Fig. 4. Spatial scaling of long-range interactions.

(A) Experimental confirmation of enhanced energy transfer rates due to the long-range dipole-dipole interactions in a hyperbolic metamaterial (green) compared to a silver film (blue) and a SiO₂ film (red). The noise levels are denoted by dashed curves, and the numerically calculated many-body dipole-dipole interaction curves are denoted by the colored bands (no free-fitting parameters). The theoretical predictions include 10% error bands accounting for uncertainty in the independently extracted physical parameters. (B) We now compare the ideal super-Coulombic behavior to the experimental observations. The curves show the numerically simulated spatial dependence of sheet-to-slab (2D sheet of donors and thin slab of acceptors) many-body dipole-dipole interactions demonstrating an enhanced FRET rate of the effective medium model (yellow) with d⁻³ power law dependence. Multilayer lattice structures with unit cell sizes of 40, 20, and 4 nm, respectively, are also shown exhibiting an extended spatial range with enhanced Coulombic interactions beyond the scale of a wavelength. The green stars correspond to the experimentally measured data. It is seen that the ideal EMT (yellow) has excellent agreement with the numerical simulations for 4-nm unit cell sizes. The same numerical simulations show excellent agreement with experimental data points for 40-nm unit cell sizes only limited by nanofabrication of ultrathin layers. The solid gray line shows the ideal limit obtained from Eq. 1 of adsorbed quantum emitters on a hyperbolic medium, whereas the dashed black line presents the analytical scaling law, taking into account the finite distance between the emitter from the metamaterial.