Understanding complex wave scattering systems through the Generalized Wigner-Smith operator

We use the Generalized Wigner-Smith (GWS) operator Q_?=-iS^-1dS/d?, where S is the scattering matrix of a ray-chaotic enclosure and ? is an arbitrary parameter, to understand the dependence of the scattering matrix on various parameters of interest. A particular example of the GWS operator is the Wigner time delay operator where ? is frequency, and we can use the eigenvalues of that operator to determine the locations of the zeros and poles of S and to find conditions for coherent perfect absorption (CPA). By using the GWS operator, we can gather more information about how a complex scattering system interacts with incoming waves, and use this information to create conditions for CPA, create hot or cold spots in specific areas, etc. In our experimental setup, we use a nonlinear variable globally biased varactor metasurface inside a ray chaotic quarter bowtie microwave billiard. In this setup we can explore how the scattering matrix nonlinearity depends on frequency, bias voltage, rf power, etc.