Theory of optical-environment-dependent spontaneous-emission rates for emitters in thin layers

Abstract

The spontaneous-emission rates of emitters embedded in a thin layer 0 of a loss-free dielectric depend on their optical environment, i.e., on the optical properties of the two media 1 and 2 adjoining layer 0. For electric- and magnetic-dipole transitions the spontaneous-emission rates, normalized with respect to those in an infinite medium 0, are expressed in terms of the Fresnel reflection coefficients for plane and evanescent waves incident from medium 0 on the interfaces to the planar-stratified or homogeneous media 1 and 2, respectively. From this result which is valid for arbitrary layer thicknesses $d_0$ we derive an approximation for extremely thin layers of optical thickness $n_0{d}_0\ll \frac{\lambda }{8}$ (where $\lambda$ is the emission wavelength) between two homogeneous loss-free dielectric media 1 and 2. For the normalized spontaneous-emission rates as functions of the refractive indices $n_j$ of media $j=0, 1, \mathrm{and} 2$ analytical expressions are obtained. We have used these expressions previously without proof [Phys. Rev. B 21, 4814 (1980)] to analyze experimentally observed changes in the fluorescence lifetime of ${\mathrm{Eu}}^{3+}$ ions in varying optical environments, which yielded the quantum efficiency of the emitting state.