  We establish the theoretical foundation of the Floquet graphene antidot
lattice, whereby massless Dirac fermions are driven periodically by a
circularly polarized electromagnetic field, while having their motion excluded
from an array of nanoholes. The properties of interest are encoded in the
quasienergy spectra, which are computed non-perturbatively within the Floquet
formalism. We find that a rich Floquet phase diagram emerges as the amplitude
of the drive field is varied. Notably, the Dirac dispersion can be restored in
real time relative to the gapped equilibrium state, which may enable the
creation of an optoelectronic switch or a dynamically tunable electronic
waveguide. As the amplitude is increased, the ability to shift the quasienergy
gap between high-symmetry points can change which crystal momenta dominate in
the scattering processes relevant to electronic transport and optical emission.
Furthermore, the bands can be flattened near the $\Gamma$ point, which is
indicative of selective dynamical localization. Lastly, quadratic and linear
dispersions emerge in orthogonal directions at the $M$ point, signaling a
Floquet semi-Dirac material. Importantly, all our predictions are valid for
experimentally accessible near-IR radiation, which corresponds to the above
bandwidth limit for the graphene antidot lattice. Cycling between engineered
Floquet electronic phases may play a key role in the development of
next-generation on-chip devices for optoelectronic applications.
