Enforcing symmetries in boundary element formulation of plasmonic and second-harmonic scattering problems

The study of metal nanoparticles and metamaterials has increased the demand for accurate and efficient numerical methods for solving electromagnetic scattering problems. The boundary element method, and especially its Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) formulation, has received growing interest lately due to its accuracy and stability at plasmon resonance conditions. Consequently, this formulation has been used to model second-harmonic generation (SHG) in plasmonic nanoparticles, which is an area of increasing importance. Many nanostructures exhibit geometrical symmetries, whose identification is often crucial for the qualitative understanding of SHG. In this work, we present the theory and details to take advantage of these symmetries in the PMCHWT formulation. We show that, importantly, the symmetry of the medium can be exploited even though the excitation source does not exhibit a well-defined symmetry. We estimate the obtainable computational benefits and apply the method to the study of the linear and second-order nonlinear properties of multiply split gold ring resonators.