  Here, we explore the combination of sub-wavelength, two-dimensional atomic
arrays and Rydberg interactions as a powerful platform to realize strong,
coherent interactions between individual photons with high fidelity. In
particular, the spatial ordering of the atoms guarantees efficient atom-light
interactions without the possibility of scattering light into unwanted
directions, for example, allowing the array to act as a perfect mirror for
individual photons. In turn, Rydberg interactions enable single photons to
alter the optical response of the array within a potentially large blockade
radius $R_b$, which can effectively punch a large "hole" for subsequent
photons. We show that such a system enables a coherent photon-photon gate or
switch, with an error scaling $\sim R_b^{-4}$ that is significantly better than
the best known scaling in a disordered ensemble. We also investigate the
optical properties of the system in the limit of strong input intensities.
Although this a priori represents a complicated, many-body quantum driven
dissipative system, we find that the behavior can be captured well by a
semi-classical model based on holes punched in a classical mirror.
