  Sub-wavelength confinement of mid-infrared light can be achieved exploiting
the metal-like optical response of polar dielectric crystals in their
Reststrahlen spectral region, where they support evanescent modes termed
surface phonon polaritons. In the past few years the investigation of phonon
polaritons localised in nanoresonators and layered heterostructures has enjoyed
remarkable success, highlighting them as a promising platform for mid-infrared
nanophotonic applications. Here we prove that the standard local dielectric
description of phonon polaritons in nanometric objects fails due to the
nonlocal nature of the phonon response and we develop the corresponding
nonlocal theory. Application of our general theory to both dielectric
nanospheres and thin films demonstrates that polar dielectrics exhibit a rich
nonlocal phenomenology, qualitatively different from the one of plasmonic
systems, due to the negative dispersion of phononic optical modes.
