  The ability to use photonic quasiparticles to control electromagnetic energy
far below the diffraction limit is a defining paradigm in nanophotonics. An
important recent development in this field is the measurement and manipulation
of extremely confined phonon-polariton modes in polar dielectrics such as
silicon carbide and hexagonal boron nitride, which pave the way for
nanophotonics and extreme light-matter interactions in the mid-IR to THz
frequency range. To further advance this promising field, it is of great
interest to predict the optical response of recently discovered and
yet-to-be-synthesized polaritonic materials alike. Here we develop a unified
framework based on quantum linear response theory to calculate the spatially
non-local dielectric function of a polar lattice in arbitrary dimensions. In
the case of a three-dimensional bulk material, the spatially local limit of our
calculation reproduces standard results for the dielectric response of a polar
lattice. Using this framework, we provide ab initio calculations of the
dielectric permittivity of important bulk polar dielectrics such as silicon
carbide and hexagonal boron nitride in good agreement with experiments. From
the ab initio theory, we are able to develop a microscopic understanding of
which phonon modes contribute to each component of the dielectric function, as
well as predict features in the dielectric function that are a result of weak
TO phonons. This formalism also identifies regime(s) where quantum nonlocal
effects may correct the phonon polariton dispersion, extremely relevant in
recent atomic-scale experiments which confine electromagnetic fields to the
scale of 1~nm. Finally, our work points the way towards first principles
descriptions of the effect of interface phonons, phonon strong coupling, and
chiral phonons on the properties of phonon polaritons.
