Surface Plasmon Polaritons in a Class of Quasiperiodic Nanoparticle Gauss Chains

We theoretically consider surface-plasmon (SP) polaritons (SPP) on chains of noncontacting metal nanoparticles (NP) along the z-axis at \(z=md\) $z=md$ with \(m\in \{ 0,1,2,\ldots \}\) $m\in \{0,1,2,\dots \}$ and d the underlying lattice constant. When \(m=j^n\) $m=j^n$ for \(j\in \{ 0,1,2,\ldots \}\) $j\in \{0,1,2,\dots \}$ and given \(n\in \{ 2, 3, 4, \ldots \}\) $n\in \{2,3,4,\dots \}$ , we term it a Gauss site where such nanoparticles differ in their surface-plasmon frequency from the other nanoparticles. Such Gauss chains are quasiperiodic and may exhibit a self-similar hierarchy of minibands and gaps. Our treatment here is carried out within a tight-binding model with the Gauss modulation on the diagonal elements and includes the retarded electromagnetic coupling between all pairs of nanoparticles. The case \(n=2\) $n=2$ exhibits a clear hierarchy of minibands and gaps, and these characteristics are preserved in the presence of long-range retarded dipole-dipole coupling, whereas for \(n>2\) $n>2$ , one observes at most weak features of such self-similar behavior. The special nature of the case \(n=2\) $n=2$ is reflected in the spatial distribution of the SPPs on the Gauss chain compared with \(n>2\) $n>2$ .