Theory of pump-probe experiments of metallic metamaterials coupled to the gain medium

We establish a new approach for pump-probe simulations of metallic metamaterials coupled to the gain materials. It is of vital importance to understand the mechanism of the coupling of metamaterials with the gain medium. Using a four-level gain system, we have studied light amplification of arrays of metallic split-ring resonators (SRRs) with a gain layer underneath. We find that that the differential transmittance \Delta T/T can be negative for SRRs on the top of the gain substrate, which is not expected, and \Delta T/T is positive for the gain substrate alone. These simulations agree with pump-probe experiments and can help to design new experiments to compensate the losses of metamaterials.