  Photonic crystal slabs (PCSs) are a well-studied class of devices known to
support optical Fano resonances for light normally incident to the slab, useful
for narrowband filters, modulators, and nonlinear photonic devices. In
shallow-etched PCSs the linewidth of the resonances is easily controlled by
tuning the etching depth. This design strength comes at the cost of large
device footprint due to the poor in-plane localization of optical energy. In
fully-etched PCSs realized in high index contrast material systems, the
in-plane localization is greatly improved, but the command over linewidth
suffers. This disadvantage in fully-etched PCSs, also known as high contrast
gratings (HCGs), can be overcome by accessing symmetry-protected Bound States
in the Continuum (BICs). By perturbing an HCG, the BIC may be excited from the
free space with quality factor showing an inverse squared dependence on the
magnitude of the perturbation, while inheriting the excellent in-plane
localization of their unperturbed counterparts. Here, we report an exhaustive
catalog of the selection rules (if and to which free space polarization
coupling occurs) of symmetry-protected BICs controlled by in-plane symmetry
breaking in six types of two-dimensional PCS lattices. The chosen lattices
allow access to the three highest symmetry mode classes of unperturbed square
and hexagonal PCSs. The restriction to in-plane symmetry breaking allows for
manufacturing devices with simple lithographic fabrication techniques in
comparison to out-of-plane symmetry breaking, useful for practical
applications. The approach reported provides a high-level roadmap for designing
PCSs supporting controllable sharp spectral features with minimal device
footprints using a mature fabrication platform.
