  Of late there has been a surge of interest in localised phonon polariton
resonators which allow for sub-diffraction confinement of light in the
mid-infrared spectral region by coupling to optical phonons at the surface of
polar dielectrics. Resonators are generally etched on deep substrates which
support propagative surface phonon polariton resonances. Recent experimental
work has shown that understanding the coupling between localised and
propagative surface phonon polaritons in these systems is vital to correctly
describe the system resonances. In this paper we comprehensively investigate
resonators composed of arrays of cylindrical SiC resonators on SiC substrates.
Our bottom-up approach, starting from the resonances of single, free standing
cylinders and isolated substrates, and exploiting both numerical and analytical
techniques, allows us to develop a consistent understanding of the parameter
space of those resonators, putting on firmer ground this blossoming technology.
