On a dedicated volume integral equation solver for Nanoparticle-on-Mirror (NPoM) structures

Abstract

Nowadays nanotechnology allows us to size down every single object in our daily life at the nanoscale. Especially, conventional microwave antennas/scatterers are reproduced at nanoscale and have found many applications, including bio-sensing [1], energy harvesting [2], nanoscale communications [3, 4], etc. Most recently, a Nanoparticle-on-Mirror (NPoM) topology with a sub-nm gap size (See Fig. 1, a nanosphere-on-mirror, as an example) is proposed [5], soon draws many interests due to its capability of confining extremely intensive field to the sub-nm gap between the nanoparticle and the gold mirror, and promises a new generation of Surface-enhanced Raman Scattering (SERS) substrates. Computationally, the NPoM structure is more often than not handled by an Finite difference Time Domain (FDTD) implementation, for example, Lumerical [5–7]. However, applying a FDTD solver to the analysis of the NPoM system may not be an optimal choice and can have several disadvantages. First of all, to model the subnanometric nature of the gap and the detailed geometric feature of a nanoparticle whose diameter is around tens of nanometers, an extremely fine mesh has to be employed, which inevitably increases the computational time. Secondly, for the NPoM system, the optical response of an oblique incident wave is of much greater interests than the usual normal incidence. Since in an FDTD algorithm, an oblique incident wave suffers from frequency/wavelength dispersion, in order to understand the optical response of the NPoM system over a wide wavelength range, multiple narrow band simulations have to be run, which worsens the computational time consideration mentioned previously and leads to inaccurate simulation results as the FDTD is ideal for a wide band simulation. Last but not least, a FDTD run renders very little physical understanding to the considered NPoM system where the following two questions are of great interests: 1) which modes are excited by the incident field and 2) how the presence of the mirror and the nanometric/sub-nanometric gap size affects the optical response of the system. All the above issues can be resolved by employing an integral equation formalism. In this work, we focus on the Electric Field Volume Integral Equation (EF-VIE) and its solution given by a Volumetric Method of Moments (V-MoM) algorithm. When we construct the solver, we have the following two design goals. Firstly, this solver should be able to deliver physical understandings as much as possible. Secondly, under the condition that the first design goal is fulfilled, this solver must be as much computationally efficient as possible. The first goal is achieved by clearly separating the contributions from primary wave and reflected wave from the mirror. Since s-(TE) and p-(TE) waves are the eigenmodes of a multilayer structure [8], their effects are further distinguished in the reflected wave contribution. Considering the computational efficiency, we extract the near field behavior for the s- and p-waves in closed form and finish the following volume integral as analytically as possible.