Role of aperiodic order in the spectral, localization, and scaling properties of plasmon modes for the design of nanoparticle arrays

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.