Plasmon spectrum and plasmon-mediated energy transfer in a multi-connected geometry

Surface plasmon spectrum of a metallic hyperbola can be found analytically with the separation of variables in elliptic coordinates. The spectrum consists of two branches: symmetric, low-frequency branch, $\omega <\omega_0/\sqrt{2}$, and antisymmetric high-frequency branch, $\omega >\omega_0/\sqrt{2}$, where $\omega_0$ is the bulk plasmon frequency. The frequency width of the plasmon band increases with decreasing the angle between the asymptotes of the hyperbola. For the simplest multi-connected geometry of two hyperbolas separated by an air spacer the plasmon spectrum contains two low-frequency branches and two high-frequency branches. Most remarkably, the lower of two low-frequency branches exists at $\omega \rightarrow 0$, i.e., unlike a single hyperbola, it is "thresholdless." We study how the complex structure of the plasmon spectrum affects the energy transfer between two emitters located on the surface of the same hyperbola and on the surfaces of different hyperbolas.