Short-distance expansion for the electromagnetic half-space Green's tensor: general results and an application to radiative lifetime computations

Abstract

We obtain a short-distance expansion for the half-space, frequency domain electromagnetic Green's tensor. The small parameter of the theory is , where ω is the frequency, ϵ1 is the permittivity of the upper half-space, in which both the source and the point of observation are located, and which is assumed to be transparent, c is the speed of light in vacuum and is a characteristic length, defined as the distance from the point of observation to the reflected (with respect to the planar interface) position of the source. In the case when the lower half-space (the substrate) is characterized by a complex permittivity ϵ2, we compute the expansion to third order. For the case when the substrate is a transparent dielectric, we compute the imaginary part of the Green's tensor to seventh order. The analytical calculations are verified numerically. The practical utility of the obtained expansion is demonstrated by computing the radiative lifetime of two electromagnetically interacting molecules in the vicinity of a transparent dielectric substrate. The computation is performed in the strong interaction regime when the quasi-particle pole approximation is inapplicable. In this regime, the integral representation for the half-space Green's tensor is difficult to use while its electrostatic limiting expression is grossly inadequate. However, the analytical expansion derived in this paper can be used directly and efficiently. The results of this study are also relevant to nano-optics and near-field imaging, especially when tomographic image reconstruction is involved.