  Optical antennas made of low-loss dielectrics have several advantages over
plasmonic antennas, including high radiative quantum efficiency, negligible
heating and excellent photostability. However, due to weak spatial confinement,
conventional dielectric antennas fail to offer light-matter interaction
strengths on par with those of plasmonic antennas. We propose here an
all-dielectric antenna configuration that can support strongly confined modes
($V\sim10^{-4}\lambda_{0}^3$) while maintaining unity antenna quantum
efficiency. This configuration consists of a high-index pillar structure with a
transverse gap that is filled with a low-index material, where the contrast of
indices induces a strong enhancement of the electric field perpendicular to the
gap. We provide a detailed explanation of the operation principle of such
Photonic Gap Antennas (PGAs) based on the dispersion relation of symmetric and
asymmetric horizontal slot-waveguides. To discuss the properties of PGAs, we
consider silicon pillars with air or CYTOP as the gap-material. We show by
full-wave simulations that PGAs with an emitter embedded in the gap can enhance
the spontaneous emission rate by a factor of $\sim$1000 for air gaps and
$\sim$400 for CYTOP gaps over a spectral bandwidth of $\Delta\lambda\approx300$
nm at $\lambda=1.25$ \textmu m. Furthermore, the PGAs can be designed to
provide unidirectional out-of-plane radiation across a substantial portion of
their spectral bandwidth. This is achieved by setting the position of the gap
at an optimized off-centered position of the pillar so as to properly break the
vertical symmetry of the structure. We also demonstrate that, when acting as
receivers, PGAs can lead to a near-field intensity enhancement by a factor of
$\sim$3000 for air gaps and $\sim$1200 for CYTOP gaps.
