A Photonic Floquet Scattering Matrix for Wavefront-Shaping in Time-Periodic Media

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.