  Motivated by the quest for experimentally accessible dynamical probes of
Floquet topological insulators, we formulate the linear response theory of a
periodically driven system. We illustrate the applications of this formalism by
giving general expressions for optical conductivity of Floquet systems,
including its homodyne and heterodyne components and beyond. We obtain the
Floquet optical conductivity of specific driven models, including
two-dimensional Dirac material such as the surface of a topological insulator,
graphene, and the Haldane model irradiated with circularly or linearly
polarized laser, as well as semiconductor quantum well driven by an ac
potential. We obtain approximate analytical expressions and perform numerically
exact calculations of the Floquet optical conductivity in different scenarios
of the occupation of the Floquet bands, in particular, the diagonal Floquet
distribution and the distribution obtained after a quench. We comment on
experimental signatures and detection of Floquet topological phases using
optical probes.
