  In this work we theoretically study, using Floquet-Bloch theory, the
influence of circularly and linearly polarized light on two-dimensional band
structures with Dirac and quadratic band touching points, and flat bands,
taking the nearest neighbor hopping model on the kagome lattice as an example.
We find circularly polarized light can invert the ordering of this three band
model, while leaving the flat-band dispersionless. We find a small gap is also
opened at the quadratic band touching point by 2-photon and higher order
processes. By contrast, linearly polarized light splits the quadratic band
touching point (into two Dirac points) by an amount that depends only on the
amplitude and polarization direction of the light, independent of the
frequency, and generally renders dispersion to the flat band. The splitting is
perpendicular to the direction of the polarization of the light. We derive an
effective low-energy theory that captures these key results. Finally, we
compute the frequency dependence of the optical conductivity for this 3-band
model and analyze the various interband contributions of the Floquet modes. Our
results suggest strategies for optically controlling band structure and
interaction strength in real systems.
