We present a theory of parametric mixing within the coplanar waveguide (CPW) of a superconducting nonlinear kinetic-inductance traveling-wave (KIT) amplifier engineered with periodic dispersion loadings. This is done by first developing a metamaterial band theory of the dispersion-engineered KIT using a Floquet-Bloch construction and then applying it to the description of mixing of the nonlinear RF traveling waves. Our theory allows us to calculate signal gain vs. signal frequency in the presence of a frequency stop gap, based solely on loading design. We present results for both three-wave mixing (3WM), with applied DC bias, and four-wave mixing (4WM), without DC. Our theory predicts an intrinsic and deterministic origin to undulations of 4WM signal gain with signal frequency, apart from extrinsic sources, such as impedance mismatch, and shows that such undulations are absent from 3WM signal gain achievable with DC. Our theory is extensible to amplifiers based on Josephson junctions in a lumped LC transmission line (TWPA).
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