Intermolecular resonance energy transfer in the presence of a dielectric cylinder

Using a Green’s tensor method, we investigate the rate of resonance electronic energy transfer between two molecules near a dielectric cylinder. Both the case of a real and frequency-independent dielectric constant $ϵ$ and the case of a Drude-Lorentz model for $ϵ\left(\omega \right)$ are considered, the latter case including dispersion and absorption. If the donor is placed at a fixed position near the cylinder, we find that the energy transfer rate to the acceptor is enhanced compared to its vacuum value in a number of discrete hotspots, centered at the cylinder’s surface. In the absence of dispersion and absorption the rate of energy transfer may be enhanced at most a few times. On the other hand, for the Drude-Lorentz model the enhancement may be huge (up to ${10}^6$) and the hotspots are sharply localized at the surface. We show that these observations can be explained from the fact that in the resonance region of the Drude-Lorentz dielectric surface plasmons occur, which play the dominant role in transferring the electronic energy between the donor and the acceptor. The dependence of the energy transfer rate on the molecular transition frequency is investigated as well. For small intermolecular distances, the cylinder hardly affects the transfer rate, independent of frequency. For larger distances, the frequency dependence is quite strong, in particular in the stop-gap region. The role of the intermolecular distance in the frequency dependence may be explained qualitatively using Heisenberg’s uncertainty principle to calculate the spread in the frequencies of the intermediate photons.

Figure 1

Time-ordered Feynman diagrams for virtual photon exchange.

Figure 2

Integration contour in the complex $k_z$ plane.

Figure 3

(Color online) Total energy transfer rate, normalized to free space, between two molecules placed near a dielectric cylinder $\left(ϵ=2\right)$ with (a) ${\rho }_c=0.2{\lambda }_0$, ${\rho }_s=0.22{\lambda }_0$, (b) ${\rho }_c=0.5{\lambda }_0$, ${\rho }_s=0.55{\lambda }_0$, (c) ${\rho }_c=1.0{\lambda }_0$, ${\rho }_s=1.05{\lambda }_0$ and (d) ${\rho }_c=2.0{\lambda }_0$, ${\rho }_s=2.2{\lambda }_0$. Both the donor and the acceptor transition dipole moments are oriented along the $z$ axis.

Figure 4

(Color online) Energy transfer rate through individual multipole terms between two molecules placed near a cylinder of radius ${\rho }_c=0.2{\lambda }_0$, normalized to the total energy transfer rate in free space. The dielectric constant is $ϵ=2.0$. (a) The multipole term of order $n=3$; (b) The multipole term of order $n=5$.

Figure 5

(Color online) Normalized energy transfer rate between two molecules placed near a cylinder with a Drude-Lorentz dielectric function [Eq. (47)] with ${\omega }_L=1.2{\omega }_T$, $\gamma ={10}^{-3}{\omega }_T$, and ${\omega }_P=\sqrt{{\omega }_L^2-{\omega }_T^2}\approx 0.663{\omega }_T$. Both the donor and the acceptor are oriented along the $z$ axis. (a) ${\rho }_c=0.2{\lambda }_T$, ${\rho }_s=0.22{\lambda }_T$ and (b) ${\rho }_c=1.0{\lambda }_T$, ${\rho }_s=1.05{\lambda }_T$. Panels (c) and (d) are the logarithmic correspondents of (a) and (b), respectively.

Figure 6

Relative energy transfer rate as a function of frequency for fixed distance to the cylinder’s interface $\left(\rho ={\rho }_s=0.22{\lambda }_T\right)$ and several intermolecular distances. Other relevant parameters: ${\rho }_c=0.2{\lambda }_T$, $\gamma =0.05{\omega }_T$, and ${\omega }_L=1.2{\omega }_T$.