  Spontaneous emission is one of the most fundamental out-of-equilibrium
processes in which an excited quantum emitter relaxes to the ground state due
to quantum fluctuations. In this process, a photon is emitted that can interact
with other nearby emitters and establish quantum correlations between them,
e.g., via super and subradiance effects. One way to modify these
photon-mediated interactions is to alter the dipole radiation patterns of the
emitter, e.g., by placing photonic crystals near them. One recent example is
the generation of strong directional emission patterns-key to enhancing super
and subradiance effects-in two dimensions by employing photonic crystals with
band structures characterized by linear isofrequency contours and
saddle-points. However, these studies have predominantly used oversimplified
toy models, overlooking the electromagnetic field's intricacies in actual
materials, including aspects like geometrical dependencies, emitter positions,
and polarization. Our study delves into the interaction between these
directional emission patterns and the aforementioned variables, revealing the
untapped potential to fine-tune collective quantum optical phenomena.
