Field Determination Near Plasmonic Structures by the Feynman–Kac Stochastic Representation

Feynman-Kac stochastic representation for elliptic equations is used to construct the solution of Helmholtz equation in domains containing plasmonic structures. The method involves initiation of Brownian motion at an interior point in the plasmonic structure and relating the field there to that on the structure boundary (equivalent surface) through the Feynman-Kac formula. The field exterior to the plasmonic structure can be related to the field on the equivalent surface through a variety of numerical methods depending on the complexity of the exterior domain. Finally, the field on the equivalent surface is determined by the imposition of boundary conditions. Comparison is shown here with the exact solution for the total field on the surface of a circular nanocylinder and a nanosphere driven by localized sources and where the permittivity is described by the Drude model.