  The operating principle of traveling-wave parametric amplifiers is typically
understood in terms of the standard coupled mode theory, which describes the
evolution of forward propagating waves without any reflections, i.e. for
perfect impedance matching. However, in practice, superconducting microwave
amplifiers are unmatched nonlinear finite-length devices, where the reflecting
waves undergo complex parametric processes, not described by the standard
coupled mode theory. Here, we present an analytical solution for the TWPA gain,
which includes the interaction of reflected waves. These reflections result in
corrections to the well-known results of the standard coupled mode theory,
which are obtained for both 3-wave and 4-wave mixing processes. Due to these
reflections, gain is enhanced and unwanted nonlinear phase modulations are
suppressed. Predictions of the model are experimentally demonstrated on two
types of unmatched TWPA, based on coplanar waveguides with a central wire
consisting of i) a high kinetic inductance superconductor, and ii) an array of
2000 Josephson junctions.
