  Observing signatures of light-induced Floquet topological states in materials
has been shown to be very challenging. Angle-resolved photoemission
spectroscopy (ARPES) is well suited for the investigation of Floquet physics,
as it allows to directly probe the dressed electronic states of driven solids.
Depending on the system, scattering and decoherence can play an important role,
hampering the emergence of Floquet states. Another challenge is to disentangle
Floquet side bands from laser-assisted photoemission (LAPE), since both lead to
similar signatures in ARPES spectra. Here, we investigate the emergence of
Floquet state in the transition metal dichalcogenide $2H$-WSe$_2$, one of the
most promising systems for observing Floquet physics. We discuss how the
Floquet topological state manifests in characteristic features in the circular
dichroism in photoelectron angular distributions (CDAD) that is determined by
the transient band structure modifications and the associated texture of the
orbital angular momentum. Combining highly accurate modeling of the
photoemission matrix elements with an \emph{ab initio} description of the
light-matter interaction, we investigate regimes which can be realized in
current state-of-the-art experimental setups. The predicted features are robust
against scattering effects and are expected to be observed in forthcoming
experiments.
