  The physics of waves in time-varying media provides numerous opportunities
for wave control that are unattainable with static media. In particular,
Floquet systems with a periodic time modulation are currently of considerable
interest. Here, we demonstrate how the scattering properties of a finite
Floquet medium can be correctly described by a static Floquet scattering
matrix, which satisfies a pseudo-unitary relation. This algebraic property is a
consequence of the conservation of wave action for which we formulate here a
continuity equation. Using this Floquet scattering matrix, we further
demonstrate how it can be used to generalize also the seminal Wigner-Smith
operator from static to Floquet systems. The eigenstates of the corresponding
Floquet Wigner-Smith matrix are shown to be light pulses that are optimally
shaped both in their spatial and temporal degrees of freedom for the optical
micromanipulation of time-varying media.
