Linear-frequency conversion with time-varying metasurfaces

We present a versatile mechanism utilizing time-varying metasurfaces for achieving linear frequency conversion, historically governed by nonlinear interactions. Our approach, rooted in invoking linear equations, demonstrates the feasibility of single-frequency conversion through a metasurface, likened to parametric processes found in time-varying systems. Leveraging a generalized time-inhomogeneous convolution product, we introduce an effective nonlinearity furnished by external memory effects, which presents a path that provably adheres to the principles of causality and energy conservation. We explicit the double time-variable electric and magnetic susceptibilities which allow frequency conversion from one frequency to another. The approach can be extended to accommodate multi-frequency conversion by reflection or transmission, as well as full coherence for each monochromatic input. As we anticipate exploring numerical solutions and extension into the quantum regime, we believe this introductory result prepares the ground for future work.