  Photonic band structures are a typical fingerprint of periodic optical
structures, and are usually observed in spectroscopic quantities such as
transmission, reflection and absorption. Here we show that also the
chiro-optical response of a metasurface constituted by a lattice of
non-centrosymmetric, L-shaped holes in a dielectric slab shows a band
structure, where intrinsic and extrinsic chirality effects are clearly
recognized and connected to localized and delocalized resonances. Superchiral
near-fields can be excited in correspondence to these resonances, and anomalous
behaviors as a function of the incidence polarization occur. Moreover, we
introduce a singular value decomposition (SVD) approach to show that the above
mentioned effects are connected to specific fingerprints of the SVD spectra.
Finally, we demonstrate by means of an inverse design technique that the
metasurface based on an L-shaped hole array is a minimal one. Indeed, its unit
cell geometry depends on the smallest number of parameters needed to implement
arbitrary transmission matrices compliant with the general symmetries for
2d-chiral structures. These observations enable more powerful wave operations
in a lossless photonic environment.
