A Review on the Application of Integral Equation-Based Computational Methods to Scattering Problems in Plasmonics

Many computational methods have been developed and used for modeling, understanding, and tailoring extreme optical effects at the nanoscale. Among them, this review focuses on the integral equation-based methods: within the local response limit, a potential-based boundary integral equation (BIE) formalism and a field-based volume integral equation (VIE) formalism; within the nonlocal hydrodynamic model, a potential-based BIE formalism. These formalisms are derived from macroscopic electrodynamics (together with appropriate constitutive relations). The derivations are based on three pillars: the Green function, the field relation(s) (for the VIE formalism, the incident-scattered-total field relation; for the BIE formalism, the interface conditions connecting the fields at two sides of the interface), and the field equivalence principle (for the VIE formalism, the volume equivalence principle; for the BIE formalism, the Huygens principle). By applying the method of moments (MoM) algorithm, the derived integral equations are converted into matrix equations, with possible problems in the implementation being discussed. Levels of solutions, including the eigenmode and natural mode solutions, and group representation theory are introduced as powerful post-processing steps. Many examples are shown to demonstrate the effectiveness of the reviewed algorithms.