Optimal matrix-based spatio-temporal wave control for virtual perfect absorption, energy deposition and scattering invariant modes in disordered systems

We present and experimentally verify a matrix approach for determining how to optimally sculpt an input wavefront both in space and time for any desired wave-control functionality, irrespective of the complexity of the wave scattering. We leverage a singular value decomposition of the transport matrix that fully captures how both the spatial and temporal degrees of freedom available to shape the input wavefront impact the output wavefront's spatial and temporal form. In our experiments in the microwave domain, we use our formalism to successfully tackle three iconic wave-control tasks in a disordered cavity: (i) reflectionless transient excitation ("virtual perfect absorption"), (ii) optimal energy deposition, and (iii) scattering-invariant time-varying states.