Quasi-static potential created by an oscillating dipole in the vicinity of two nanospheres (dimer): inversion transformation method

Abstract

We present a method based on the inversion transformation for obtaining the quasi-static potential created by an oscillating dipole in the vicinity of two dielectric nanospheres (dimer). In the inversion space, a Poisson equation for another potential must be solved in which the source is an effective charge density composed of a point dipole and a point charge; the charge of this point charge depends on the radial component of the dipole moment with respect to the inversion center. We particularly derive the solution of the potential when the dipole is on the dimer axis. Our formalism can provide an alternative physical insight for explaining the radiative properties of localized emitters (molecules) as well as intermolecular interactions mediated by electromagnetic fields. In addition, this method can yield to a less computational effort for calculating the electric field. To illustrate our method, the spatial distributions of the electric field created by a dipole for two particular setups are obtained.

Electrostatic images and T-matrices for simple geometries

“…This method is equivalent to the multi-particle T-matrix method where each sphere has its own T-matrix and the fields produced by each sphere are expanded on offset spherical bases [15,120]. The problem can also be approached using a radial inversion transformation to make the spheres concentric, then solve the problem using spherical harmonics [121]. The problem is also naturally formulated using bispherical harmonics which are partially separable solutions to Laplace's equation, based on the bispherical geometry.…”

“…It is a common technique for solving problems in complicated geometries that are related to simpler geometries through inversion, for example it was used in Ref. [121] to solve the electrostatics problem of two spheres illuminated by a point dipole. The geometry was inverted about one of the foci of the corresponding bispherical coordinate system; this transformation complicates the boundary conditions, but makes the spheres concentric, allowing for the problem to be solved using spherical harmonics.…”