localized modes in aperiodic chains. These modal properties are therefore uniquely determined by the aperiodic geometry of the arrays for a given shape of the nanoparticles. The proposed method, which can be easily extended in order to incorporate retardation effects and higher multipolar orders, explains in very clear terms the role of aperiodicity in the particle arrangement, the effect of particle shapes, incoming field polarization, material dispersion, and optical losses. Our method is applied to the simple case of linear arrays generated according to the Fibonacci sequence, which is the chief example of deterministic quasiperiodic order. The conditions for the resonant excitation of dipolar modes in Fibonacci chains are systematically investigated. In particular, we study the scaling of localized dipolar modes, the enhancement of near fields, and the formation of Fibonacci pseudodispersion diagrams for chains with different interparticle separations and particle numbers. Far-field scattering cross sections are also discussed in detail. All results are compared with the well-known case of periodic linear chains of metal nanoparticles, which can be derived as a special application of our general model. Our theory enables the quantitative and predictive understanding of band-gap positions, field enhancement, scattering, and localization properties of aperiodic arrays of resonant nanoparticles in terms of their geometry. This is central to the design of metallic resonant arrays that, when excited by an external electromagnetic wave, manifest strongly localized and enhanced near fields.