Three-Wave Mixing as the Limit of Nonlinear Dynamics Theory for Nonlinear Transmission Line-Type Metamaterials

Using nonlinear dynamics (NLD) theory, a linear time periodic equivalent circuit of a sinusoidally pumped distributed transmission line-type metamaterial is proposed. For small pump power and sufficient number of unit cells, it is shown that three-wave mixing, coupled with phase matching, can be used to determine the frequencies of the autonomous components and their propagation behavior. The resulting signal and idler waves satisfy the well-known Manley-Rowe relation. However, unlike NLD theory and stability analysis, three-wave mixing can be inaccurate for relatively large input power levels and/or short transmission lines. In particular, the waveform profile and consequently radiation of the autonomous frequencies located inside the light cone cannot be described using three-wave mixing and NLD and/or stability analysis must be applied.