Fig. 1. Comparison of dipole-dipole interactions (V_dd) in metallic waveguides, photonic crystal band-edge structures, and hyperbolic metamaterials.

Here, r is the distance between interacting emitters, v_g is the group velocity of the waveguide mode with wave vector k, ω_cutoff is the cutoff frequency of the metallic waveguide mode or photonic crystal, and ξ is an interaction range. (A) When the transition frequencies of interacting atoms lie above the cutoff, they will have a sinusoidal-type interaction. (B) On the other hand, at the band edge of a photonic crystal, there occur interactions with a divergent strength and range. (C) Hyperbolic media exhibit fundamentally different Coulombic long-range interactions, which diverge for specific angular directions in the low-loss effective medium limit.