  Motivated by the recent experimental realization of twisted trilayer graphene
and the observed superconductivity that is associated with its flat bands at
specific angles, we study trilayer graphene under the influence of different
forms of light in the non-interacting limit. Specifically, we study four
different types of stacking configurations with a single twisted layer. In all
four cases, we study the impact of circularly polarized light and longitudinal
light coming from a waveguide. We derive effective time-independent Floquet
Hamiltonians and review light-induced changes to the band structure. For
circularly polarized light, we find band flattening effects as well as band gap
openings. We emphasize that there is a rich band topology, which we summarize
in Chern number maps that are different for all four studied lattice
configurations. The case of a so-called ABC stacking with top layer twist is
especially rich and shows a different phase diagram depending on the handedness
of the circularly polarized light. Consequently, we propose an experiment where
this difference in typologies could be captured via optical conductivity
measurements. In contrast for the case of longitudinal light that is coming
from a waveguide, we find that the band structure is very closely related to
the equilibrium one but the magic angles can be tuned in-situ by varying the
intensity of the incident beam of light.
