Nonlocal extensions of the electromagnetic response of plasmonic and metamaterial structures

Nonlocal effects, requiring wave-vector- $(q$-) dependent dielectric response functions, are becoming increasingly important in studies of plasmonic and metamaterial structures. The phenomenological hydrodynamic approximation is the simplest and most often used model but with limited applicability to problems involving surface plasmons. We show here that the $d$-function formalism, exact to first order in $q$, is a powerful and simple-to-use alternative, which allows for exact nonlocal extensions of local calculation schemes, e.g., finite-difference time-domain methods, without code changes. It is also extendable to order ${q}²$, and we demonstrate this by comparing with various earlier ab initio calculations and experiments as well as by performing our own random-phase-approximation calculations (valid for all $q)$ of the surface-plasmon dispersions for simple metals with various electron-gas densities. Finally we show that this hydrodynamic-extended $d$-function formalism can also be applied to arbitrary plasmonic/metamaterial structures as long as the nonflat interfaces can be modeled as effective media films.